∫ (d+ex)7∕2 ___________________

3.2295 \(\int \frac {(d+e x)^{7/2}}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=691 \[ \frac {e \sqrt {d+e x} \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e (d+e x)^{3/2} (2 c d-b e)}{c \left (b^2-4 a c\right )}+\frac {\left (-2 c^3 d^2 e \left (-d \sqrt {b^2-4 a c}-18 a e+8 b d\right )+c^2 e^2 \left (-3 b d \left (d \sqrt {b^2-4 a c}+12 a e\right )+2 a e \left (13 d \sqrt {b^2-4 a c}-10 a e\right )+3 b^2 d^2\right )+b c e^3 \left (-5 b d \sqrt {b^2-4 a c}-13 a e \sqrt {b^2-4 a c}+19 a b e+5 b^2 d\right )-3 b^3 e^4 \left (b-\sqrt {b^2-4 a c}\right )+8 c^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-2 c^3 d^2 e \left (d \sqrt {b^2-4 a c}-18 a e+8 b d\right )+c^2 e^2 \left (3 b d \left (d \sqrt {b^2-4 a c}-12 a e\right )-2 a e \left (13 d \sqrt {b^2-4 a c}+10 a e\right )+3 b^2 d^2\right )+b c e^3 \left (5 b d \sqrt {b^2-4 a c}+13 a e \sqrt {b^2-4 a c}+19 a b e+5 b^2 d\right )-3 b^3 e^4 \left (\sqrt {b^2-4 a c}+b\right )+8 c^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \]

[Out]

e*(-b*e+2*c*d)*(e*x+d)^(3/2)/c/(-4*a*c+b^2)-(e*x+d)^(5/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a

)+e*(2*c^2*d^2+3*b^2*e^2-2*c*e*(5*a*e+b*d))*(e*x+d)^(1/2)/c^2/(-4*a*c+b^2)+1/2*arctanh(2^(1/2)*c^(1/2)*(e*x+d)

^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*(8*c^4*d^4-3*b^3*e^4*(b-(-4*a*c+b^2)^(1/2))-2*c^3*d^2*e*(8*b*d-

18*a*e-d*(-4*a*c+b^2)^(1/2))+b*c*e^3*(5*b^2*d+19*a*b*e-5*b*d*(-4*a*c+b^2)^(1/2)-13*a*e*(-4*a*c+b^2)^(1/2))+c^2

*e^2*(3*b^2*d^2-3*b*d*(12*a*e+d*(-4*a*c+b^2)^(1/2))+2*a*e*(-10*a*e+13*d*(-4*a*c+b^2)^(1/2))))/c^(5/2)/(-4*a*c+

b^2)^(3/2)*2^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-1/2*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(

b+(-4*a*c+b^2)^(1/2)))^(1/2))*(8*c^4*d^4-3*b^3*e^4*(b+(-4*a*c+b^2)^(1/2))-2*c^3*d^2*e*(8*b*d-18*a*e+d*(-4*a*c+

b^2)^(1/2))+b*c*e^3*(5*b^2*d+19*a*b*e+5*b*d*(-4*a*c+b^2)^(1/2)+13*a*e*(-4*a*c+b^2)^(1/2))+c^2*e^2*(3*b^2*d^2+3

*b*d*(-12*a*e+d*(-4*a*c+b^2)^(1/2))-2*a*e*(10*a*e+13*d*(-4*a*c+b^2)^(1/2))))/c^(5/2)/(-4*a*c+b^2)^(3/2)*2^(1/2

)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A] time = 15.96, antiderivative size = 691, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {738, 824, 826, 1166, 208} \[ \frac {e \sqrt {d+e x} \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac {\left (c^2 e^2 \left (-3 b d \left (d \sqrt {b^2-4 a c}+12 a e\right )+2 a e \left (13 d \sqrt {b^2-4 a c}-10 a e\right )+3 b^2 d^2\right )-2 c^3 d^2 e \left (-d \sqrt {b^2-4 a c}-18 a e+8 b d\right )+b c e^3 \left (-5 b d \sqrt {b^2-4 a c}-13 a e \sqrt {b^2-4 a c}+19 a b e+5 b^2 d\right )-3 b^3 e^4 \left (b-\sqrt {b^2-4 a c}\right )+8 c^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (c^2 e^2 \left (3 b d \left (d \sqrt {b^2-4 a c}-12 a e\right )-2 a e \left (13 d \sqrt {b^2-4 a c}+10 a e\right )+3 b^2 d^2\right )-2 c^3 d^2 e \left (d \sqrt {b^2-4 a c}-18 a e+8 b d\right )+b c e^3 \left (5 b d \sqrt {b^2-4 a c}+13 a e \sqrt {b^2-4 a c}+19 a b e+5 b^2 d\right )-3 b^3 e^4 \left (\sqrt {b^2-4 a c}+b\right )+8 c^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e (d+e x)^{3/2} (2 c d-b e)}{c \left (b^2-4 a c\right )} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(7/2)/(a + b*x + c*x^2)^2,x]

[Out]

(e*(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e))*Sqrt[d + e*x])/(c^2*(b^2 - 4*a*c)) + (e*(2*c*d - b*e)*(d + e*

x)^(3/2))/(c*(b^2 - 4*a*c)) - ((d + e*x)^(5/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^

2)) + ((8*c^4*d^4 - 3*b^3*(b - Sqrt[b^2 - 4*a*c])*e^4 - 2*c^3*d^2*e*(8*b*d - Sqrt[b^2 - 4*a*c]*d - 18*a*e) + b

*c*e^3*(5*b^2*d - 5*b*Sqrt[b^2 - 4*a*c]*d + 19*a*b*e - 13*a*Sqrt[b^2 - 4*a*c]*e) + c^2*e^2*(3*b^2*d^2 + 2*a*e*

(13*Sqrt[b^2 - 4*a*c]*d - 10*a*e) - 3*b*d*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e

*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2

- 4*a*c])*e]) - ((8*c^4*d^4 - 3*b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 2*c^3*d^2*e*(8*b*d + Sqrt[b^2 - 4*a*c]*d -

18*a*e) + b*c*e^3*(5*b^2*d + 5*b*Sqrt[b^2 - 4*a*c]*d + 19*a*b*e + 13*a*Sqrt[b^2 - 4*a*c]*e) + c^2*e^2*(3*b^2*d

^2 + 3*b*d*(Sqrt[b^2 - 4*a*c]*d - 12*a*e) - 2*a*e*(13*Sqrt[b^2 - 4*a*c]*d + 10*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]

*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b

+ Sqrt[b^2 - 4*a*c])*e])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,

b}, x] && NegQ[a/b]

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(

d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -

4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*

c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &

& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,

e, m, p, x]

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g

*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])

/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*

e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,

Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /

; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di

st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +

q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^

2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (4 c d^2-7 b d e+10 a e^2\right )-\frac {3}{2} e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac {e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (4 c^2 d^3-3 a b e^3-c d e (7 b d-16 a e)\right )-\frac {1}{2} e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) x\right )}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) \sqrt {d+e x}}{c^2 \left (b^2-4 a c\right )}+\frac {e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (4 c^3 d^4+3 a b^2 e^4-c^2 d^2 e (7 b d-18 a e)-5 a c e^3 (b d+2 a e)\right )+\frac {1}{2} e (2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{c^2 \left (b^2-4 a c\right )}\\ &=\frac {e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) \sqrt {d+e x}}{c^2 \left (b^2-4 a c\right )}+\frac {e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {-\frac {1}{2} d e (2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right )+\frac {1}{2} e \left (4 c^3 d^4+3 a b^2 e^4-c^2 d^2 e (7 b d-18 a e)-5 a c e^3 (b d+2 a e)\right )+\frac {1}{2} e (2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c^2 \left (b^2-4 a c\right )}\\ &=\frac {e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) \sqrt {d+e x}}{c^2 \left (b^2-4 a c\right )}+\frac {e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^4 d^4-3 b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-2 c^3 d^2 e \left (8 b d+\sqrt {b^2-4 a c} d-18 a e\right )+b c e^3 \left (5 b^2 d+5 b \sqrt {b^2-4 a c} d+19 a b e+13 a \sqrt {b^2-4 a c} e\right )+c^2 e^2 \left (3 b^2 d^2+3 b d \left (\sqrt {b^2-4 a c} d-12 a e\right )-2 a e \left (13 \sqrt {b^2-4 a c} d+10 a e\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (8 c^4 d^4-3 b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-2 c^3 d^2 e \left (8 b d-\sqrt {b^2-4 a c} d-18 a e\right )+b c e^3 \left (5 b^2 d-5 b \sqrt {b^2-4 a c} d+19 a b e-13 a \sqrt {b^2-4 a c} e\right )+c^2 e^2 \left (3 b^2 d^2+2 a e \left (13 \sqrt {b^2-4 a c} d-10 a e\right )-3 b d \left (\sqrt {b^2-4 a c} d+12 a e\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}\\ &=\frac {e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) \sqrt {d+e x}}{c^2 \left (b^2-4 a c\right )}+\frac {e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^4 d^4-3 b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-2 c^3 d^2 e \left (8 b d-\sqrt {b^2-4 a c} d-18 a e\right )+b c e^3 \left (5 b^2 d-5 b \sqrt {b^2-4 a c} d+19 a b e-13 a \sqrt {b^2-4 a c} e\right )+c^2 e^2 \left (3 b^2 d^2+2 a e \left (13 \sqrt {b^2-4 a c} d-10 a e\right )-3 b d \left (\sqrt {b^2-4 a c} d+12 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 c^4 d^4-3 b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-2 c^3 d^2 e \left (8 b d+\sqrt {b^2-4 a c} d-18 a e\right )+b c e^3 \left (5 b^2 d+5 b \sqrt {b^2-4 a c} d+19 a b e+13 a \sqrt {b^2-4 a c} e\right )+c^2 e^2 \left (3 b^2 d^2+3 b d \left (\sqrt {b^2-4 a c} d-12 a e\right )-2 a e \left (13 \sqrt {b^2-4 a c} d+10 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [A] time = 6.50, size = 1116, normalized size = 1.62 \[ -\frac {\left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right ) (d+e x)^{9/2}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {\frac {2 \left (\frac {2 \left (\frac {2 \left (\frac {4 \left (\frac {\sqrt {2 c d-b e-\sqrt {b^2-4 a c} e} \left (\frac {105}{32} c^2 e (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right )-\frac {2 c \left (\frac {105}{32} c^2 e \left (c d^2-b e d+a e^2\right ) \left (4 c^3 d^4-7 b c^2 e d^3+18 a c^2 e^2 d^2-5 a b c e^3 d+3 a b^2 e^4-10 a^2 c e^4\right )-\frac {105}{32} c^2 d e (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right )\right )-\frac {105}{32} c^2 e (2 c d-b e) (b e-2 c d) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right )}{\sqrt {b^2-4 a c} e}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2} \sqrt {c} \left (-2 c d+b e+\sqrt {b^2-4 a c} e\right )}+\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} \left (\frac {105}{32} e (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right ) c^2+\frac {2 c \left (\frac {105}{32} c^2 e \left (c d^2-b e d+a e^2\right ) \left (4 c^3 d^4-7 b c^2 e d^3+18 a c^2 e^2 d^2-5 a b c e^3 d+3 a b^2 e^4-10 a^2 c e^4\right )-\frac {105}{32} c^2 d e (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right )\right )-\frac {105}{32} c^2 e (2 c d-b e) (b e-2 c d) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right )}{\sqrt {b^2-4 a c} e}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2} \sqrt {c} \left (-2 c d+b e-\sqrt {b^2-4 a c} e\right )}\right )}{c}-\frac {105}{8} c e \left (c d^2-b e d+a e^2\right ) \left (2 c^2 d^2-2 b c e d+3 b^2 e^2-10 a c e^2\right ) \sqrt {d+e x}\right )}{3 c}-\frac {35}{4} c e (2 c d-b e) \left (c d^2-e (b d-a e)\right ) (d+e x)^{3/2}\right )}{5 c}-7 c e \left (c d^2-e (b d-a e)\right ) (d+e x)^{5/2}\right )}{7 c}-e (2 c d-b e) (d+e x)^{7/2}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^2,x]

[Out]

-(((d + e*x)^(9/2)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a +

b*x + c*x^2))) - (-(e*(2*c*d - b*e)*(d + e*x)^(7/2)) + (2*(-7*c*e*(c*d^2 - e*(b*d - a*e))*(d + e*x)^(5/2) + (2

*((-35*c*e*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))*(d + e*x)^(3/2))/4 + (2*((-105*c*e*(c*d^2 - b*d*e + a*e^2)*(2

*c^2*d^2 - 2*b*c*d*e + 3*b^2*e^2 - 10*a*c*e^2)*Sqrt[d + e*x])/8 + (4*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]

*((105*c^2*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 - ((-105*c

^2*e*(2*c*d - b*e)*(-2*c*d + b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + 2

*c*((-105*c^2*d*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + (10

5*c^2*e*(c*d^2 - b*d*e + a*e^2)*(4*c^3*d^4 - 7*b*c^2*d^3*e + 18*a*c^2*d^2*e^2 - 5*a*b*c*d*e^3 + 3*a*b^2*e^4 -

10*a^2*c*e^4))/32))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2

- 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*

e]*((105*c^2*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + ((-105

*c^2*e*(2*c*d - b*e)*(-2*c*d + b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 +

2*c*((-105*c^2*d*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + (

105*c^2*e*(c*d^2 - b*d*e + a*e^2)*(4*c^3*d^4 - 7*b*c^2*d^3*e + 18*a*c^2*d^2*e^2 - 5*a*b*c*d*e^3 + 3*a*b^2*e^4

- 10*a^2*c*e^4))/32))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b

^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e))))/c))/(3*c)))/(5*c)))/(7*c))/((b^2 - 4

*a*c)*(c*d^2 - b*d*e + a*e^2))

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fricas [B] time = 22.28, size = 10077, normalized size = 14.58 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

-1/2*(sqrt(1/2)*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*sqrt((32*c^7*d^7 -

112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b

^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 +

150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 + (b^6*c^5

- 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 1

1*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e

^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^

2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81

*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a

^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8))*log(sqrt(1/2)*(70*(b^4*c^

6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^6*e^4 - 210*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^5*e^5 + 21*(13*b^6*c^4 -

106*a*b^4*c^5 + 224*a^2*b^2*c^6 - 32*a^3*c^7)*d^4*e^6 - 28*(7*b^7*c^3 - 59*a*b^5*c^4 + 136*a^2*b^3*c^5 - 48*a^

3*b*c^6)*d^3*e^7 - 6*(3*b^8*c^2 - 146*a*b^6*c^3 + 1289*a^2*b^4*c^4 - 4072*a^3*b^2*c^5 + 4240*a^4*c^6)*d^2*e^8

+ 3*(27*b^9*c - 474*a*b^7*c^2 + 3026*a^2*b^5*c^3 - 8368*a^3*b^3*c^4 + 8480*a^4*b*c^5)*d*e^9 - (27*b^10 - 459*a

*b^8*c + 2961*a^2*b^6*c^2 - 8818*a^3*b^4*c^3 + 11360*a^4*b^2*c^4 - 4000*a^5*c^5)*e^10 - (8*(b^6*c^8 - 12*a*b^4

*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^3 - 12*(b^7*c^7 - 12*a*b^5*c^8 + 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*d^2*e

- 2*(b^8*c^6 - 28*a*b^6*c^7 + 240*a^2*b^4*c^8 - 832*a^3*b^2*c^9 + 1024*a^4*c^10)*d*e^2 + (3*b^9*c^5 - 52*a*b^

7*c^6 + 336*a^2*b^5*c^7 - 960*a^3*b^3*c^8 + 1024*a^4*b*c^9)*e^3)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 +

980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 -

1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^

4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*

b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 1

2*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)

*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 +

5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^

7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8

)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6

)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*

a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*

c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550

*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a

*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)) - 2*(560*c^8*d^10*e^3 - 2800*b*c^7*d^9*e^4 + 7*(647*b^2*c^6 + 1012*a*

c^7)*d^8*e^5 - 28*(47*b^3*c^5 + 1012*a*b*c^6)*d^7*e^6 - (3329*b^4*c^4 - 35844*a*b^2*c^5 - 27488*a^2*c^6)*d^6*e

^7 + (2833*b^5*c^3 - 8356*a*b^3*c^4 - 82464*a^2*b*c^5)*d^5*e^8 + (9*b^6*c^2 - 14273*a*b^4*c^3 + 77982*a^2*b^2*

c^4 + 33464*a^3*c^5)*d^4*e^9 - (675*b^7*c - 9414*a*b^5*c^2 + 18524*a^2*b^3*c^3 + 66928*a^3*b*c^4)*d^3*e^10 + (

189*b^8 - 999*a*b^6*c - 8127*a^2*b^4*c^2 + 40196*a^3*b^2*c^3 + 10000*a^4*c^4)*d^2*e^11 - (378*a*b^7 - 3645*a^2

*b^5*c + 6732*a^3*b^3*c^2 + 10000*a^4*b*c^3)*d*e^12 + (189*a^2*b^6 - 1971*a^3*b^4*c + 5625*a^4*b^2*c^2 - 2500*

a^5*c^3)*e^13)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*

c^3)*x)*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4

*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*

(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*

a^3*b*c^3)*e^7 + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^

7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2

*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 11

16*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 5

50*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c

^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8))

*log(-sqrt(1/2)*(70*(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^6*e^4 - 210*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*

d^5*e^5 + 21*(13*b^6*c^4 - 106*a*b^4*c^5 + 224*a^2*b^2*c^6 - 32*a^3*c^7)*d^4*e^6 - 28*(7*b^7*c^3 - 59*a*b^5*c^

4 + 136*a^2*b^3*c^5 - 48*a^3*b*c^6)*d^3*e^7 - 6*(3*b^8*c^2 - 146*a*b^6*c^3 + 1289*a^2*b^4*c^4 - 4072*a^3*b^2*c

^5 + 4240*a^4*c^6)*d^2*e^8 + 3*(27*b^9*c - 474*a*b^7*c^2 + 3026*a^2*b^5*c^3 - 8368*a^3*b^3*c^4 + 8480*a^4*b*c^

5)*d*e^9 - (27*b^10 - 459*a*b^8*c + 2961*a^2*b^6*c^2 - 8818*a^3*b^4*c^3 + 11360*a^4*b^2*c^4 - 4000*a^5*c^5)*e^

10 - (8*(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^3 - 12*(b^7*c^7 - 12*a*b^5*c^8 + 48*a^2*b^3

*c^9 - 64*a^3*b*c^10)*d^2*e - 2*(b^8*c^6 - 28*a*b^6*c^7 + 240*a^2*b^4*c^8 - 832*a^3*b^2*c^9 + 1024*a^4*c^10)*d

*e^2 + (3*b^9*c^5 - 52*a*b^7*c^6 + 336*a^2*b^5*c^7 - 960*a^3*b^3*c^8 + 1024*a^4*b*c^9)*e^3)*sqrt((1225*c^8*d^8

*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*

b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11

+ 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 +

1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a

^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e

+ 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^

5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 -

120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 + (b^6*c^5 - 12*a*b^4*c^6 +

48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8

- 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*

c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*

c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*

c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*

a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)) - 2*(560*c^8*d^10*e^3 - 2800*b*c^7*d^9*e^4

+ 7*(647*b^2*c^6 + 1012*a*c^7)*d^8*e^5 - 28*(47*b^3*c^5 + 1012*a*b*c^6)*d^7*e^6 - (3329*b^4*c^4 - 35844*a*b^2

*c^5 - 27488*a^2*c^6)*d^6*e^7 + (2833*b^5*c^3 - 8356*a*b^3*c^4 - 82464*a^2*b*c^5)*d^5*e^8 + (9*b^6*c^2 - 14273

*a*b^4*c^3 + 77982*a^2*b^2*c^4 + 33464*a^3*c^5)*d^4*e^9 - (675*b^7*c - 9414*a*b^5*c^2 + 18524*a^2*b^3*c^3 + 66

928*a^3*b*c^4)*d^3*e^10 + (189*b^8 - 999*a*b^6*c - 8127*a^2*b^4*c^2 + 40196*a^3*b^2*c^3 + 10000*a^4*c^4)*d^2*e

^11 - (378*a*b^7 - 3645*a^2*b^5*c + 6732*a^3*b^3*c^2 + 10000*a^4*b*c^3)*d*e^12 + (189*a^2*b^6 - 1971*a^3*b^4*c

+ 5625*a^4*b^2*c^2 - 2500*a^5*c^3)*e^13)*sqrt(e*x + d)) + sqrt(1/2)*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c

^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35

*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 -

60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5

*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 - (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*

c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 1

4*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^

3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5

*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3

+ 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48

*a^2*b^2*c^7 - 64*a^3*c^8))*log(sqrt(1/2)*(70*(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^6*e^4 - 210*(b^5*c^5 - 8*

a*b^3*c^6 + 16*a^2*b*c^7)*d^5*e^5 + 21*(13*b^6*c^4 - 106*a*b^4*c^5 + 224*a^2*b^2*c^6 - 32*a^3*c^7)*d^4*e^6 - 2

8*(7*b^7*c^3 - 59*a*b^5*c^4 + 136*a^2*b^3*c^5 - 48*a^3*b*c^6)*d^3*e^7 - 6*(3*b^8*c^2 - 146*a*b^6*c^3 + 1289*a^

2*b^4*c^4 - 4072*a^3*b^2*c^5 + 4240*a^4*c^6)*d^2*e^8 + 3*(27*b^9*c - 474*a*b^7*c^2 + 3026*a^2*b^5*c^3 - 8368*a

^3*b^3*c^4 + 8480*a^4*b*c^5)*d*e^9 - (27*b^10 - 459*a*b^8*c + 2961*a^2*b^6*c^2 - 8818*a^3*b^4*c^3 + 11360*a^4*

b^2*c^4 - 4000*a^5*c^5)*e^10 + (8*(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^3 - 12*(b^7*c^7 -

12*a*b^5*c^8 + 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*d^2*e - 2*(b^8*c^6 - 28*a*b^6*c^7 + 240*a^2*b^4*c^8 - 832*a^3*

b^2*c^9 + 1024*a^4*c^10)*d*e^2 + (3*b^9*c^5 - 52*a*b^7*c^6 + 336*a^2*b^5*c^7 - 960*a^3*b^3*c^8 + 1024*a^4*b*c^

9)*e^3)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a

*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 -

1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(2

7*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2

- 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt((32*

c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^

3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b

^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 -

(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^

2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c

^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 55

32*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e

^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^

11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)) - 2*(560*c^8*d^1

0*e^3 - 2800*b*c^7*d^9*e^4 + 7*(647*b^2*c^6 + 1012*a*c^7)*d^8*e^5 - 28*(47*b^3*c^5 + 1012*a*b*c^6)*d^7*e^6 - (

3329*b^4*c^4 - 35844*a*b^2*c^5 - 27488*a^2*c^6)*d^6*e^7 + (2833*b^5*c^3 - 8356*a*b^3*c^4 - 82464*a^2*b*c^5)*d^

5*e^8 + (9*b^6*c^2 - 14273*a*b^4*c^3 + 77982*a^2*b^2*c^4 + 33464*a^3*c^5)*d^4*e^9 - (675*b^7*c - 9414*a*b^5*c^

2 + 18524*a^2*b^3*c^3 + 66928*a^3*b*c^4)*d^3*e^10 + (189*b^8 - 999*a*b^6*c - 8127*a^2*b^4*c^2 + 40196*a^3*b^2*

c^3 + 10000*a^4*c^4)*d^2*e^11 - (378*a*b^7 - 3645*a^2*b^5*c + 6732*a^3*b^3*c^2 + 10000*a^4*b*c^3)*d*e^12 + (18

9*a^2*b^6 - 1971*a^3*b^4*c + 5625*a^4*b^2*c^2 - 2500*a^5*c^3)*e^13)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2*c^2 - 4*

a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^

5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 1

4*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*

d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 - (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7

- 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5

+ 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^

3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12

- 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b

^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b

^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8))*log(-sqrt(1/2)*(70*(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*

d^6*e^4 - 210*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^5*e^5 + 21*(13*b^6*c^4 - 106*a*b^4*c^5 + 224*a^2*b^2*c^

6 - 32*a^3*c^7)*d^4*e^6 - 28*(7*b^7*c^3 - 59*a*b^5*c^4 + 136*a^2*b^3*c^5 - 48*a^3*b*c^6)*d^3*e^7 - 6*(3*b^8*c^

2 - 146*a*b^6*c^3 + 1289*a^2*b^4*c^4 - 4072*a^3*b^2*c^5 + 4240*a^4*c^6)*d^2*e^8 + 3*(27*b^9*c - 474*a*b^7*c^2

+ 3026*a^2*b^5*c^3 - 8368*a^3*b^3*c^4 + 8480*a^4*b*c^5)*d*e^9 - (27*b^10 - 459*a*b^8*c + 2961*a^2*b^6*c^2 - 88

18*a^3*b^4*c^3 + 11360*a^4*b^2*c^4 - 4000*a^5*c^5)*e^10 + (8*(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^

3*c^11)*d^3 - 12*(b^7*c^7 - 12*a*b^5*c^8 + 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*d^2*e - 2*(b^8*c^6 - 28*a*b^6*c^7 +

240*a^2*b^4*c^8 - 832*a^3*b^2*c^9 + 1024*a^4*c^10)*d*e^2 + (3*b^9*c^5 - 52*a*b^7*c^6 + 336*a^2*b^5*c^7 - 960*

a^3*b^3*c^8 + 1024*a^4*b*c^9)*e^3)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^

6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(6

6*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 110

0*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*

a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12

- 64*a^3*c^13)))*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*

b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^

2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*

c^2 - 420*a^3*b*c^3)*e^7 - (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 490

0*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 -

2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^

6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b

^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^

14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64

*a^3*c^8)) - 2*(560*c^8*d^10*e^3 - 2800*b*c^7*d^9*e^4 + 7*(647*b^2*c^6 + 1012*a*c^7)*d^8*e^5 - 28*(47*b^3*c^5

+ 1012*a*b*c^6)*d^7*e^6 - (3329*b^4*c^4 - 35844*a*b^2*c^5 - 27488*a^2*c^6)*d^6*e^7 + (2833*b^5*c^3 - 8356*a*b^

3*c^4 - 82464*a^2*b*c^5)*d^5*e^8 + (9*b^6*c^2 - 14273*a*b^4*c^3 + 77982*a^2*b^2*c^4 + 33464*a^3*c^5)*d^4*e^9 -

(675*b^7*c - 9414*a*b^5*c^2 + 18524*a^2*b^3*c^3 + 66928*a^3*b*c^4)*d^3*e^10 + (189*b^8 - 999*a*b^6*c - 8127*a

^2*b^4*c^2 + 40196*a^3*b^2*c^3 + 10000*a^4*c^4)*d^2*e^11 - (378*a*b^7 - 3645*a^2*b^5*c + 6732*a^3*b^3*c^2 + 10

000*a^4*b*c^3)*d*e^12 + (189*a^2*b^6 - 1971*a^3*b^4*c + 5625*a^4*b^2*c^2 - 2500*a^5*c^3)*e^13)*sqrt(e*x + d))

+ 2*(b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - 2*(b^2*c - 4*a*c^2)*e^3*x^2 - (3*a*b^2 - 10*a^2*c)*e^3 + (2*c

^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (3*b^3 - 11*a*b*c)*e^3)*x)*sqrt(e*x + d))/(a*b^2*c^2 - 4*

a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)

________________________________________________________________________________________

giac [B] time = 3.23, size = 1797, normalized size = 2.60 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*e^3/c^2 - (2*(x*e + d)^(3/2)*c^3*d^3*e - 2*sqrt(x*e + d)*c^3*d^4*e - 3*(x*e + d)^(3/2)*b*c^2*d

^2*e^2 + 4*sqrt(x*e + d)*b*c^2*d^3*e^2 + 3*(x*e + d)^(3/2)*b^2*c*d*e^3 - 6*(x*e + d)^(3/2)*a*c^2*d*e^3 - 3*sqr

t(x*e + d)*b^2*c*d^2*e^3 - (x*e + d)^(3/2)*b^3*e^4 + 3*(x*e + d)^(3/2)*a*b*c*e^4 + sqrt(x*e + d)*b^3*d*e^4 - s

qrt(x*e + d)*a*b^2*e^5 + 2*sqrt(x*e + d)*a^2*c*e^5)/((b^2*c^2 - 4*a*c^3)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*

d^2 + (x*e + d)*b*e - b*d*e + a*e^2)) - (16*(b^2*c^9 - 4*a*c^10)*d^5*e - 40*(b^3*c^8 - 4*a*b*c^9)*d^4*e^2 + 2*

(11*b^4*c^7 - 8*a*b^2*c^8 - 144*a^2*c^9)*d^3*e^3 + (7*b^5*c^6 - 136*a*b^3*c^7 + 432*a^2*b*c^8)*d^2*e^4 - (2*c^

3*d^3*e - 3*b*c^2*d^2*e^2 - (5*b^2*c - 26*a*c^2)*d*e^3 + (3*b^3 - 13*a*b*c)*e^4)*(b^2*c^2*e - 4*a*c^3*e)^2 - (

11*b^6*c^5 - 118*a*b^4*c^6 + 336*a^2*b^2*c^7 - 160*a^3*c^8)*d*e^5 - 2*(2*sqrt(b^2 - 4*a*c)*c^6*d^4*e - 4*sqrt(

b^2 - 4*a*c)*b*c^5*d^3*e^2 + (5*b^2*c^4 - 8*a*c^5)*sqrt(b^2 - 4*a*c)*d^2*e^3 - (3*b^3*c^3 - 8*a*b*c^4)*sqrt(b^

2 - 4*a*c)*d*e^4 + (3*a*b^2*c^3 - 10*a^2*c^4)*sqrt(b^2 - 4*a*c)*e^5)*abs(b^2*c^2*e - 4*a*c^3*e) + (3*b^7*c^4 -

31*a*b^5*c^5 + 96*a^2*b^3*c^6 - 80*a^3*b*c^7)*e^6)*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c^3*d - 8*a*

c^4*d - b^3*c^2*e + 4*a*b*c^3*e + sqrt((2*b^2*c^3*d - 8*a*c^4*d - b^3*c^2*e + 4*a*b*c^3*e)^2 - 4*(b^2*c^3*d^2

- 4*a*c^4*d^2 - b^3*c^2*d*e + 4*a*b*c^3*d*e + a*b^2*c^2*e^2 - 4*a^2*c^3*e^2)*(b^2*c^3 - 4*a*c^4)))/(b^2*c^3 -

4*a*c^4)))/(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^2*c^4 - 4*a*c^5)*sqrt(b^2 - 4*a*c)*d - (b^4

*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5 + (b^3*c^3 - 4*a*b*c^4)*sqrt(b^2 - 4*a*c))*e)*abs(b^2*c^2*e - 4*a*c^3*e)*abs(c

)) + (16*(b^2*c^9 - 4*a*c^10)*d^5*e - 40*(b^3*c^8 - 4*a*b*c^9)*d^4*e^2 + 2*(11*b^4*c^7 - 8*a*b^2*c^8 - 144*a^2

*c^9)*d^3*e^3 + (7*b^5*c^6 - 136*a*b^3*c^7 + 432*a^2*b*c^8)*d^2*e^4 - (2*c^3*d^3*e - 3*b*c^2*d^2*e^2 - (5*b^2*

c - 26*a*c^2)*d*e^3 + (3*b^3 - 13*a*b*c)*e^4)*(b^2*c^2*e - 4*a*c^3*e)^2 - (11*b^6*c^5 - 118*a*b^4*c^6 + 336*a^

2*b^2*c^7 - 160*a^3*c^8)*d*e^5 + 2*(2*sqrt(b^2 - 4*a*c)*c^6*d^4*e - 4*sqrt(b^2 - 4*a*c)*b*c^5*d^3*e^2 + (5*b^2

*c^4 - 8*a*c^5)*sqrt(b^2 - 4*a*c)*d^2*e^3 - (3*b^3*c^3 - 8*a*b*c^4)*sqrt(b^2 - 4*a*c)*d*e^4 + (3*a*b^2*c^3 - 1

0*a^2*c^4)*sqrt(b^2 - 4*a*c)*e^5)*abs(b^2*c^2*e - 4*a*c^3*e) + (3*b^7*c^4 - 31*a*b^5*c^5 + 96*a^2*b^3*c^6 - 80

*a^3*b*c^7)*e^6)*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c^3*d - 8*a*c^4*d - b^3*c^2*e + 4*a*b*c^3*e - s

qrt((2*b^2*c^3*d - 8*a*c^4*d - b^3*c^2*e + 4*a*b*c^3*e)^2 - 4*(b^2*c^3*d^2 - 4*a*c^4*d^2 - b^3*c^2*d*e + 4*a*b

*c^3*d*e + a*b^2*c^2*e^2 - 4*a^2*c^3*e^2)*(b^2*c^3 - 4*a*c^4)))/(b^2*c^3 - 4*a*c^4)))/(sqrt(-4*c^2*d + 2*(b*c

+ sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^2*c^4 - 4*a*c^5)*sqrt(b^2 - 4*a*c)*d + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5 - (

b^3*c^3 - 4*a*b*c^4)*sqrt(b^2 - 4*a*c))*e)*abs(b^2*c^2*e - 4*a*c^3*e)*abs(c))

________________________________________________________________________________________

maple [B] time = 0.32, size = 3838, normalized size = 5.55 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^2,x)

[Out]

-18*e^3*c/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh

((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*d^2-18*e^3*c/(4*a*c-b^2)/(-(4*a*c-

b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*

d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*d^2+18*e^4/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d

+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1

/2)*c)*a*b*d+18*e^4/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2

)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*b*d+8*e^2*c/(4*a*c-b^2)/(-(

4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((

-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d^3-5/2*e^4/c/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/

((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/

2))*c)^(1/2)*c)*b^3*d+8*e^2*c/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2

))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d^3-19/2*e^5/c/(4

*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)

*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*b^2-19/2*e^5/c/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/

2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c

-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*b^2-5/2*e^4/c/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a

*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*

b^3*d-e^4/c^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*b^3-6*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b

^2)*(e*x+d)^(3/2)*a*d+2*e*c/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*d^3+2*e^5/c/(c*e^2*x^2+b*e^2*x

+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*a^2-2*e*c/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*d^4-3*e^2/(c*e

^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*b*d^2+4*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2

)*b*d^3+3/2*e^5/c^2/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/

2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^4+3/2*e^5/c^2/(4*a*c-b^2

)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)

/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^4-4*e*c^2/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((

b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2)

)*c)^(1/2)*c)*d^4-13/2*e^4/c/(4*a*c-b^2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)

^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*b+13/2*e^4/c/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c

*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^

(1/2)*c)*a*b-3/2*e^3/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1

/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2*d^2-3/2*e^3/(4*a*c-b^

2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2

)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2*d^2-5/2*e^3/c/(4*a*c-b^2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c

-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2*

d+5/2*e^3/c/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/

((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2*d-4*e*c^2/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/

((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^

(1/2))*c)^(1/2)*c)*d^4+2*e^3/c^2*(e*x+d)^(1/2)+e^4/c^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*b^3

*d+13*e^3/(4*a*c-b^2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*

e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*d-13*e^3/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(

1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*d+3*e^4/c/(c

*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*a*b-e*c/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^

(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^3+e*c/(4*a*

c-b^2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*

c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^3-3/2*e^4/c^2/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(

1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^3+3/2*e^4/c^2/(4*a*c-b

^2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b

^2)*e^2)^(1/2))*c)^(1/2)*c)*b^3+10*e^5/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)

*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a^2+10*

e^5/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)

^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a^2-3/2*e^2/(4*a*c-b^2)*2^(1/2)/((b*e-2*c*d+(

-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*

c)*b*d^2+3/2*e^2/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(

1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d^2+3*e^3/c/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e

*x+d)^(3/2)*b^2*d-3*e^3/c/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*b^2*d^2-e^5/c^2/(c*e^2*x^2+b*e^2

*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*a*b^2

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x + a)^2, x)

________________________________________________________________________________________

mupad [B] time = 3.88, size = 32541, normalized size = 47.09 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)/(a + b*x + c*x^2)^2,x)

[Out]

(2*e^3*(d + e*x)^(1/2))/c^2 - atan(((((2560*a^5*c^7*e^7 + 12*a*b^8*c^3*e^7 - 12*b^9*c^3*d*e^6 - 184*a^2*b^6*c^

4*e^7 + 1056*a^3*b^4*c^5*e^7 - 2688*a^4*b^2*c^6*e^7 - 512*a^3*c^9*d^4*e^3 + 2048*a^4*c^8*d^2*e^5 + 8*b^6*c^6*d

^4*e^3 - 16*b^7*c^5*d^3*e^4 + 20*b^8*c^4*d^2*e^5 + 384*a^2*b^2*c^8*d^4*e^3 - 768*a^2*b^3*c^7*d^3*e^4 + 1344*a^

2*b^4*c^6*d^2*e^5 - 2816*a^3*b^2*c^7*d^2*e^5 + 176*a*b^7*c^4*d*e^6 - 2048*a^4*b*c^7*d*e^6 - 96*a*b^4*c^7*d^4*e

^3 + 192*a*b^5*c^6*d^3*e^4 - 272*a*b^6*c^5*d^2*e^5 - 960*a^2*b^5*c^5*d*e^6 + 1024*a^3*b*c^8*d^3*e^4 + 2304*a^3

*b^3*c^6*d*e^6)/(64*a^3*c^6 - b^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5) - (2*(d + e*x)^(1/2)*((32*b^6*c^7*d^7 -

2048*a^3*c^10*d^7 - 9*b^13*e^7 - 9*b^4*e^7*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^8*d^7 - 26880*a^6*b*c^6*e^7

+ 53760*a^6*c^7*d*e^6 - 112*b^7*c^6*d^6*e + 1536*a^2*b^2*c^9*d^7 - 2077*a^2*b^9*c^2*e^7 + 10656*a^3*b^7*c^3*e

^7 - 30240*a^4*b^5*c^4*e^7 + 44800*a^5*b^3*c^5*e^7 - 25*a^2*c^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - 17920*a^4*c^9*d

^5*e^2 - 35840*a^5*c^8*d^3*e^4 + 98*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 70*b^10*c^3*d^3*e^4 + 14*b^11*c^2*d

^2*e^5 + 35*c^4*d^4*e^3*(-(4*a*c - b^2)^9)^(1/2) + 213*a*b^11*c*e^7 + 21*b^12*c*d*e^6 + 1344*a^2*b^4*c^7*d^5*e

^2 + 10080*a^2*b^5*c^6*d^4*e^3 - 7840*a^2*b^6*c^5*d^3*e^4 - 1008*a^2*b^7*c^4*d^2*e^5 + 7168*a^3*b^2*c^8*d^5*e^

2 - 35840*a^3*b^3*c^7*d^4*e^3 + 17920*a^3*b^4*c^6*d^3*e^4 + 12544*a^3*b^5*c^5*d^2*e^5 - 44800*a^4*b^3*c^6*d^2*

e^5 + 14*b^2*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 51*a*b^2*c*e^7*(-(4*a*c - b^2)^9)^(1/2) + 1344*a*b^5*c^7*d

^6*e - 532*a*b^10*c^2*d*e^6 + 7168*a^3*b*c^9*d^6*e + 21*b^3*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) - 896*a*b^6*c^6*d

^5*e^2 - 1120*a*b^7*c^5*d^4*e^3 + 1260*a*b^8*c^4*d^3*e^4 - 98*a*b^9*c^3*d^2*e^5 - 5376*a^2*b^3*c^8*d^6*e + 541

8*a^2*b^8*c^3*d*e^6 - 28224*a^3*b^6*c^4*d*e^6 + 44800*a^4*b*c^8*d^4*e^3 + 78400*a^4*b^4*c^5*d*e^6 + 53760*a^5*

b*c^7*d^2*e^5 - 107520*a^5*b^2*c^6*d*e^6 + 154*a*c^3*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 70*b*c^3*d^3*e^4*(-(4*

a*c - b^2)^9)^(1/2) - 154*a*b*c^2*d*e^6*(-(4*a*c - b^2)^9)^(1/2))/(8*(4096*a^6*c^11 + b^12*c^5 - 24*a*b^10*c^6

+ 240*a^2*b^8*c^7 - 1280*a^3*b^6*c^8 + 3840*a^4*b^4*c^9 - 6144*a^5*b^2*c^10)))^(1/2)*(4*b^7*c^5*e^3 - 48*a*b^

5*c^6*e^3 - 256*a^3*b*c^8*e^3 + 512*a^3*c^9*d*e^2 - 8*b^6*c^6*d*e^2 + 192*a^2*b^3*c^7*e^3 + 96*a*b^4*c^7*d*e^2

- 384*a^2*b^2*c^8*d*e^2))/(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4))*((32*b^6*c^7*d^7 - 2048*a^3*c^10*d^7 - 9*b^13

*e^7 - 9*b^4*e^7*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^8*d^7 - 26880*a^6*b*c^6*e^7 + 53760*a^6*c^7*d*e^6 - 11

2*b^7*c^6*d^6*e + 1536*a^2*b^2*c^9*d^7 - 2077*a^2*b^9*c^2*e^7 + 10656*a^3*b^7*c^3*e^7 - 30240*a^4*b^5*c^4*e^7

+ 44800*a^5*b^3*c^5*e^7 - 25*a^2*c^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - 17920*a^4*c^9*d^5*e^2 - 35840*a^5*c^8*d^3*

e^4 + 98*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 70*b^10*c^3*d^3*e^4 + 14*b^11*c^2*d^2*e^5 + 35*c^4*d^4*e^3*(-(

4*a*c - b^2)^9)^(1/2) + 213*a*b^11*c*e^7 + 21*b^12*c*d*e^6 + 1344*a^2*b^4*c^7*d^5*e^2 + 10080*a^2*b^5*c^6*d^4*

e^3 - 7840*a^2*b^6*c^5*d^3*e^4 - 1008*a^2*b^7*c^4*d^2*e^5 + 7168*a^3*b^2*c^8*d^5*e^2 - 35840*a^3*b^3*c^7*d^4*e

^3 + 17920*a^3*b^4*c^6*d^3*e^4 + 12544*a^3*b^5*c^5*d^2*e^5 - 44800*a^4*b^3*c^6*d^2*e^5 + 14*b^2*c^2*d^2*e^5*(-

(4*a*c - b^2)^9)^(1/2) + 51*a*b^2*c*e^7*(-(4*a*c - b^2)^9)^(1/2) + 1344*a*b^5*c^7*d^6*e - 532*a*b^10*c^2*d*e^6

+ 7168*a^3*b*c^9*d^6*e + 21*b^3*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) - 896*a*b^6*c^6*d^5*e^2 - 1120*a*b^7*c^5*d^4

*e^3 + 1260*a*b^8*c^4*d^3*e^4 - 98*a*b^9*c^3*d^2*e^5 - 5376*a^2*b^3*c^8*d^6*e + 5418*a^2*b^8*c^3*d*e^6 - 28224

*a^3*b^6*c^4*d*e^6 + 44800*a^4*b*c^8*d^4*e^3 + 78400*a^4*b^4*c^5*d*e^6 + 53760*a^5*b*c^7*d^2*e^5 - 107520*a^5*

b^2*c^6*d*e^6 + 154*a*c^3*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 70*b*c^3*d^3*e^4*(-(4*a*c - b^2)^9)^(1/2) - 154*a

*b*c^2*d*e^6*(-(4*a*c - b^2)^9)^(1/2))/(8*(4096*a^6*c^11 + b^12*c^5 - 24*a*b^10*c^6 + 240*a^2*b^8*c^7 - 1280*a

^3*b^6*c^8 + 3840*a^4*b^4*c^9 - 6144*a^5*b^2*c^10)))^(1/2) - (2*(d + e*x)^(1/2)*(9*b^8*e^10 + 200*a^4*c^4*e^10

+ 32*c^8*d^8*e^2 + 280*a*c^7*d^6*e^4 - 128*b*c^7*d^7*e^3 + 481*a^2*b^4*c^2*e^10 - 718*a^3*b^2*c^3*e^10 + 280*

a^2*c^6*d^4*e^6 - 2072*a^3*c^5*d^2*e^8 + 154*b^2*c^6*d^6*e^4 - 14*b^3*c^5*d^5*e^5 - 105*b^4*c^4*d^4*e^6 + 84*b

^5*c^3*d^3*e^7 + 7*b^6*c^2*d^2*e^8 - 114*a*b^6*c*e^10 - 30*b^7*c*d*e^9 + 1974*a^2*b^2*c^4*d^2*e^8 - 840*a*b*c^

6*d^5*e^5 + 406*a*b^5*c^2*d*e^9 + 2072*a^3*b*c^4*d*e^9 + 910*a*b^2*c^5*d^4*e^6 - 420*a*b^3*c^4*d^3*e^7 - 336*a

*b^4*c^3*d^2*e^8 - 560*a^2*b*c^5*d^3*e^7 - 1694*a^2*b^3*c^3*d*e^9))/(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4))*((32

*b^6*c^7*d^7 - 2048*a^3*c^10*d^7 - 9*b^13*e^7 - 9*b^4*e^7*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^8*d^7 - 26880

*a^6*b*c^6*e^7 + 53760*a^6*c^7*d*e^6 - 112*b^7*c^6*d^6*e + 1536*a^2*b^2*c^9*d^7 - 2077*a^2*b^9*c^2*e^7 + 10656

*a^3*b^7*c^3*e^7 - 30240*a^4*b^5*c^4*e^7 + 44800*a^5*b^3*c^5*e^7 - 25*a^2*c^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - 1

7920*a^4*c^9*d^5*e^2 - 35840*a^5*c^8*d^3*e^4 + 98*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 70*b^10*c^3*d^3*e^4 +

14*b^11*c^2*d^2*e^5 + 35*c^4*d^4*e^3*(-(4*a*c - b^2)^9)^(1/2) + 213*a*b^11*c*e^7 + 21*b^12*c*d*e^6 + 1344*a^2

*b^4*c^7*d^5*e^2 + 10080*a^2*b^5*c^6*d^4*e^3 - 7840*a^2*b^6*c^5*d^3*e^4 - 1008*a^2*b^7*c^4*d^2*e^5 + 7168*a^3*

b^2*c^8*d^5*e^2 - 35840*a^3*b^3*c^7*d^4*e^3 + 17920*a^3*b^4*c^6*d^3*e^4 + 12544*a^3*b^5*c^5*d^2*e^5 - 44800*a^

4*b^3*c^6*d^2*e^5 + 14*b^2*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 51*a*b^2*c*e^7*(-(4*a*c - b^2)^9)^(1/2) + 13

44*a*b^5*c^7*d^6*e - 532*a*b^10*c^2*d*e^6 + 7168*a^3*b*c^9*d^6*e + 21*b^3*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) - 8

96*a*b^6*c^6*d^5*e^2 - 1120*a*b^7*c^5*d^4*e^3 + 1260*a*b^8*c^4*d^3*e^4 - 98*a*b^9*c^3*d^2*e^5 - 5376*a^2*b^3*c

^8*d^6*e + 5418*a^2*b^8*c^3*d*e^6 - 28224*a^3*b^6*c^4*d*e^6 + 44800*a^4*b*c^8*d^4*e^3 + 78400*a^4*b^4*c^5*d*e^

6 + 53760*a^5*b*c^7*d^2*e^5 - 107520*a^5*b^2*c^6*d*e^6 + 154*a*c^3*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 70*b*c^3

*d^3*e^4*(-(4*a*c - b^2)^9)^(1/2) - 154*a*b*c^2*d*e^6*(-(4*a*c - b^2)^9)^(1/2))/(8*(4096*a^6*c^11 + b^12*c^5 -

24*a*b^10*c^6 + 240*a^2*b^8*c^7 - 1280*a^3*b^6*c^8 + 3840*a^4*b^4*c^9 - 6144*a^5*b^2*c^10)))^(1/2)*1i - (((25

60*a^5*c^7*e^7 + 12*a*b^8*c^3*e^7 - 12*b^9*c^3*d*e^6 - 184*a^2*b^6*c^4*e^7 + 1056*a^3*b^4*c^5*e^7 - 2688*a^4*b

^2*c^6*e^7 - 512*a^3*c^9*d^4*e^3 + 2048*a^4*c^8*d^2*e^5 + 8*b^6*c^6*d^4*e^3 - 16*b^7*c^5*d^3*e^4 + 20*b^8*c^4*

d^2*e^5 + 384*a^2*b^2*c^8*d^4*e^3 - 768*a^2*b^3*c^7*d^3*e^4 + 1344*a^2*b^4*c^6*d^2*e^5 - 2816*a^3*b^2*c^7*d^2*

e^5 + 176*a*b^7*c^4*d*e^6 - 2048*a^4*b*c^7*d*e^6 - 96*a*b^4*c^7*d^4*e^3 + 192*a*b^5*c^6*d^3*e^4 - 272*a*b^6*c^

5*d^2*e^5 - 960*a^2*b^5*c^5*d*e^6 + 1024*a^3*b*c^8*d^3*e^4 + 2304*a^3*b^3*c^6*d*e^6)/(64*a^3*c^6 - b^6*c^3 + 1

2*a*b^4*c^4 - 48*a^2*b^2*c^5) + (2*(d + e*x)^(1/2)*((32*b^6*c^7*d^7 - 2048*a^3*c^10*d^7 - 9*b^13*e^7 - 9*b^4*e

^7*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^8*d^7 - 26880*a^6*b*c^6*e^7 + 53760*a^6*c^7*d*e^6 - 112*b^7*c^6*d^6*

e + 1536*a^2*b^2*c^9*d^7 - 2077*a^2*b^9*c^2*e^7 + 10656*a^3*b^7*c^3*e^7 - 30240*a^4*b^5*c^4*e^7 + 44800*a^5*b^

3*c^5*e^7 - 25*a^2*c^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - 17920*a^4*c^9*d^5*e^2 - 35840*a^5*c^8*d^3*e^4 + 98*b^8*c

^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 70*b^10*c^3*d^3*e^4 + 14*b^11*c^2*d^2*e^5 + 35*c^4*d^4*e^3*(-(4*a*c - b^2)^9

)^(1/2) + 213*a*b^11*c*e^7 + 21*b^12*c*d*e^6 + 1344*a^2*b^4*c^7*d^5*e^2 + 10080*a^2*b^5*c^6*d^4*e^3 - 7840*a^2

*b^6*c^5*d^3*e^4 - 1008*a^2*b^7*c^4*d^2*e^5 + 7168*a^3*b^2*c^8*d^5*e^2 - 35840*a^3*b^3*c^7*d^4*e^3 + 17920*a^3

*b^4*c^6*d^3*e^4 + 12544*a^3*b^5*c^5*d^2*e^5 - 44800*a^4*b^3*c^6*d^2*e^5 + 14*b^2*c^2*d^2*e^5*(-(4*a*c - b^2)^

9)^(1/2) + 51*a*b^2*c*e^7*(-(4*a*c - b^2)^9)^(1/2) + 1344*a*b^5*c^7*d^6*e - 532*a*b^10*c^2*d*e^6 + 7168*a^3*b*

c^9*d^6*e + 21*b^3*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) - 896*a*b^6*c^6*d^5*e^2 - 1120*a*b^7*c^5*d^4*e^3 + 1260*a*

b^8*c^4*d^3*e^4 - 98*a*b^9*c^3*d^2*e^5 - 5376*a^2*b^3*c^8*d^6*e + 5418*a^2*b^8*c^3*d*e^6 - 28224*a^3*b^6*c^4*d

*e^6 + 44800*a^4*b*c^8*d^4*e^3 + 78400*a^4*b^4*c^5*d*e^6 + 53760*a^5*b*c^7*d^2*e^5 - 107520*a^5*b^2*c^6*d*e^6

+ 154*a*c^3*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 70*b*c^3*d^3*e^4*(-(4*a*c - b^2)^9)^(1/2) - 154*a*b*c^2*d*e^6*(

-(4*a*c - b^2)^9)^(1/2))/(8*(4096*a^6*c^11 + b^12*c^5 - 24*a*b^10*c^6 + 240*a^2*b^8*c^7 - 1280*a^3*b^6*c^8 + 3

840*a^4*b^4*c^9 - 6144*a^5*b^2*c^10)))^(1/2)*(4*b^7*c^5*e^3 - 48*a*b^5*c^6*e^3 - 256*a^3*b*c^8*e^3 + 512*a^3*c

^9*d*e^2 - 8*b^6*c^6*d*e^2 + 192*a^2*b^3*c^7*e^3 + 96*a*b^4*c^7*d*e^2 - 384*a^2*b^2*c^8*d*e^2))/(16*a^2*c^5 +

b^4*c^3 - 8*a*b^2*c^4))*((32*b^6*c^7*d^7 - 2048*a^3*c^10*d^7 - 9*b^13*e^7 - 9*b^4*e^7*(-(4*a*c - b^2)^9)^(1/2)

- 384*a*b^4*c^8*d^7 - 26880*a^6*b*c^6*e^7 + 53760*a^6*c^7*d*e^6 - 112*b^7*c^6*d^6*e + 1536*a^2*b^2*c^9*d^7 -

2077*a^2*b^9*c^2*e^7 + 10656*a^3*b^7*c^3*e^7 - 30240*a^4*b^5*c^4*e^7 + 44800*a^5*b^3*c^5*e^7 - 25*a^2*c^2*e^7*

(-(4*a*c - b^2)^9)^(1/2) - 17920*a^4*c^9*d^5*e^2 - 35840*a^5*c^8*d^3*e^4 + 98*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4

*e^3 - 70*b^10*c^3*d^3*e^4 + 14*b^11*c^2*d^2*e^5 + 35*c^4*d^4*e^3*(-(4*a*c - b^2)^9)^(1/2) + 213*a*b^11*c*e^7

+ 21*b^12*c*d*e^6 + 1344*a^2*b^4*c^7*d^5*e^2 + 10080*a^2*b^5*c^6*d^4*e^3 - 7840*a^2*b^6*c^5*d^3*e^4 - 1008*a^2

*b^7*c^4*d^2*e^5 + 7168*a^3*b^2*c^8*d^5*e^2 - 35840*a^3*b^3*c^7*d^4*e^3 + 17920*a^3*b^4*c^6*d^3*e^4 + 12544*a^

3*b^5*c^5*d^2*e^5 - 44800*a^4*b^3*c^6*d^2*e^5 + 14*b^2*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 51*a*b^2*c*e^7*(

-(4*a*c - b^2)^9)^(1/2) + 1344*a*b^5*c^7*d^6*e - 532*a*b^10*c^2*d*e^6 + 7168*a^3*b*c^9*d^6*e + 21*b^3*c*d*e^6*

(-(4*a*c - b^2)^9)^(1/2) - 896*a*b^6*c^6*d^5*e^2 - 1120*a*b^7*c^5*d^4*e^3 + 1260*a*b^8*c^4*d^3*e^4 - 98*a*b^9*

c^3*d^2*e^5 - 5376*a^2*b^3*c^8*d^6*e + 5418*a^2*b^8*c^3*d*e^6 - 28224*a^3*b^6*c^4*d*e^6 + 44800*a^4*b*c^8*d^4*

e^3 + 78400*a^4*b^4*c^5*d*e^6 + 53760*a^5*b*c^7*d^2*e^5 - 107520*a^5*b^2*c^6*d*e^6 + 154*a*c^3*d^2*e^5*(-(4*a*

c - b^2)^9)^(1/2) - 70*b*c^3*d^3*e^4*(-(4*a*c - b^2)^9)^(1/2) - 154*a*b*c^2*d*e^6*(-(4*a*c - b^2)^9)^(1/2))/(8

*(4096*a^6*c^11 + b^12*c^5 - 24*a*b^10*c^6 + 240*a^2*b^8*c^7 - 1280*a^3*b^6*c^8 + 3840*a^4*b^4*c^9 - 6144*a^5*

b^2*c^10)))^(1/2) + (2*(d + e*x)^(1/2)*(9*b^8*e^10 + 200*a^4*c^4*e^10 + 32*c^8*d^8*e^2 + 280*a*c^7*d^6*e^4 - 1

28*b*c^7*d^7*e^3 + 481*a^2*b^4*c^2*e^10 - 718*a^3*b^2*c^3*e^10 + 280*a^2*c^6*d^4*e^6 - 2072*a^3*c^5*d^2*e^8 +

154*b^2*c^6*d^6*e^4 - 14*b^3*c^5*d^5*e^5 - 105*b^4*c^4*d^4*e^6 + 84*b^5*c^3*d^3*e^7 + 7*b^6*c^2*d^2*e^8 - 114*

a*b^6*c*e^10 - 30*b^7*c*d*e^9 + 1974*a^2*b^2*c^4*d^2*e^8 - 840*a*b*c^6*d^5*e^5 + 406*a*b^5*c^2*d*e^9 + 2072*a^

3*b*c^4*d*e^9 + 910*a*b^2*c^5*d^4*e^6 - 420*a*b^3*c^4*d^3*e^7 - 336*a*b^4*c^3*d^2*e^8 - 560*a^2*b*c^5*d^3*e^7

- 1694*a^2*b^3*c^3*d*e^9))/(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4))*((32*b^6*c^7*d^7 - 2048*a^3*c^10*d^7 - 9*b^13