3.23.95 \(\int \frac {(d+e x)^{7/2}}{(a+b x+c x^2)^2} \, dx\) [2295]

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

[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 = 12.21, 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 = {752, 838, 840, 1180, 214} \begin {gather*} \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 )}} \end {gather*}

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 214

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

Rule 752

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 838

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 840

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 1180

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 \text {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 ) \text {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 ) \text {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 [C] Result contains complex when optimal does not.
time = 9.67, size = 771, normalized size = 1.12 \begin {gather*} \frac {-\frac {2 \sqrt {c} \sqrt {d+e x} \left (-3 b^3 e^3 x+b^2 e^2 (-3 a e+c x (3 d-2 e x))+b c \left (c d^2 (d-3 e x)+a e^2 (3 d+11 e x)\right )+2 c \left (5 a^2 e^3+c^2 d^3 x+a c e \left (-3 d^2-3 d e x+4 e^2 x^2\right )\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {\left (8 i c^4 d^4-3 b^3 \left (i b+\sqrt {-b^2+4 a c}\right ) e^4-2 c^3 d^2 e \left (8 i b d+\sqrt {-b^2+4 a c} d-18 i a e\right )+b c e^3 \left (5 i b^2 d+5 b \sqrt {-b^2+4 a c} d+19 i a b e+13 a \sqrt {-b^2+4 a c} e\right )+c^2 e^2 \left (3 i b^2 d^2+2 a e \left (-13 \sqrt {-b^2+4 a c} d-10 i a e\right )+3 b d \left (\sqrt {-b^2+4 a c} d-12 i a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\left (-b^2+4 a c\right )^{3/2} \sqrt {-c d+\frac {1}{2} \left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {\left (-8 i c^4 d^4-3 b^3 \left (-i b+\sqrt {-b^2+4 a c}\right ) e^4-2 c^3 d^2 e \left (-8 i b d+\sqrt {-b^2+4 a c} d+18 i a e\right )+b c e^3 \left (-5 i b^2 d+5 b \sqrt {-b^2+4 a c} d-19 i a b e+13 a \sqrt {-b^2+4 a c} e\right )+c^2 e^2 \left (-3 i b^2 d^2+2 a e \left (-13 \sqrt {-b^2+4 a c} d+10 i a e\right )+3 b d \left (\sqrt {-b^2+4 a c} d+12 i a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\left (-b^2+4 a c\right )^{3/2} \sqrt {-c d+\frac {1}{2} \left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{2 c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Sqrt[c]*Sqrt[d + e*x]*(-3*b^3*e^3*x + b^2*e^2*(-3*a*e + c*x*(3*d - 2*e*x)) + b*c*(c*d^2*(d - 3*e*x) + a*e
^2*(3*d + 11*e*x)) + 2*c*(5*a^2*e^3 + c^2*d^3*x + a*c*e*(-3*d^2 - 3*d*e*x + 4*e^2*x^2))))/((b^2 - 4*a*c)*(a +
x*(b + c*x))) - (((8*I)*c^4*d^4 - 3*b^3*(I*b + Sqrt[-b^2 + 4*a*c])*e^4 - 2*c^3*d^2*e*((8*I)*b*d + Sqrt[-b^2 +
4*a*c]*d - (18*I)*a*e) + b*c*e^3*((5*I)*b^2*d + 5*b*Sqrt[-b^2 + 4*a*c]*d + (19*I)*a*b*e + 13*a*Sqrt[-b^2 + 4*a
*c]*e) + c^2*e^2*((3*I)*b^2*d^2 + 2*a*e*(-13*Sqrt[-b^2 + 4*a*c]*d - (10*I)*a*e) + 3*b*d*(Sqrt[-b^2 + 4*a*c]*d
- (12*I)*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/((-b^2 +
4*a*c)^(3/2)*Sqrt[-(c*d) + ((b - I*Sqrt[-b^2 + 4*a*c])*e)/2]) - (((-8*I)*c^4*d^4 - 3*b^3*((-I)*b + Sqrt[-b^2 +
4*a*c])*e^4 - 2*c^3*d^2*e*((-8*I)*b*d + Sqrt[-b^2 + 4*a*c]*d + (18*I)*a*e) + b*c*e^3*((-5*I)*b^2*d + 5*b*Sqrt
[-b^2 + 4*a*c]*d - (19*I)*a*b*e + 13*a*Sqrt[-b^2 + 4*a*c]*e) + c^2*e^2*((-3*I)*b^2*d^2 + 2*a*e*(-13*Sqrt[-b^2
+ 4*a*c]*d + (10*I)*a*e) + 3*b*d*(Sqrt[-b^2 + 4*a*c]*d + (12*I)*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/
Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/((-b^2 + 4*a*c)^(3/2)*Sqrt[-(c*d) + ((b + I*Sqrt[-b^2 + 4*a*c])*
e)/2]))/(2*c^(5/2))

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Maple [A]
time = 0.91, size = 945, normalized size = 1.37

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {\frac {-\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a -b^{3} e^{3}+3 b^{2} d \,e^{2} c -3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (2 e^{4} a^{2} c -a \,b^{2} e^{4}+b^{3} d \,e^{3}-3 b^{2} c \,d^{2} e^{2}+4 d^{3} e b \,c^{2}-2 d^{4} c^{3}\right ) \sqrt {e x +d}}{2 e^{2} \left (4 a c -b^{2}\right )}}{\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}}+\frac {2 c \left (\frac {\left (-20 e^{4} a^{2} c^{2}+19 a \,b^{2} c \,e^{4}-36 a b \,c^{2} d \,e^{3}+36 d^{2} e^{2} c^{3} a -3 b^{4} e^{4}+5 b^{3} c d \,e^{3}+3 b^{2} c^{2} d^{2} e^{2}-16 b \,c^{3} d^{3} e +8 c^{4} d^{4}+13 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-26 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+5 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (20 e^{4} a^{2} c^{2}-19 a \,b^{2} c \,e^{4}+36 a b \,c^{2} d \,e^{3}-36 d^{2} e^{2} c^{3} a +3 b^{4} e^{4}-5 b^{3} c d \,e^{3}-3 b^{2} c^{2} d^{2} e^{2}+16 b \,c^{3} d^{3} e -8 c^{4} d^{4}+13 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-26 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+5 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}}{c^{2}}\right )\)	\(945\)
default	\(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {\frac {-\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a -b^{3} e^{3}+3 b^{2} d \,e^{2} c -3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (2 e^{4} a^{2} c -a \,b^{2} e^{4}+b^{3} d \,e^{3}-3 b^{2} c \,d^{2} e^{2}+4 d^{3} e b \,c^{2}-2 d^{4} c^{3}\right ) \sqrt {e x +d}}{2 e^{2} \left (4 a c -b^{2}\right )}}{\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}}+\frac {2 c \left (\frac {\left (-20 e^{4} a^{2} c^{2}+19 a \,b^{2} c \,e^{4}-36 a b \,c^{2} d \,e^{3}+36 d^{2} e^{2} c^{3} a -3 b^{4} e^{4}+5 b^{3} c d \,e^{3}+3 b^{2} c^{2} d^{2} e^{2}-16 b \,c^{3} d^{3} e +8 c^{4} d^{4}+13 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-26 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+5 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (20 e^{4} a^{2} c^{2}-19 a \,b^{2} c \,e^{4}+36 a b \,c^{2} d \,e^{3}-36 d^{2} e^{2} c^{3} a +3 b^{4} e^{4}-5 b^{3} c d \,e^{3}-3 b^{2} c^{2} d^{2} e^{2}+16 b \,c^{3} d^{3} e -8 c^{4} d^{4}+13 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-26 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+5 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}}{c^{2}}\right )\)	\(945\)
risch	\(\text {Expression too large to display}\)	\(3838\)

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,method=_RETURNVERBOSE)

[Out]

2*e^3*(1/c^2*(e*x+d)^(1/2)-1/c^2*((-1/2*(3*a*b*c*e^3-6*a*c^2*d*e^2-b^3*e^3+3*b^2*c*d*e^2-3*b*c^2*d^2*e+2*c^3*d
^3)/e^2/(4*a*c-b^2)*(e*x+d)^(3/2)-1/2*(2*a^2*c*e^4-a*b^2*e^4+b^3*d*e^3-3*b^2*c*d^2*e^2+4*b*c^2*d^3*e-2*c^3*d^4
)/e^2/(4*a*c-b^2)*(e*x+d)^(1/2))/((e*x+d)^2*c+b*e*(e*x+d)-2*c*d*(e*x+d)+e^2*a-b*d*e+c*d^2)+2/e^2/(4*a*c-b^2)*c
*(1/8*(-20*e^4*a^2*c^2+19*a*b^2*c*e^4-36*a*b*c^2*d*e^3+36*d^2*e^2*c^3*a-3*b^4*e^4+5*b^3*c*d*e^3+3*b^2*c^2*d^2*
e^2-16*b*c^3*d^3*e+8*c^4*d^4+13*(-e^2*(4*a*c-b^2))^(1/2)*a*b*c*e^3-26*(-e^2*(4*a*c-b^2))^(1/2)*a*c^2*d*e^2-3*(
-e^2*(4*a*c-b^2))^(1/2)*b^3*e^3+5*(-e^2*(4*a*c-b^2))^(1/2)*b^2*c*d*e^2+3*(-e^2*(4*a*c-b^2))^(1/2)*b*c^2*d^2*e-
2*(-e^2*(4*a*c-b^2))^(1/2)*c^3*d^3)/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c
)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))-1/8*(20*e^4*a^2*c^2-19*
a*b^2*c*e^4+36*a*b*c^2*d*e^3-36*d^2*e^2*c^3*a+3*b^4*e^4-5*b^3*c*d*e^3-3*b^2*c^2*d^2*e^2+16*b*c^3*d^3*e-8*c^4*d
^4+13*(-e^2*(4*a*c-b^2))^(1/2)*a*b*c*e^3-26*(-e^2*(4*a*c-b^2))^(1/2)*a*c^2*d*e^2-3*(-e^2*(4*a*c-b^2))^(1/2)*b^
3*e^3+5*(-e^2*(4*a*c-b^2))^(1/2)*b^2*c*d*e^2+3*(-e^2*(4*a*c-b^2))^(1/2)*b*c^2*d^2*e-2*(-e^2*(4*a*c-b^2))^(1/2)
*c^3*d^3)/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)
^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9850 vs. \(2 (646) = 1292\).
time = 35.94, size = 9850, normalized size = 14.25 \begin {gather*} \text {Too large to display} \end {gather*}

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(x*e + 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 + ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1797 vs. \(2 (646) = 1292\).
time = 2.04, size = 1797, normalized size = 2.60 \begin {gather*} \frac {2 \, \sqrt {x e + d} e^{3}}{c^{2}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e - 2 \, \sqrt {x e + d} c^{3} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{2} + 4 \, \sqrt {x e + d} b c^{2} d^{3} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d e^{3} - 6 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d e^{3} - 3 \, \sqrt {x e + d} b^{2} c d^{2} e^{3} - {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{4} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} a b c e^{4} + \sqrt {x e + d} b^{3} d e^{4} - \sqrt {x e + d} a b^{2} e^{5} + 2 \, \sqrt {x e + d} a^{2} c e^{5}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e + a e^{2}\right )}} - \frac {{\left (16 \, {\left (b^{2} c^{9} - 4 \, a c^{10}\right )} d^{5} e - 40 \, {\left (b^{3} c^{8} - 4 \, a b c^{9}\right )} d^{4} e^{2} + 2 \, {\left (11 \, b^{4} c^{7} - 8 \, a b^{2} c^{8} - 144 \, a^{2} c^{9}\right )} d^{3} e^{3} + {\left (7 \, b^{5} c^{6} - 136 \, a b^{3} c^{7} + 432 \, a^{2} b c^{8}\right )} d^{2} e^{4} - {\left (2 \, c^{3} d^{3} e - 3 \, b c^{2} d^{2} e^{2} - {\left (5 \, b^{2} c - 26 \, a c^{2}\right )} d e^{3} + {\left (3 \, b^{3} - 13 \, a b c\right )} e^{4}\right )} {\left (b^{2} c^{2} e - 4 \, a c^{3} e\right )}^{2} - {\left (11 \, b^{6} c^{5} - 118 \, a b^{4} c^{6} + 336 \, a^{2} b^{2} c^{7} - 160 \, a^{3} c^{8}\right )} d e^{5} - 2 \, {\left (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} + {\left (5 \, b^{2} c^{4} - 8 \, a c^{5}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e^{3} - {\left (3 \, b^{3} c^{3} - 8 \, a b c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d e^{4} + {\left (3 \, a b^{2} c^{3} - 10 \, a^{2} c^{4}\right )} \sqrt {b^{2} - 4 \, a c} e^{5}\right )} {\left | b^{2} c^{2} e - 4 \, a c^{3} e \right |} + {\left (3 \, b^{7} c^{4} - 31 \, a b^{5} c^{5} + 96 \, a^{2} b^{3} c^{6} - 80 \, a^{3} b c^{7}\right )} e^{6}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{2} c^{3} d - 8 \, a c^{4} d - b^{3} c^{2} e + 4 \, a b c^{3} e + \sqrt {{\left (2 \, b^{2} c^{3} d - 8 \, a c^{4} d - b^{3} c^{2} e + 4 \, a b c^{3} e\right )}^{2} - 4 \, {\left (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}\right )} {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}}{b^{2} c^{3} - 4 \, a c^{4}}}}\right )}{\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt {b^{2} - 4 \, a c} d - {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {b^{2} - 4 \, a c}\right )} e\right )} {\left | b^{2} c^{2} e - 4 \, a c^{3} e \right |} {\left | c \right |}} + \frac {{\left (16 \, {\left (b^{2} c^{9} - 4 \, a c^{10}\right )} d^{5} e - 40 \, {\left (b^{3} c^{8} - 4 \, a b c^{9}\right )} d^{4} e^{2} + 2 \, {\left (11 \, b^{4} c^{7} - 8 \, a b^{2} c^{8} - 144 \, a^{2} c^{9}\right )} d^{3} e^{3} + {\left (7 \, b^{5} c^{6} - 136 \, a b^{3} c^{7} + 432 \, a^{2} b c^{8}\right )} d^{2} e^{4} - {\left (2 \, c^{3} d^{3} e - 3 \, b c^{2} d^{2} e^{2} - {\left (5 \, b^{2} c - 26 \, a c^{2}\right )} d e^{3} + {\left (3 \, b^{3} - 13 \, a b c\right )} e^{4}\right )} {\left (b^{2} c^{2} e - 4 \, a c^{3} e\right )}^{2} - {\left (11 \, b^{6} c^{5} - 118 \, a b^{4} c^{6} + 336 \, a^{2} b^{2} c^{7} - 160 \, a^{3} c^{8}\right )} d e^{5} + 2 \, {\left (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} + {\left (5 \, b^{2} c^{4} - 8 \, a c^{5}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e^{3} - {\left (3 \, b^{3} c^{3} - 8 \, a b c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d e^{4} + {\left (3 \, a b^{2} c^{3} - 10 \, a^{2} c^{4}\right )} \sqrt {b^{2} - 4 \, a c} e^{5}\right )} {\left | b^{2} c^{2} e - 4 \, a c^{3} e \right |} + {\left (3 \, b^{7} c^{4} - 31 \, a b^{5} c^{5} + 96 \, a^{2} b^{3} c^{6} - 80 \, a^{3} b c^{7}\right )} e^{6}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{2} c^{3} d - 8 \, a c^{4} d - b^{3} c^{2} e + 4 \, a b c^{3} e - \sqrt {{\left (2 \, b^{2} c^{3} d - 8 \, a c^{4} d - b^{3} c^{2} e + 4 \, a b c^{3} e\right )}^{2} - 4 \, {\left (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}\right )} {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}}{b^{2} c^{3} - 4 \, a c^{4}}}}\right )}{\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt {b^{2} - 4 \, a c} d + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5} - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {b^{2} - 4 \, a c}\right )} e\right )} {\left | b^{2} c^{2} e - 4 \, a c^{3} e \right |} {\left | c \right |}} \end {gather*}

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))

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Mupad [B]
time = 3.88, size = 2500, normalized size = 3.62 \begin {gather*} \text {Too large to display} \end {gather*}

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 ...

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