∫ (b+2cx)(d+ex)7∕2 __________________________

3.1623 \(\int \frac {(b+2 c x) (d+e x)^{7/2}}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=543 \[ \frac {7 e \left (-2 c^2 d e \left (-d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b d \sqrt {b^2-4 a c}+3 a e \sqrt {b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {7 e \left (-2 c^2 d e \left (d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt {b^2-4 a c}-3 a e \sqrt {b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {7 e^2 \sqrt {d+e x} (2 c d-b e)}{4 c \left (b^2-4 a c\right )}-\frac {7 e (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

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

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

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

*d-4*a*b*e+b*d*(-4*a*c+b^2)^(1/2)-3*a*e*(-4*a*c+b^2)^(1/2)))/c^(3/2)/(-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 = 5.72, antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {768, 738, 824, 826, 1166, 208} \[ \frac {7 e \left (-2 c^2 d e \left (-d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b d \sqrt {b^2-4 a c}+3 a e \sqrt {b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {7 e \left (-2 c^2 d e \left (d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt {b^2-4 a c}-3 a e \sqrt {b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {7 e^2 \sqrt {d+e x} (2 c d-b e)}{4 c \left (b^2-4 a c\right )}-\frac {7 e (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully veriﬁed.

[In]

Int[((b + 2*c*x)*(d + e*x)^(7/2))/(a + b*x + c*x^2)^3,x]

[Out]

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

e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (7*e*(8*c^3*d^3 + b^2*(b - S

qrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 2*c*e^2*(b^2*d - b*Sqrt[b^2 - 4*a*c]

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

4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (7*e*(8*c^3*d^3

+ b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 2*c*e^2*(b^2*d + b*Sqrt

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

+ Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(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 768

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

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

1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]

&& GtQ[m, 0]

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 {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{4} (7 e) \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(7 e) \int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (4 c d^2-5 b d e+6 a e^2\right )-\frac {1}{2} e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{4 \left (b^2-4 a c\right )}\\ &=\frac {7 e^2 (2 c d-b e) \sqrt {d+e x}}{4 c \left (b^2-4 a c\right )}-\frac {(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(7 e) \int \frac {\frac {1}{2} \left (4 c^2 d^3-a b e^3-c d e (5 b d-8 a e)\right )+\frac {1}{2} e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{4 c \left (b^2-4 a c\right )}\\ &=\frac {7 e^2 (2 c d-b e) \sqrt {d+e x}}{4 c \left (b^2-4 a c\right )}-\frac {(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(7 e) \operatorname {Subst}\left (\int \frac {\frac {1}{2} e \left (4 c^2 d^3-a b e^3-c d e (5 b d-8 a e)\right )-\frac {1}{2} d e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )+\frac {1}{2} e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 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 )}{2 c \left (b^2-4 a c\right )}\\ &=\frac {7 e^2 (2 c d-b e) \sqrt {d+e x}}{4 c \left (b^2-4 a c\right )}-\frac {(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (7 e \left (8 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d-4 a b e-3 a \sqrt {b^2-4 a c} 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 )}{8 c \left (b^2-4 a c\right )^{3/2}}-\frac {\left (7 e \left (8 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+3 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+4 a e\right )\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 )}{8 c \left (b^2-4 a c\right )^{3/2}}\\ &=\frac {7 e^2 (2 c d-b e) \sqrt {d+e x}}{4 c \left (b^2-4 a c\right )}-\frac {(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 e (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {7 e \left (8 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d-b \sqrt {b^2-4 a c} d-4 a b e+3 a \sqrt {b^2-4 a c} e\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 )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {7 e \left (8 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d-4 a b e-3 a \sqrt {b^2-4 a c} e\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 )}{4 \sqrt {2} c^{3/2} \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 [B] time = 6.75, size = 1403, normalized size = 2.58 \[ -\frac {\left (-2 a c (2 c d-b e)+b \left (-e b^2+c d b+2 a c e\right )+c (b (2 c d-b e)-2 c (b d-2 a e)) x\right ) (d+e x)^{9/2}}{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 )^2}-\frac {-\frac {\left (-\frac {3}{2} a c \left (b^2-4 a c\right ) (2 c d-b e) e^2-\frac {1}{2} \left (b^2-4 a c\right ) (7 c d-5 b e) \left (-e b^2+c d b+2 a c e\right ) e+c \left (-\frac {3}{2} c \left (b^2-4 a c\right ) (b d-2 a e) e^2-\frac {1}{2} \left (b^2-4 a c\right ) (7 c d-5 b e) (2 c d-b e) e\right ) 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 {1}{2} \left (b^2-4 a c\right ) e^2 \left (14 c^2 d^2+5 b^2 e^2-2 c e (7 b d+3 a e)\right ) (d+e x)^{7/2}+\frac {2 \left (\frac {49}{4} c \left (b^2-4 a c\right ) e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right ) (d+e x)^{5/2}+\frac {2 \left (\frac {245}{4} c^2 \left (b^2-4 a c\right ) e^2 (d+e x)^{3/2} \left (c d^2-e (b d-a e)\right )^2+\frac {2 \left (\frac {735}{16} c^2 \left (b^2-4 a c\right ) e^2 (2 c d-b e) \sqrt {d+e x} \left (c d^2-e (b d-a e)\right )^2+\frac {4 \left (\frac {\sqrt {2 c d-b e-\sqrt {b^2-4 a c} e} \left (-\frac {735}{64} \left (b^2-4 a c\right ) e^2 \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2 c^3-\frac {\frac {735}{64} \left (b^2-4 a c\right ) e^2 (b e-2 c d) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2 c^3+2 \left (\frac {735}{64} c^3 \left (b^2-4 a c\right ) d e^2 \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2-\frac {735}{64} c^3 \left (b^2-4 a c\right ) e^2 \left (4 c^2 d^3-c e (5 b d-8 a e) d-a b e^3\right ) \left (c d^2-e (b d-a e)\right )^2\right ) c}{\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 {\frac {735}{64} \left (b^2-4 a c\right ) e^2 (b e-2 c d) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2 c^3+2 \left (\frac {735}{64} c^3 \left (b^2-4 a c\right ) d e^2 \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2-\frac {735}{64} c^3 \left (b^2-4 a c\right ) e^2 \left (4 c^2 d^3-c e (5 b d-8 a e) d-a b e^3\right ) \left (c d^2-e (b d-a e)\right )^2\right ) c}{\sqrt {b^2-4 a c} e}-\frac {735}{64} c^3 \left (b^2-4 a c\right ) e^2 \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \left (c d^2-e (b d-a e)\right )^2\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}\right )}{3 c}\right )}{5 c}\right )}{7 c}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{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[((b + 2*c*x)*(d + e*x)^(7/2))/(a + b*x + c*x^2)^3,x]

[Out]

-1/2*((d + e*x)^(9/2)*(-2*a*c*(2*c*d - b*e) + b*(b*c*d - b^2*e + 2*a*c*e) + c*(-2*c*(b*d - 2*a*e) + b*(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)^2) - (-(((d + e*x)^(9/2)*((-3*a*c*(b^2 - 4

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

*a*c)*e^2*(b*d - 2*a*e))/2 - ((b^2 - 4*a*c)*e*(7*c*d - 5*b*e)*(2*c*d - b*e))/2)*x))/((b^2 - 4*a*c)*(c*d^2 - b*

d*e + a*e^2)*(a + b*x + c*x^2))) - (((b^2 - 4*a*c)*e^2*(14*c^2*d^2 + 5*b^2*e^2 - 2*c*e*(7*b*d + 3*a*e))*(d + e

*x)^(7/2))/2 + (2*((49*c*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))*(d + e*x)^(5/2))/4 + (2*((245

*c^2*(b^2 - 4*a*c)*e^2*(c*d^2 - e*(b*d - a*e))^2*(d + e*x)^(3/2))/4 + (2*((735*c^2*(b^2 - 4*a*c)*e^2*(2*c*d -

b*e)*(c*d^2 - e*(b*d - a*e))^2*Sqrt[d + e*x])/16 + (4*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*((-735*c^3*(b^

2 - 4*a*c)*e^2*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 - ((735*c^3*(b^2 - 4*

a*c)*e^2*(-2*c*d + b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 + 2*c*((-735

*c^3*(b^2 - 4*a*c)*e^2*(4*c^2*d^3 - a*b*e^3 - c*d*e*(5*b*d - 8*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 + (735*c^3*

(b^2 - 4*a*c)*d*e^2*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64))/(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]*((-735*c^3*(b^2 - 4*a*c)*e^2*(2

*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 + ((735*c^3*(b^2 - 4*a*c)*e^2*(-2*c*d

+ b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 + 2*c*((-735*c^3*(b^2 - 4*a*c

)*e^2*(4*c^2*d^3 - a*b*e^3 - c*d*e*(5*b*d - 8*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64 + (735*c^3*(b^2 - 4*a*c)*d*e

^2*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*(c*d^2 - e*(b*d - a*e))^2)/64))/(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 - Sqr

t[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)))/(2*(b^2 - 4*a*c)*(c*d

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

________________________________________________________________________________________

fricas [B] time = 1.75, size = 5701, normalized size = 10.50 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/8*(7*sqrt(1/2)*(a^2*b^2*c - 4*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*x^3 + (b^4*c - 2*a

*b^2*c^2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*x)*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c

^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5

- 15*a*b^3*c + 60*a^2*b*c^2)*e^7 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^1

0 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c +

81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^

2*c^5 - 64*a^3*c^6))*log(343/2*sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^6 - 15*(b^5*c^2 - 8*a*

b^3*c^3 + 16*a^2*b*c^4)*d^2*e^7 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^8 + (b^7 - 17*a*b^

5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^9 - (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*

(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640

*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 +

10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^

8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 3

6*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^7 + (b^6

*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6

*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c

^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 343*(80*c^5*d^6*

e^6 - 240*b*c^4*d^5*e^7 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^8 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^9 - 6*(6*b^4*c -

47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^10 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^11 + (5*a*b^4 - 81*a^2*b^2*c

+ 324*a^3*c^2)*e^12)*sqrt(e*x + d)) - 7*sqrt(1/2)*(a^2*b^2*c - 4*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c

^2 - 4*a*b*c^3)*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*x)*sqrt((32*c^5*d^5*e^

2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b

^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^7 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5

- 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*

c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(

b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(-343/2*sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^

2*c^5)*d^3*e^6 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^7 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 +

96*a^3*c^4)*d*e^8 + (b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^9 - (8*(b^6*c^5 - 12*a*b^4*c^6 + 48

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

4*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11

+ 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6

*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 +

12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15

*a*b^3*c + 60*a^2*b*c^2)*e^7 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 -

50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a

^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^

5 - 64*a^3*c^6)) + 343*(80*c^5*d^6*e^6 - 240*b*c^4*d^5*e^7 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^8 + 2*(b^3*c^2 -

404*a*b*c^3)*d^3*e^9 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^10 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^

2)*d*e^11 + (5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*e^12)*sqrt(e*x + d)) + 7*sqrt(1/2)*(a^2*b^2*c - 4*a^3*c^2 +

(b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c -

4*a^2*b*c^2)*x)*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*

a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^7 - (b^6*c

^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a

*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7

+ 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(343/2*sqrt(1/2)*

(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^6 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^7 + 3*(b^6*

c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^8 + (b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^9

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

- 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25

*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 1

8*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5*e^2 -

80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c

^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^7 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64

*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)

*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*

c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 343*(80*c^5*d^6*e^6 - 240*b*c^4*d^5*e^7 + (199*b^2*c^3 +

404*a*c^4)*d^4*e^8 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^9 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^10 - (

5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^11 + (5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*e^12)*sqrt(e*x + d)) - 7*

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

2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*x)*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12

*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*

b^3*c + 60*a^2*b*c^2)*e^7 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*

b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*

c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 -

64*a^3*c^6))*log(-343/2*sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^6 - 15*(b^5*c^2 - 8*a*b^3*c^

3 + 16*a^2*b*c^4)*d^2*e^7 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^8 + (b^7 - 17*a*b^5*c +

88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^9 + (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c

^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b

^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^12 + 10*(b

^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64

*a^3*c^9)))*sqrt((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^4 + 5*(b^3*c^2 - 36*a*b*

c^3)*d^2*e^5 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^6 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^7 - (b^6*c^3 -

12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^10 - 50*b*c^3*d^3*e^11 + 15*(b^2*c^2 + 6*a*c^3

)*d^2*e^12 + 10*(b^3*c - 9*a*b*c^2)*d*e^13 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^14)/(b^6*c^6 - 12*a*b^4*c^7 + 4

8*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 343*(80*c^5*d^6*e^6 -

240*b*c^4*d^5*e^7 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^8 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^9 - 6*(6*b^4*c - 47*a*

b^2*c^2 - 108*a^2*c^3)*d^2*e^10 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^11 + (5*a*b^4 - 81*a^2*b^2*c + 324

*a^3*c^2)*e^12)*sqrt(e*x + d)) - 2*(7*a*b*c*d^2*e - 28*a^2*c*d*e^2 + 7*a^2*b*e^3 + 2*(b^2*c - 4*a*c^2)*d^3 + (

14*c^3*d^2*e - 14*b*c^2*d*e^2 + (9*b^2*c - 22*a*c^2)*e^3)*x^3 + (21*b*c^2*d^2*e - 4*(2*b^2*c + 13*a*c^2)*d*e^2

+ 7*(b^3 - a*b*c)*e^3)*x^2 - (42*a*b*c*d*e^2 - (13*b^2*c - 10*a*c^2)*d^2*e - 14*(a*b^2 - a^2*c)*e^3)*x)*sqrt(

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

2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*x)

________________________________________________________________________________________

giac [B] time = 6.05, size = 1749, normalized size = 3.22 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

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

6)*d^2*e^4 - (2*c^2*d^2*e^2 - 2*b*c*d*e^3 - (b^2 - 6*a*c)*e^4)*(b^2*c*e - 4*a*c^2*e)^2 - 32*(a*b^3*c^4 - 4*a^2

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

*c^2*e^5 + (b^2*c^2 + 2*a*c^3)*sqrt(b^2 - 4*a*c)*d*e^4)*abs(-b^2*c*e + 4*a*c^2*e) - (b^6*c^2 - 12*a*b^4*c^3 +

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

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

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

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

- 4*a*b*c^2)*sqrt(b^2 - 4*a*c))*e)*abs(-b^2*c*e + 4*a*c^2*e)*abs(c)) + 7/4*(16*(b^2*c^6 - 4*a*c^7)*d^4*e^2 - 3

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

3 - (b^2 - 6*a*c)*e^4)*(b^2*c*e - 4*a*c^2*e)^2 - 32*(a*b^3*c^4 - 4*a^2*b*c^5)*d*e^5 + 2*(2*sqrt(b^2 - 4*a*c)*c

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

- 4*a*c)*d*e^4)*abs(-b^2*c*e + 4*a*c^2*e) - (b^6*c^2 - 12*a*b^4*c^3 + 32*a^2*b^2*c^4)*e^6)*arctan(2*sqrt(1/2)*

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

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

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

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

b^2*c*e + 4*a*c^2*e)*abs(c)) - 1/4*(14*(x*e + d)^(7/2)*c^3*d^2*e^2 - 42*(x*e + d)^(5/2)*c^3*d^3*e^2 + 42*(x*e

+ d)^(3/2)*c^3*d^4*e^2 - 14*sqrt(x*e + d)*c^3*d^5*e^2 - 14*(x*e + d)^(7/2)*b*c^2*d*e^3 + 63*(x*e + d)^(5/2)*b*

c^2*d^2*e^3 - 84*(x*e + d)^(3/2)*b*c^2*d^3*e^3 + 35*sqrt(x*e + d)*b*c^2*d^4*e^3 + 9*(x*e + d)^(7/2)*b^2*c*e^4

- 22*(x*e + d)^(7/2)*a*c^2*e^4 - 35*(x*e + d)^(5/2)*b^2*c*d*e^4 + 14*(x*e + d)^(5/2)*a*c^2*d*e^4 + 56*(x*e + d

)^(3/2)*b^2*c*d^2*e^4 + 28*(x*e + d)^(3/2)*a*c^2*d^2*e^4 - 28*sqrt(x*e + d)*b^2*c*d^3*e^4 - 28*sqrt(x*e + d)*a

*c^2*d^3*e^4 + 7*(x*e + d)^(5/2)*b^3*e^5 - 7*(x*e + d)^(5/2)*a*b*c*e^5 - 14*(x*e + d)^(3/2)*b^3*d*e^5 - 28*(x*

e + d)^(3/2)*a*b*c*d*e^5 + 7*sqrt(x*e + d)*b^3*d^2*e^5 + 42*sqrt(x*e + d)*a*b*c*d^2*e^5 + 14*(x*e + d)^(3/2)*a

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

(x*e + d)*a^2*b*e^7)/(((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)^2*(b^2*c - 4*a

*c^2))

________________________________________________________________________________________

maple [B] time = 0.15, size = 3503, normalized size = 6.45 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x)

[Out]

-14*e^4/(4*a*c-b^2)*c/(-(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+21/2*e^3/(4*a*c-b^2)*c/(-(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^2-14*e^4/(4*a*c-b^2)*c/(-(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+21/2*e^3/(4*a*c-b^2)*c/(-(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^2+21/4*e^4/(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-35/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(5/2)*b^2*d-11/2*e^4/(c*e

^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(7/2)*a*c+14*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(

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

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

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

*b^2*d^3+21/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a*c-b^2)*(e*x+d)^(3/2)*d^4-7/2*e^2/(c*e^2*x^2+b*e^2*x+a*e

^2)^2*c^2/(4*a*c-b^2)*(e*x+d)^(1/2)*d^5-7/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(5/2)*a*b+7/2*

e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(7/2)*c^2*d^2+7/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b

^2)/c*(e*x+d)^(5/2)*b^3-21/4*e^4/(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-21/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2

/(4*a*c-b^2)*c^2*(e*x+d)^(5/2)*d^3-7/2*e^6/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*a^2*d-7/4*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)*b*d+7/8*e^4/(4*a*c-b^2)/c*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+7/4*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)*b*d-7/4*e^2/(4*a*c-b^2)*c*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^2+63/4*e^3/(c*e^2*x^2+b*

e^2*x+a*e^2)^2/(4*a*c-b^2)*c*(e*x+d)^(5/2)*b*d^2+7/2*e^6/(c*e^2*x^2+b*e^2*x+a*e^2)^2/c/(4*a*c-b^2)*(e*x+d)^(3/

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

2)^2/(4*a*c-b^2)*c*(e*x+d)^(5/2)*a*d-21*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/(4*a*c-b^2)*(e*x+d)^(3/2)*b*d^3+7/4*

e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/c/(4*a*c-b^2)*(e*x+d)^(1/2)*b^3*d^2+21/2*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*

c-b^2)*(e*x+d)^(1/2)*a*b*d^2-7*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)*a*b*d+7/4*e^2/(4*a*c-

b^2)*c*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^2-7/8*e^4/(4*a*c-b^2)/c*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+7/4*e^7/(c*e^2*x^2+b*e^2*

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

^2-7*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/(4*a*c-b^2)*(e*x+d)^(1/2)*a*d^3+35/4*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/

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

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((2*c*x + b)*(e*x + d)^(7/2)/(c*x^2 + b*x + a)^3, x)

________________________________________________________________________________________

mupad [B] time = 6.07, size = 22151, normalized size = 40.79 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(d + e*x)^(7/2))/(a + b*x + c*x^2)^3,x)

[Out]

(((d + e*x)^(7/2)*(9*b^2*e^4 + 14*c^2*d^2*e^2 - 22*a*c*e^4 - 14*b*c*d*e^3))/(4*(4*a*c - b^2)) - (7*(d + e*x)^(

1/2)*(2*c^3*d^5*e^2 - b^3*d^2*e^5 - a^2*b*e^7 + 4*a*c^2*d^3*e^4 - 5*b*c^2*d^4*e^3 + 4*b^2*c*d^3*e^4 + 2*a*b^2*

d*e^6 + 2*a^2*c*d*e^6 - 6*a*b*c*d^2*e^5))/(4*c*(4*a*c - b^2)) + (7*(d + e*x)^(3/2)*(a*b^2*e^6 - a^2*c*e^6 - b^

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

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

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

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

2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2) + atan(((((7*(4096*a^4*b*c^5*e^7 - 64*a*b^7*c^2*e^7 - 8192*

a^4*c^6*d*e^6 + 64*b^8*c^2*d*e^6 + 768*a^2*b^5*c^3*e^7 - 3072*a^3*b^3*c^4*e^7 - 8192*a^3*c^7*d^3*e^4 + 128*b^6

*c^4*d^3*e^4 - 192*b^7*c^3*d^2*e^5 + 6144*a^2*b^2*c^6*d^3*e^4 - 9216*a^2*b^3*c^5*d^2*e^5 - 640*a*b^6*c^3*d*e^6

- 1536*a*b^4*c^5*d^3*e^4 + 2304*a*b^5*c^4*d^2*e^5 + 1536*a^2*b^4*c^4*d*e^6 + 12288*a^3*b*c^6*d^2*e^5 + 2048*a

^3*b^2*c^5*d*e^6))/(64*(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3)) - ((d + e*x)^(1/2)*((49*(b^2*e^7*

(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3

*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^

7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e

^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2

*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^

6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 48

0*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4

*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6))/(128*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 24

0*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(64*b^7*c^3*e^3 - 768*a*b^5*c^

4*e^3 - 4096*a^3*b*c^6*e^3 + 8192*a^3*c^7*d*e^2 - 128*b^6*c^4*d*e^2 + 3072*a^2*b^3*c^5*e^3 + 1536*a*b^4*c^5*d*

e^2 - 6144*a^2*b^2*c^6*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))*((49*(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2)

- b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b

^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^

3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c -

b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b

^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*

c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240

*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*

d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6))/(128*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*

b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2) - ((d + e*x)^(1/2)*(49*b^6*e^10 - 3528*a^3*c^3*e^10 + 1

568*c^6*d^6*e^4 + 5880*a*c^5*d^4*e^6 - 4704*b*c^5*d^5*e^5 + 3626*a^2*b^2*c^2*e^10 + 3920*a^2*c^4*d^2*e^8 + 441

0*b^2*c^4*d^4*e^6 - 980*b^3*c^3*d^3*e^7 - 490*b^4*c^2*d^2*e^8 - 784*a*b^4*c*e^10 + 196*b^5*c*d*e^9 - 11760*a*b

*c^4*d^3*e^7 - 980*a*b^3*c^2*d*e^9 - 3920*a^2*b*c^3*d*e^9 + 6860*a*b^2*c^3*d^2*e^8))/(8*(b^4*c + 16*a^2*c^3 -

8*a*b^2*c^2)))*((49*(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 2

88*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4

+ 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c -

b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c -

b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*

c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2

+ 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*

c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6))/(128*(4096*a^6*c^9 +

b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1

i - (((7*(4096*a^4*b*c^5*e^7 - 64*a*b^7*c^2*e^7 - 8192*a^4*c^6*d*e^6 + 64*b^8*c^2*d*e^6 + 768*a^2*b^5*c^3*e^7

- 3072*a^3*b^3*c^4*e^7 - 8192*a^3*c^7*d^3*e^4 + 128*b^6*c^4*d^3*e^4 - 192*b^7*c^3*d^2*e^5 + 6144*a^2*b^2*c^6*d

^3*e^4 - 9216*a^2*b^3*c^5*d^2*e^5 - 640*a*b^6*c^3*d*e^6 - 1536*a*b^4*c^5*d^3*e^4 + 2304*a*b^5*c^4*d^2*e^5 + 15

36*a^2*b^4*c^4*d*e^6 + 12288*a^3*b*c^6*d^2*e^5 + 2048*a^3*b^2*c^5*d*e^6))/(64*(b^6*c - 64*a^3*c^4 - 12*a*b^4*c

^2 + 48*a^2*b^2*c^3)) + ((d + e*x)^(1/2)*((49*(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^

7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*

e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5

- 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^

6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c

^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e

^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b

^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e

^6))/(128*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6

144*a^5*b^2*c^8)))^(1/2)*(64*b^7*c^3*e^3 - 768*a*b^5*c^4*e^3 - 4096*a^3*b*c^6*e^3 + 8192*a^3*c^7*d*e^2 - 128*b

^6*c^4*d*e^2 + 3072*a^2*b^3*c^5*e^3 + 1536*a*b^4*c^5*d*e^2 - 6144*a^2*b^2*c^6*d*e^2))/(8*(b^4*c + 16*a^2*c^3 -

8*a*b^2*c^2)))*((49*(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) - b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 -

288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^

4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c

- b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c

- b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5

*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^

2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b

*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d^2*e^5 + 3840*a^4*b^2*c^5*d*e^6))/(128*(4096*a^6*c^9 +

b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)

+ ((d + e*x)^(1/2)*(49*b^6*e^10 - 3528*a^3*c^3*e^10 + 1568*c^6*d^6*e^4 + 5880*a*c^5*d^4*e^6 - 4704*b*c^5*d^5*e

^5 + 3626*a^2*b^2*c^2*e^10 + 3920*a^2*c^4*d^2*e^8 + 4410*b^2*c^4*d^4*e^6 - 980*b^3*c^3*d^3*e^7 - 490*b^4*c^2*d

^2*e^8 - 784*a*b^4*c*e^10 + 196*b^5*c*d*e^9 - 11760*a*b*c^4*d^3*e^7 - 980*a*b^3*c^2*d*e^9 - 3920*a^2*b*c^3*d*e

^9 + 6860*a*b^2*c^3*d^2*e^8))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))*((49*(b^2*e^7*(-(4*a*c - b^2)^9)^(1/2) -

b^11*e^7 + 3840*a^5*b*c^5*e^7 - 7680*a^5*c^6*d*e^6 - 288*a^2*b^7*c^2*e^7 + 1504*a^3*b^5*c^3*e^7 - 3840*a^4*b^

3*c^4*e^7 - 2048*a^3*c^8*d^5*e^2 - 7680*a^4*c^7*d^3*e^4 + 32*b^6*c^5*d^5*e^2 - 80*b^7*c^4*d^4*e^3 + 50*b^8*c^3

*d^3*e^4 + 5*b^9*c^2*d^2*e^5 - 5*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^7 - 9*a*c*e^7*(-(4*a*c -

b^2)^9)^(1/2) - 5*b^10*c*d*e^6 + 5*b*c*d*e^6*(-(4*a*c - b^2)^9)^(1/2) + 1536*a^2*b^2*c^7*d^5*e^2 - 3840*a^2*b^

3*c^6*d^4*e^3 + 960*a^2*b^4*c^5*d^3*e^4 + 2400*a^2*b^5*c^4*d^2*e^5 + 2560*a^3*b^2*c^6*d^3*e^4 - 8960*a^3*b^3*c

^5*d^2*e^5 + 90*a*b^8*c^2*d*e^6 - 384*a*b^4*c^6*d^5*e^2 + 960*a*b^5*c^5*d^4*e^3 - 480*a*b^6*c^4*d^3*e^4 - 240*

a*b^7*c^3*d^2*e^5 - 480*a^2*b^6*c^3*d*e^6 + 5120*a^3*b*c^7*d^4*e^3 + 320*a^3*b^4*c^4*d*e^6 + 11520*a^4*b*c^6*d

^2*e^5 + 3840*a^4*b^2*c^5*d*e^6))/(128*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b

^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1i)/((((7*(4096*a^4*b*c^5*e^7 - 64*a*b^7*c^2*e^7 - 8192*