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

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]

(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 - 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]) - (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))*ArcTan
h[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*S
qrt[2]*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi [A]  time = 11.0258, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{7 e \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b d \sqrt{b^2-4 a c}+3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{7 e \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{7 e^2 \sqrt{d+e x} (2 c d-b e)}{4 c \left (b^2-4 a c\right )}-\frac{7 e (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(d+e x)^{7/2}}{2 \left (a+b x+c x^2\right )^2} \]

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 - 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]) - (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))*ArcTan
h[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*S
qrt[2]*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 5.89269, size = 621, normalized size = 1.14 \[ \frac{1}{8} \left (\frac{2 \sqrt{d+e x} \left (7 b e \left (a^2 e^2+a c \left (d^2-6 d e x-e^2 x^2\right )+c^2 d x^2 (3 d-2 e x)\right )-2 c \left (7 a^2 e^2 (2 d+e x)+a c \left (4 d^3+5 d^2 e x+26 d e^2 x^2+11 e^3 x^3\right )-7 c^2 d^2 e x^3\right )+b^2 \left (14 a e^3 x+c \left (2 d^3+13 d^2 e x-8 d e^2 x^2+9 e^3 x^3\right )\right )+7 b^3 e^3 x^2\right )}{c \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{7 e \left (2 c^2 d e \left (d \sqrt{b^2-4 a c}+8 a e-6 b d\right )+2 c e^2 \left (-b \left (d \sqrt{b^2-4 a c}+4 a e\right )+3 a e \sqrt{b^2-4 a c}+b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{\frac{1}{2} e \left (\sqrt{b^2-4 a c}-b\right )+c d}}-\frac{7 e \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{c d-\frac{1}{2} e \left (\sqrt{b^2-4 a c}+b\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[d + e*x]*(7*b^3*e^3*x^2 + 7*b*e*(a^2*e^2 + c^2*d*x^2*(3*d - 2*e*x) + a*
c*(d^2 - 6*d*e*x - e^2*x^2)) + b^2*(14*a*e^3*x + c*(2*d^3 + 13*d^2*e*x - 8*d*e^2
*x^2 + 9*e^3*x^3)) - 2*c*(-7*c^2*d^2*e*x^3 + 7*a^2*e^2*(2*d + e*x) + a*c*(4*d^3
+ 5*d^2*e*x + 26*d*e^2*x^2 + 11*e^3*x^3))))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^
2) + (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 + Sqr
t[b^2 - 4*a*c]*d + 8*a*e) + 2*c*e^2*(b^2*d + 3*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b
^2 - 4*a*c]*d + 4*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*
e + Sqrt[b^2 - 4*a*c]*e]])/(c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[c*d + ((-b + Sqrt[b
^2 - 4*a*c])*e)/2]) - (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]])/(c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[c*d
- ((b + Sqrt[b^2 - 4*a*c])*e)/2]))/8

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Maple [B]  time = 0.13, size = 10751, normalized size = 19.8 \[ \text{output too large to display} \]

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]

result too large to display

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

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)

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Fricas [A]  time = 0.695477, size = 7696, normalized size = 14.17 \[ \text{result too large to display} \]

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^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 + 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 - 6
4*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 + 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)*sqr
t(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 + 8
1*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 - 1
2*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((2
5*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 + 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 - 4
04*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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out