3.2388 \(\int \frac{1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=621 \[ \frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )}{12 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^4}-\frac{2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-9 a e)-7 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (20 a c e^2-7 b^2 e^2+8 c^2 d^2\right )+8 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac{5 e^4 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac{e \sqrt{a+b x+c x^2} \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right )}{6 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3} \]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d
*e + a*e^2)*(d + e*x)^2*(a + b*x + c*x^2)^(3/2)) - (2*(8*a*c*e*(2*c*d - b*e)^2 -
(b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 7*b^2*e^2 + 20*a*c*e^2) - c*(2*c*d - b*e
)*(8*c^2*d^2 - 7*b^2*e^2 - 4*c*e*(2*b*d - 9*a*e))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2
- b*d*e + a*e^2)^2*(d + e*x)^2*Sqrt[a + b*x + c*x^2]) + (e*(64*c^4*d^4 - 35*b^4*
e^4 - 128*c^3*d^2*e*(b*d - 3*a*e) - 48*a*c^2*e^3*(8*b*d + 5*a*e) + 8*b^2*c*e^3*(
8*b*d + 27*a*e))*Sqrt[a + b*x + c*x^2])/(6*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^
2)^3*(d + e*x)^2) + (e*(2*c*d - b*e)*(64*c^4*d^4 - 105*b^4*e^4 - 64*c^3*d^2*e*(2
*b*d - 7*a*e) + 40*b^2*c*e^3*(2*b*d + 19*a*e) - 16*c^2*e^2*(b^2*d^2 + 28*a*b*d*e
+ 81*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(12*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^
2)^4*(d + e*x)) + (5*e^4*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[
(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*
x^2])])/(8*(c*d^2 - b*d*e + a*e^2)^(9/2))

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Rubi [A]  time = 2.86477, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )}{12 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^4}-\frac{2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-9 a e)-7 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (20 a c e^2-7 b^2 e^2+8 c^2 d^2\right )+8 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac{5 e^4 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac{e \sqrt{a+b x+c x^2} \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right )}{6 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d
*e + a*e^2)*(d + e*x)^2*(a + b*x + c*x^2)^(3/2)) - (2*(8*a*c*e*(2*c*d - b*e)^2 -
(b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 7*b^2*e^2 + 20*a*c*e^2) - c*(2*c*d - b*e
)*(8*c^2*d^2 - 7*b^2*e^2 - 4*c*e*(2*b*d - 9*a*e))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2
- b*d*e + a*e^2)^2*(d + e*x)^2*Sqrt[a + b*x + c*x^2]) + (e*(64*c^4*d^4 - 35*b^4*
e^4 - 128*c^3*d^2*e*(b*d - 3*a*e) - 48*a*c^2*e^3*(8*b*d + 5*a*e) + 8*b^2*c*e^3*(
8*b*d + 27*a*e))*Sqrt[a + b*x + c*x^2])/(6*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^
2)^3*(d + e*x)^2) + (e*(2*c*d - b*e)*(64*c^4*d^4 - 105*b^4*e^4 - 64*c^3*d^2*e*(2
*b*d - 7*a*e) + 40*b^2*c*e^3*(2*b*d + 19*a*e) - 16*c^2*e^2*(b^2*d^2 + 28*a*b*d*e
+ 81*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(12*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^
2)^4*(d + e*x)) + (5*e^4*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[
(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*
x^2])])/(8*(c*d^2 - b*d*e + a*e^2)^(9/2))

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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(1/(e*x+d)**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 7.97085, size = 675, normalized size = 1.09 \[ \frac{\sqrt{a+x (b+c x)} \left (-\frac{8 \left (e (a e-b d)+c d^2\right ) \left (2 c^2 \left (-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right )+b^2 c e \left (4 a e^2-3 c d (d-e x)\right )+b c^2 \left (3 a e^2 (e x-3 d)+c d^2 (d-3 e x)\right )+b^4 \left (-e^3\right )+b^3 c e^2 (3 d-e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{8 \left (-4 b^2 c^2 e \left (-36 a^2 e^4+3 a c d e^2 (17 d-15 e x)+c^2 d^3 (5 d-3 e x)\right )+8 b c^3 \left (6 a^2 e^4 (2 e x-7 d)+7 a c d^2 e^2 (d-3 e x)+c^2 d^4 (d-5 e x)\right )+16 c^3 \left (-3 a^3 e^5+3 a^2 c d e^3 (5 d-4 e x)+7 a c^2 d^3 e^2 x+c^3 d^5 x\right )-2 b^4 c e^3 \left (35 a e^2+c d (14 e x-13 d)\right )+2 b^3 c^2 e^2 \left (a e^2 (105 d-31 e x)+c d^2 (3 d+11 e x)\right )+9 b^6 e^5+b^5 c e^4 (9 e x-29 d)\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}-\frac{6 e^5 \left (e (a e-b d)+c d^2\right )}{(d+e x)^2}+\frac{33 e^5 (b e-2 c d)}{d+e x}\right )}{12 \left (e (a e-b d)+c d^2\right )^4}+\frac{5 e^4 \log (d+e x) \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{8 \left (e (a e-b d)+c d^2\right )^{9/2}}+\frac{5 e^4 \left (4 c e (a e+6 b d)-7 b^2 e^2-24 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{8 \left (e (a e-b d)+c d^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(Sqrt[a + x*(b + c*x)]*((-6*e^5*(c*d^2 + e*(-(b*d) + a*e)))/(d + e*x)^2 + (33*e^
5*(-2*c*d + b*e))/(d + e*x) - (8*(c*d^2 + e*(-(b*d) + a*e))*(-(b^4*e^3) + b^3*c*
e^2*(3*d - e*x) + b^2*c*e*(4*a*e^2 - 3*c*d*(d - e*x)) + 2*c^2*(-(a^2*e^3) + c^2*
d^3*x + 3*a*c*d*e*(d - e*x)) + b*c^2*(c*d^2*(d - 3*e*x) + 3*a*e^2*(-3*d + e*x)))
)/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (8*(9*b^6*e^5 + b^5*c*e^4*(-29*d + 9*e*x
) + 16*c^3*(-3*a^3*e^5 + c^3*d^5*x + 7*a*c^2*d^3*e^2*x + 3*a^2*c*d*e^3*(5*d - 4*
e*x)) - 4*b^2*c^2*e*(-36*a^2*e^4 + 3*a*c*d*e^2*(17*d - 15*e*x) + c^2*d^3*(5*d -
3*e*x)) + 8*b*c^3*(c^2*d^4*(d - 5*e*x) + 7*a*c*d^2*e^2*(d - 3*e*x) + 6*a^2*e^4*(
-7*d + 2*e*x)) + 2*b^3*c^2*e^2*(a*e^2*(105*d - 31*e*x) + c*d^2*(3*d + 11*e*x)) -
2*b^4*c*e^3*(35*a*e^2 + c*d*(-13*d + 14*e*x))))/((b^2 - 4*a*c)^2*(a + x*(b + c*
x)))))/(12*(c*d^2 + e*(-(b*d) + a*e))^4) + (5*e^4*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*
e*(6*b*d + a*e))*Log[d + e*x])/(8*(c*d^2 + e*(-(b*d) + a*e))^(9/2)) + (5*e^4*(-2
4*c^2*d^2 - 7*b^2*e^2 + 4*c*e*(6*b*d + a*e))*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*
x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(8*(c*d^2 + e*(-(b*
d) + a*e))^(9/2))

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Maple [B]  time = 0.033, size = 4942, normalized size = 8. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x)

[Out]

-1/2/e/(a*e^2-b*d*e+c*d^2)/(d/e+x)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b
*d*e+c*d^2)/e^2)^(3/2)+35/24*e^3/(a*e^2-b*d*e+c*d^2)^3/(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2+35/8*e^5/(a*e^2-b*d*e+c*d^2)^4/(c*(
d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-5/6*e*c/(a*e^2
-b*d*e+c*d^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2
)-5/2*e^3*c/(a*e^2-b*d*e+c*d^2)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*
e+c*d^2)/e^2)^(1/2)-280*e/(a*e^2-b*d*e+c*d^2)^3*c^4/(4*a*c-b^2)^2/(c*(d/e+x)^2+(
b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d^2*b+5/2*e^3*c/(a*e^2-b*d
*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2
*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))-35/8*e^5/(a*e^2-b*d*e+c*d^2)^4/(4*
a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^4+3
5/6*e/(a*e^2-b*d*e+c*d^2)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^
2)/e^2)^(3/2)*c^2*d^2+7/4*e/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*
d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b-35/24*e^3/(a*e^2-b*d*e+c*d^2)^3/(4
*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^4+
35/2*e^3/(a*e^2-b*d*e+c*d^2)^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2)*c^2*d^2-7/2/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*d-88/(a*e^2-b*d*e+c*d^2)^2*c^3/(4
*a*c-b^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*
d+5/2*e^3*c/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-35/6*e^2/(a*e^2-b*d*e+c*d^2)^3/(c*(d/e+x)^2+
(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*c*d-35/2*e^4/(a*e^2-b*d*e
+c*d^2)^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c*
d-35/3*e^3/(a*e^2-b*d*e+c*d^2)^3*c/(4*a*c-b^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^4-35/2*e^3/(a*e^2-b*d*e+c*d^2)^4/((a*e^2-b*
d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*
e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^
2)/e^2)^(1/2))/(d/e+x))*c^2*d^2+11/2*e/(a*e^2-b*d*e+c*d^2)^2*c/(4*a*c-b^2)/(c*(d
/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2+44*e/(a*e^2-b*d
*e+c*d^2)^2*c^2/(4*a*c-b^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*
d^2)/e^2)^(1/2)*b^2+70*e^2/(a*e^2-b*d*e+c*d^2)^3*c^2/(4*a*c-b^2)^2/(c*(d/e+x)^2+
(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*d-35/12*e^3/(a*e^2-b*d*
e+c*d^2)^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^
2)^(3/2)*c*x*b^3-140*e/(a*e^2-b*d*e+c*d^2)^3*c^3/(4*a*c-b^2)^2/(c*(d/e+x)^2+(b*e
-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*d^2-35/4*e^5/(a*e^2-b*d*e+c
*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2)*x*b^3*c+70*e^2/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^4*d^3+35/2*e^4/(a*e^2-b*d*e+c*d^2)^
4/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d
/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^
2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*c*d*b+11*e/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-
b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*b+88*e/
(a*e^2-b*d*e+c*d^2)^2*c^3/(4*a*c-b^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^
2-b*d*e+c*d^2)/e^2)^(1/2)*x*b+105/4*e^4/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/
e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*c*d-105/2*e^3/(a
*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+
c*d^2)/e^2)^(1/2)*b^2*c^2*d^2+35*e^2/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x
)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c^3*d^3+5*e^3*c^2/(a*
e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2)*x*b-10*e^2*c^3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/(c*(d/e+x)^2+(
b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d-5*e^2*c^2/(a*e^2-b*d*e+c
*d^2)^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2)*b*d+35/4*e^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*
(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^3*c*d-35/2*e/(a*e^2-b*d*e+c*d^2)^3/(4*a
*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2*c^
2*d^2-70/3*e^3/(a*e^2-b*d*e+c*d^2)^3*c^2/(4*a*c-b^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^3-35/8*e^5/(a*e^2-b*d*e+c*d^2)^4/((
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x
)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*
d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^2+105/2*e^4/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/
(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2*c^2*d-10
5*e^3/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^
2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^3*d^2*b+35/2*e^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)
/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^2*x*b^2*d-3
5*e/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-
b*d*e+c*d^2)/e^2)^(3/2)*c^3*x*d^2*b+140*e^2/(a*e^2-b*d*e+c*d^2)^3*c^3/(4*a*c-b^2
)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2*d+35
/3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b
*d*e+c*d^2)/e^2)^(3/2)*b*c^3*d^3+560/3/(a*e^2-b*d*e+c*d^2)^3*c^5/(4*a*c-b^2)^2/(
c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d^3+70/3/(a*e
^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*
d^2)/e^2)^(3/2)*c^4*x*d^3+280/3/(a*e^2-b*d*e+c*d^2)^3*c^4/(4*a*c-b^2)^2/(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d^3-22/(a*e^2-b*d*e+
c*d^2)^2*c^3/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/
e^2)^(3/2)*x*d-11/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)
/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*d-176/(a*e^2-b*d*e+c*d^2)^2*c^4/(4*a
*c-b^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 14.2509, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)^3),x, algorithm="fricas")

[Out]

[-1/48*(4*(8*(b^3*c^4 - 12*a*b*c^5)*d^7 - 16*(2*b^4*c^3 - 21*a*b^2*c^4 + 12*a^2*
c^5)*d^6*e + 16*(3*b^5*c^2 - 22*a*b^3*c^3)*d^5*e^2 - 32*(b^6*c - 42*a^2*b^2*c^3
+ 64*a^3*c^4)*d^4*e^3 + 8*(b^7 + 24*a*b^5*c - 213*a^2*b^3*c^2 + 364*a^3*b*c^3)*d
^3*e^4 - 8*(10*a*b^6 - 87*a^2*b^4*c + 234*a^3*b^2*c^2 - 200*a^4*c^3)*d^2*e^5 - 3
9*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d*e^6 + 6*(a^3*b^4 - 8*a^4*b^2*c + 16*a
^5*c^2)*e^7 - (128*c^7*d^5*e^2 - 320*b*c^6*d^4*e^3 + 32*(3*b^2*c^5 + 28*a*c^6)*d
^3*e^4 + 16*(11*b^3*c^4 - 84*a*b*c^5)*d^2*e^5 - 2*(145*b^4*c^3 - 984*a*b^2*c^4 +
1296*a^2*c^5)*d*e^6 + (105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*e^7)*x^5 -
(256*c^7*d^6*e - 448*b*c^6*d^5*e^2 - 32*(9*b^2*c^5 - 56*a*c^6)*d^4*e^3 + 16*(31
*b^3*c^4 - 84*a*b*c^5)*d^3*e^4 - 8*(17*b^4*c^3 - 60*a*b^2*c^4 + 288*a^2*c^5)*d^2
*e^5 - (405*b^5*c^2 - 2872*a*b^3*c^3 + 4176*a^2*b*c^4)*d*e^6 + 6*(35*b^6*c - 265
*a*b^4*c^2 + 504*a^2*b^2*c^3 - 80*a^3*c^4)*e^7)*x^4 - (128*c^7*d^7 + 64*b*c^6*d^
6*e - 272*(3*b^2*c^5 - 4*a*c^6)*d^5*e^2 + 8*(43*b^3*c^4 + 108*a*b*c^5)*d^4*e^3 +
32*(17*b^4*c^3 - 117*a*b^2*c^4 + 96*a^2*c^5)*d^3*e^4 - 8*(93*b^5*c^2 - 658*a*b^
3*c^3 + 1152*a^2*b*c^4)*d^2*e^5 + 12*(5*b^6*c - 71*a*b^4*c^2 + 324*a^2*b^2*c^3 -
384*a^3*c^4)*d*e^6 + 3*(35*b^7 - 230*a*b^5*c + 232*a^2*b^3*c^2 + 448*a^3*b*c^3)
*e^7)*x^3 - (192*b*c^6*d^7 - 384*(b^2*c^5 - a*c^6)*d^6*e - 8*(13*b^3*c^4 - 60*a*
b*c^5)*d^5*e^2 + 16*(26*b^4*c^3 - 111*a*b^2*c^4 + 228*a^2*c^5)*d^4*e^3 - 8*(3*b^
5*c^2 - 26*a*b^3*c^3 + 336*a^2*b*c^4)*d^3*e^4 - 8*(47*b^6*c - 294*a*b^4*c^2 + 22
8*a^2*b^2*c^3 + 512*a^3*c^4)*d^2*e^5 + (175*b^7 - 1518*a*b^5*c + 3960*a^2*b^3*c^
2 - 2816*a^3*b*c^3)*d*e^6 + 4*(35*a*b^6 - 279*a^2*b^4*c + 588*a^3*b^2*c^2 - 160*
a^4*c^3)*e^7)*x^2 - (48*(b^2*c^5 + 4*a*c^6)*d^7 - 8*(17*b^3*c^4 + 36*a*b*c^5)*d^
6*e + 32*(2*b^4*c^3 + 3*a*b^2*c^4 + 36*a^2*c^5)*d^5*e^2 + 16*(9*b^5*c^2 - 58*a*b
^3*c^3 + 48*a^2*b*c^4)*d^4*e^3 - 16*(11*b^6*c - 93*a*b^4*c^2 + 243*a^2*b^2*c^3 -
148*a^3*c^4)*d^3*e^4 + 8*(7*b^7 - 126*a*b^5*c + 645*a^2*b^3*c^2 - 956*a^3*b*c^3
)*d^2*e^5 + 2*(119*a*b^6 - 969*a^2*b^4*c + 2184*a^3*b^2*c^2 - 976*a^4*c^3)*d*e^6
+ 21*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^7)*x)*sqrt(c*d^2 - b*d*e + a*e^2)
*sqrt(c*x^2 + b*x + a) + 15*(24*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^4*e
^4 - 24*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^3*e^5 + (7*a^2*b^6 - 60*a^3
*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*d^2*e^6 + (24*(b^4*c^4 - 8*a*b^2*c^5 + 16
*a^2*c^6)*d^2*e^6 - 24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d*e^7 + (7*b^6*c^2
- 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*e^8)*x^6 + 2*(24*(b^4*c^4 - 8*a*
b^2*c^5 + 16*a^2*c^6)*d^3*e^5 - (17*b^6*c^2 - 132*a*b^4*c^3 + 240*a^2*b^2*c^4 +
64*a^3*c^5)*d*e^7 + (7*b^7*c - 60*a*b^5*c^2 + 144*a^2*b^3*c^3 - 64*a^3*b*c^4)*e^
8)*x^5 + (24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^4*e^4 + 72*(b^5*c^3 - 8*a*b^
3*c^4 + 16*a^2*b*c^5)*d^3*e^5 - (65*b^6*c^2 - 564*a*b^4*c^3 + 1392*a^2*b^2*c^4 -
704*a^3*c^5)*d^2*e^6 + 4*(b^7*c - 24*a*b^5*c^2 + 144*a^2*b^3*c^3 - 256*a^3*b*c^
4)*d*e^7 + (7*b^8 - 46*a*b^6*c + 24*a^2*b^4*c^2 + 224*a^3*b^2*c^3 - 128*a^4*c^4)
*e^8)*x^4 + 2*(24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^4*e^4 + 48*(a*b^4*c^3
- 8*a^2*b^2*c^4 + 16*a^3*c^5)*d^3*e^5 - (17*b^7*c - 108*a*b^5*c^2 + 48*a^2*b^3*
c^3 + 448*a^3*b*c^4)*d^2*e^6 + (7*b^8 - 70*a*b^6*c + 216*a^2*b^4*c^2 - 160*a^3*b
^2*c^3 - 128*a^4*c^4)*d*e^7 + (7*a*b^7 - 60*a^2*b^5*c + 144*a^3*b^3*c^2 - 64*a^4
*b*c^3)*e^8)*x^3 + (24*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^4*e^4 - 24*(b^7*c
- 10*a*b^5*c^2 + 32*a^2*b^3*c^3 - 32*a^3*b*c^4)*d^3*e^5 + (7*b^8 - 142*a*b^6*c +
816*a^2*b^4*c^2 - 1504*a^3*b^2*c^3 + 256*a^4*c^4)*d^2*e^6 + 4*(7*a*b^7 - 66*a^2
*b^5*c + 192*a^3*b^3*c^2 - 160*a^4*b*c^3)*d*e^7 + (7*a^2*b^6 - 60*a^3*b^4*c + 14
4*a^4*b^2*c^2 - 64*a^5*c^3)*e^8)*x^2 + 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3
*b*c^4)*d^4*e^4 - 24*(a*b^6*c - 9*a^2*b^4*c^2 + 24*a^3*b^2*c^3 - 16*a^4*c^4)*d^3
*e^5 + 7*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2*e^6 + (7*a^2
*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*d*e^7)*x)*log(((8*a*b*d*e -
8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2
- 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2 - b*d*e + a*e^2)
+ 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*b
*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/(e^2*x^2 + 2
*d*e*x + d^2)))/(((a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c^6)*d^10 - 4*(a^2*b^5*c
^3 - 8*a^3*b^3*c^4 + 16*a^4*b*c^5)*d^9*e + 2*(3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 3
2*a^4*b^2*c^4 + 32*a^5*c^5)*d^8*e^2 - 4*(a^2*b^7*c - 5*a^3*b^5*c^2 - 8*a^4*b^3*c
^3 + 48*a^5*b*c^4)*d^7*e^3 + (a^2*b^8 + 4*a^3*b^6*c - 74*a^4*b^4*c^2 + 144*a^5*b
^2*c^3 + 96*a^6*c^4)*d^6*e^4 - 4*(a^3*b^7 - 5*a^4*b^5*c - 8*a^5*b^3*c^2 + 48*a^6
*b*c^3)*d^5*e^5 + 2*(3*a^4*b^6 - 22*a^5*b^4*c + 32*a^6*b^2*c^2 + 32*a^7*c^3)*d^4
*e^6 - 4*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d^3*e^7 + (a^6*b^4 - 8*a^7*b^2*c
+ 16*a^8*c^2)*d^2*e^8 + ((b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^8*e^2 - 4*(b^5*
c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^7*e^3 + 2*(3*b^6*c^4 - 22*a*b^4*c^5 + 32*a^2
*b^2*c^6 + 32*a^3*c^7)*d^6*e^4 - 4*(b^7*c^3 - 5*a*b^5*c^4 - 8*a^2*b^3*c^5 + 48*a
^3*b*c^6)*d^5*e^5 + (b^8*c^2 + 4*a*b^6*c^3 - 74*a^2*b^4*c^4 + 144*a^3*b^2*c^5 +
96*a^4*c^6)*d^4*e^6 - 4*(a*b^7*c^2 - 5*a^2*b^5*c^3 - 8*a^3*b^3*c^4 + 48*a^4*b*c^
5)*d^3*e^7 + 2*(3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^4*b^2*c^4 + 32*a^5*c^5)*d^
2*e^8 - 4*(a^3*b^5*c^2 - 8*a^4*b^3*c^3 + 16*a^5*b*c^4)*d*e^9 + (a^4*b^4*c^2 - 8*
a^5*b^2*c^3 + 16*a^6*c^4)*e^10)*x^6 + 2*((b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^
9*e - 3*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^8*e^2 + 2*(b^6*c^4 - 6*a*b^4*c^
5 + 32*a^3*c^7)*d^7*e^3 + 2*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*
c^6)*d^6*e^4 - 3*(b^8*c^2 - 8*a*b^6*c^3 + 14*a^2*b^4*c^4 + 16*a^3*b^2*c^5 - 32*a
^4*c^6)*d^5*e^5 + (b^9*c - 54*a^2*b^5*c^3 + 176*a^3*b^3*c^4 - 96*a^4*b*c^5)*d^4*
e^6 - 2*(2*a*b^8*c - 13*a^2*b^6*c^2 + 6*a^3*b^4*c^3 + 64*a^4*b^2*c^4 - 32*a^5*c^
5)*d^3*e^7 + 6*(a^2*b^7*c - 8*a^3*b^5*c^2 + 16*a^4*b^3*c^3)*d^2*e^8 - (4*a^3*b^6
*c - 33*a^4*b^4*c^2 + 72*a^5*b^2*c^3 - 16*a^6*c^4)*d*e^9 + (a^4*b^5*c - 8*a^5*b^
3*c^2 + 16*a^6*b*c^3)*e^10)*x^5 + ((b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^10 - 3
*(3*b^6*c^4 - 26*a*b^4*c^5 + 64*a^2*b^2*c^6 - 32*a^3*c^7)*d^8*e^2 + 4*(4*b^7*c^3
- 33*a*b^5*c^4 + 72*a^2*b^3*c^5 - 16*a^3*b*c^6)*d^7*e^3 - (9*b^8*c^2 - 52*a*b^6
*c^3 - 30*a^2*b^4*c^4 + 432*a^3*b^2*c^5 - 224*a^4*c^6)*d^6*e^4 + 12*(2*a*b^7*c^2
- 17*a^2*b^5*c^3 + 40*a^3*b^3*c^4 - 16*a^4*b*c^5)*d^5*e^5 + (b^10 - 10*a*b^8*c
+ 20*a^2*b^6*c^2 + 80*a^3*b^4*c^3 - 320*a^4*b^2*c^4 + 256*a^5*c^5)*d^4*e^6 - 4*(
a*b^9 - 9*a^2*b^7*c + 27*a^3*b^5*c^2 - 40*a^4*b^3*c^3 + 48*a^5*b*c^4)*d^3*e^7 +
3*(2*a^2*b^8 - 16*a^3*b^6*c + 35*a^4*b^4*c^2 - 24*a^5*b^2*c^3 + 48*a^6*c^4)*d^2*
e^8 - 4*(a^3*b^7 - 7*a^4*b^5*c + 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d*e^9 + (a^4*b^6
- 6*a^5*b^4*c + 32*a^7*c^3)*e^10)*x^4 + 2*((b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7
)*d^10 - (3*b^6*c^4 - 26*a*b^4*c^5 + 64*a^2*b^2*c^6 - 32*a^3*c^7)*d^9*e + (2*b^7
*c^3 - 19*a*b^5*c^4 + 56*a^2*b^3*c^5 - 48*a^3*b*c^6)*d^8*e^2 + 2*(b^8*c^2 - 8*a*
b^6*c^3 + 20*a^2*b^4*c^4 - 32*a^3*b^2*c^5 + 64*a^4*c^6)*d^7*e^3 - (3*b^9*c - 22*
a*b^7*c^2 + 46*a^2*b^5*c^3 - 80*a^3*b^3*c^4 + 224*a^4*b*c^5)*d^6*e^4 + (b^10 - 2
*a*b^8*c - 26*a^2*b^6*c^2 + 60*a^3*b^4*c^3 + 192*a^5*c^5)*d^5*e^5 - (3*a*b^9 - 2
2*a^2*b^7*c + 46*a^3*b^5*c^2 - 80*a^4*b^3*c^3 + 224*a^5*b*c^4)*d^4*e^6 + 2*(a^2*
b^8 - 8*a^3*b^6*c + 20*a^4*b^4*c^2 - 32*a^5*b^2*c^3 + 64*a^6*c^4)*d^3*e^7 + (2*a
^3*b^7 - 19*a^4*b^5*c + 56*a^5*b^3*c^2 - 48*a^6*b*c^3)*d^2*e^8 - (3*a^4*b^6 - 26
*a^5*b^4*c + 64*a^6*b^2*c^2 - 32*a^7*c^3)*d*e^9 + (a^5*b^5 - 8*a^6*b^3*c + 16*a^
7*b*c^2)*e^10)*x^3 + ((b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*d^10 - 4*(b^7*c^3 - 7
*a*b^5*c^4 + 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*d^9*e + 3*(2*b^8*c^2 - 16*a*b^6*c^3 +
35*a^2*b^4*c^4 - 24*a^3*b^2*c^5 + 48*a^4*c^6)*d^8*e^2 - 4*(b^9*c - 9*a*b^7*c^2
+ 27*a^2*b^5*c^3 - 40*a^3*b^3*c^4 + 48*a^4*b*c^5)*d^7*e^3 + (b^10 - 10*a*b^8*c +
20*a^2*b^6*c^2 + 80*a^3*b^4*c^3 - 320*a^4*b^2*c^4 + 256*a^5*c^5)*d^6*e^4 + 12*(
2*a^2*b^7*c - 17*a^3*b^5*c^2 + 40*a^4*b^3*c^3 - 16*a^5*b*c^4)*d^5*e^5 - (9*a^2*b
^8 - 52*a^3*b^6*c - 30*a^4*b^4*c^2 + 432*a^5*b^2*c^3 - 224*a^6*c^4)*d^4*e^6 + 4*
(4*a^3*b^7 - 33*a^4*b^5*c + 72*a^5*b^3*c^2 - 16*a^6*b*c^3)*d^3*e^7 - 3*(3*a^4*b^
6 - 26*a^5*b^4*c + 64*a^6*b^2*c^2 - 32*a^7*c^3)*d^2*e^8 + (a^6*b^4 - 8*a^7*b^2*c
+ 16*a^8*c^2)*e^10)*x^2 + 2*((a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*d^10 -
(4*a*b^6*c^3 - 33*a^2*b^4*c^4 + 72*a^3*b^2*c^5 - 16*a^4*c^6)*d^9*e + 6*(a*b^7*c^
2 - 8*a^2*b^5*c^3 + 16*a^3*b^3*c^4)*d^8*e^2 - 2*(2*a*b^8*c - 13*a^2*b^6*c^2 + 6*
a^3*b^4*c^3 + 64*a^4*b^2*c^4 - 32*a^5*c^5)*d^7*e^3 + (a*b^9 - 54*a^3*b^5*c^2 + 1
76*a^4*b^3*c^3 - 96*a^5*b*c^4)*d^6*e^4 - 3*(a^2*b^8 - 8*a^3*b^6*c + 14*a^4*b^4*c
^2 + 16*a^5*b^2*c^3 - 32*a^6*c^4)*d^5*e^5 + 2*(a^3*b^7 - 12*a^4*b^5*c + 48*a^5*b
^3*c^2 - 64*a^6*b*c^3)*d^4*e^6 + 2*(a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*d^3*e^7
- 3*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d^2*e^8 + (a^6*b^4 - 8*a^7*b^2*c + 16
*a^8*c^2)*d*e^9)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), -1/24*(2*(8*(b^3*c^4 - 12*a*b*
c^5)*d^7 - 16*(2*b^4*c^3 - 21*a*b^2*c^4 + 12*a^2*c^5)*d^6*e + 16*(3*b^5*c^2 - 22
*a*b^3*c^3)*d^5*e^2 - 32*(b^6*c - 42*a^2*b^2*c^3 + 64*a^3*c^4)*d^4*e^3 + 8*(b^7
+ 24*a*b^5*c - 213*a^2*b^3*c^2 + 364*a^3*b*c^3)*d^3*e^4 - 8*(10*a*b^6 - 87*a^2*b
^4*c + 234*a^3*b^2*c^2 - 200*a^4*c^3)*d^2*e^5 - 39*(a^2*b^5 - 8*a^3*b^3*c + 16*a
^4*b*c^2)*d*e^6 + 6*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*e^7 - (128*c^7*d^5*e^2
- 320*b*c^6*d^4*e^3 + 32*(3*b^2*c^5 + 28*a*c^6)*d^3*e^4 + 16*(11*b^3*c^4 - 84*a*
b*c^5)*d^2*e^5 - 2*(145*b^4*c^3 - 984*a*b^2*c^4 + 1296*a^2*c^5)*d*e^6 + (105*b^5
*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*e^7)*x^5 - (256*c^7*d^6*e - 448*b*c^6*d^5
*e^2 - 32*(9*b^2*c^5 - 56*a*c^6)*d^4*e^3 + 16*(31*b^3*c^4 - 84*a*b*c^5)*d^3*e^4
- 8*(17*b^4*c^3 - 60*a*b^2*c^4 + 288*a^2*c^5)*d^2*e^5 - (405*b^5*c^2 - 2872*a*b^
3*c^3 + 4176*a^2*b*c^4)*d*e^6 + 6*(35*b^6*c - 265*a*b^4*c^2 + 504*a^2*b^2*c^3 -
80*a^3*c^4)*e^7)*x^4 - (128*c^7*d^7 + 64*b*c^6*d^6*e - 272*(3*b^2*c^5 - 4*a*c^6)
*d^5*e^2 + 8*(43*b^3*c^4 + 108*a*b*c^5)*d^4*e^3 + 32*(17*b^4*c^3 - 117*a*b^2*c^4
+ 96*a^2*c^5)*d^3*e^4 - 8*(93*b^5*c^2 - 658*a*b^3*c^3 + 1152*a^2*b*c^4)*d^2*e^5
+ 12*(5*b^6*c - 71*a*b^4*c^2 + 324*a^2*b^2*c^3 - 384*a^3*c^4)*d*e^6 + 3*(35*b^7
- 230*a*b^5*c + 232*a^2*b^3*c^2 + 448*a^3*b*c^3)*e^7)*x^3 - (192*b*c^6*d^7 - 38
4*(b^2*c^5 - a*c^6)*d^6*e - 8*(13*b^3*c^4 - 60*a*b*c^5)*d^5*e^2 + 16*(26*b^4*c^3
- 111*a*b^2*c^4 + 228*a^2*c^5)*d^4*e^3 - 8*(3*b^5*c^2 - 26*a*b^3*c^3 + 336*a^2*
b*c^4)*d^3*e^4 - 8*(47*b^6*c - 294*a*b^4*c^2 + 228*a^2*b^2*c^3 + 512*a^3*c^4)*d^
2*e^5 + (175*b^7 - 1518*a*b^5*c + 3960*a^2*b^3*c^2 - 2816*a^3*b*c^3)*d*e^6 + 4*(
35*a*b^6 - 279*a^2*b^4*c + 588*a^3*b^2*c^2 - 160*a^4*c^3)*e^7)*x^2 - (48*(b^2*c^
5 + 4*a*c^6)*d^7 - 8*(17*b^3*c^4 + 36*a*b*c^5)*d^6*e + 32*(2*b^4*c^3 + 3*a*b^2*c
^4 + 36*a^2*c^5)*d^5*e^2 + 16*(9*b^5*c^2 - 58*a*b^3*c^3 + 48*a^2*b*c^4)*d^4*e^3
- 16*(11*b^6*c - 93*a*b^4*c^2 + 243*a^2*b^2*c^3 - 148*a^3*c^4)*d^3*e^4 + 8*(7*b^
7 - 126*a*b^5*c + 645*a^2*b^3*c^2 - 956*a^3*b*c^3)*d^2*e^5 + 2*(119*a*b^6 - 969*
a^2*b^4*c + 2184*a^3*b^2*c^2 - 976*a^4*c^3)*d*e^6 + 21*(a^2*b^5 - 8*a^3*b^3*c +
16*a^4*b*c^2)*e^7)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 15*(2
4*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^4*e^4 - 24*(a^2*b^5*c - 8*a^3*b^3
*c^2 + 16*a^4*b*c^3)*d^3*e^5 + (7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*
a^5*c^3)*d^2*e^6 + (24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^2*e^6 - 24*(b^5*c^
3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d*e^7 + (7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*
c^4 - 64*a^3*c^5)*e^8)*x^6 + 2*(24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^3*e^5
- (17*b^6*c^2 - 132*a*b^4*c^3 + 240*a^2*b^2*c^4 + 64*a^3*c^5)*d*e^7 + (7*b^7*c -
60*a*b^5*c^2 + 144*a^2*b^3*c^3 - 64*a^3*b*c^4)*e^8)*x^5 + (24*(b^4*c^4 - 8*a*b^
2*c^5 + 16*a^2*c^6)*d^4*e^4 + 72*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^3*e^5
- (65*b^6*c^2 - 564*a*b^4*c^3 + 1392*a^2*b^2*c^4 - 704*a^3*c^5)*d^2*e^6 + 4*(b^7
*c - 24*a*b^5*c^2 + 144*a^2*b^3*c^3 - 256*a^3*b*c^4)*d*e^7 + (7*b^8 - 46*a*b^6*c
+ 24*a^2*b^4*c^2 + 224*a^3*b^2*c^3 - 128*a^4*c^4)*e^8)*x^4 + 2*(24*(b^5*c^3 - 8
*a*b^3*c^4 + 16*a^2*b*c^5)*d^4*e^4 + 48*(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5)
*d^3*e^5 - (17*b^7*c - 108*a*b^5*c^2 + 48*a^2*b^3*c^3 + 448*a^3*b*c^4)*d^2*e^6 +
(7*b^8 - 70*a*b^6*c + 216*a^2*b^4*c^2 - 160*a^3*b^2*c^3 - 128*a^4*c^4)*d*e^7 +
(7*a*b^7 - 60*a^2*b^5*c + 144*a^3*b^3*c^2 - 64*a^4*b*c^3)*e^8)*x^3 + (24*(b^6*c^
2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^4*e^4 - 24*(b^7*c - 10*a*b^5*c^2 + 32*a^2*b^3*c^
3 - 32*a^3*b*c^4)*d^3*e^5 + (7*b^8 - 142*a*b^6*c + 816*a^2*b^4*c^2 - 1504*a^3*b^
2*c^3 + 256*a^4*c^4)*d^2*e^6 + 4*(7*a*b^7 - 66*a^2*b^5*c + 192*a^3*b^3*c^2 - 160
*a^4*b*c^3)*d*e^7 + (7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*e^
8)*x^2 + 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d^4*e^4 - 24*(a*b^6*c
- 9*a^2*b^4*c^2 + 24*a^3*b^2*c^3 - 16*a^4*c^4)*d^3*e^5 + 7*(a*b^7 - 12*a^2*b^5*c
+ 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2*e^6 + (7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*
b^2*c^2 - 64*a^5*c^3)*d*e^7)*x)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(b*d -
2*a*e + (2*c*d - b*e)*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a))))/(((a^
2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c^6)*d^10 - 4*(a^2*b^5*c^3 - 8*a^3*b^3*c^4 +
16*a^4*b*c^5)*d^9*e + 2*(3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^4*b^2*c^4 + 32*a^
5*c^5)*d^8*e^2 - 4*(a^2*b^7*c - 5*a^3*b^5*c^2 - 8*a^4*b^3*c^3 + 48*a^5*b*c^4)*d^
7*e^3 + (a^2*b^8 + 4*a^3*b^6*c - 74*a^4*b^4*c^2 + 144*a^5*b^2*c^3 + 96*a^6*c^4)*
d^6*e^4 - 4*(a^3*b^7 - 5*a^4*b^5*c - 8*a^5*b^3*c^2 + 48*a^6*b*c^3)*d^5*e^5 + 2*(
3*a^4*b^6 - 22*a^5*b^4*c + 32*a^6*b^2*c^2 + 32*a^7*c^3)*d^4*e^6 - 4*(a^5*b^5 - 8
*a^6*b^3*c + 16*a^7*b*c^2)*d^3*e^7 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*d^2*e^
8 + ((b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^8*e^2 - 4*(b^5*c^5 - 8*a*b^3*c^6 + 1
6*a^2*b*c^7)*d^7*e^3 + 2*(3*b^6*c^4 - 22*a*b^4*c^5 + 32*a^2*b^2*c^6 + 32*a^3*c^7
)*d^6*e^4 - 4*(b^7*c^3 - 5*a*b^5*c^4 - 8*a^2*b^3*c^5 + 48*a^3*b*c^6)*d^5*e^5 + (
b^8*c^2 + 4*a*b^6*c^3 - 74*a^2*b^4*c^4 + 144*a^3*b^2*c^5 + 96*a^4*c^6)*d^4*e^6 -
4*(a*b^7*c^2 - 5*a^2*b^5*c^3 - 8*a^3*b^3*c^4 + 48*a^4*b*c^5)*d^3*e^7 + 2*(3*a^2
*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^4*b^2*c^4 + 32*a^5*c^5)*d^2*e^8 - 4*(a^3*b^5*c^
2 - 8*a^4*b^3*c^3 + 16*a^5*b*c^4)*d*e^9 + (a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*
c^4)*e^10)*x^6 + 2*((b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^9*e - 3*(b^5*c^5 - 8*
a*b^3*c^6 + 16*a^2*b*c^7)*d^8*e^2 + 2*(b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*d^7*e
^3 + 2*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^6*e^4 - 3*(b^8
*c^2 - 8*a*b^6*c^3 + 14*a^2*b^4*c^4 + 16*a^3*b^2*c^5 - 32*a^4*c^6)*d^5*e^5 + (b^
9*c - 54*a^2*b^5*c^3 + 176*a^3*b^3*c^4 - 96*a^4*b*c^5)*d^4*e^6 - 2*(2*a*b^8*c -
13*a^2*b^6*c^2 + 6*a^3*b^4*c^3 + 64*a^4*b^2*c^4 - 32*a^5*c^5)*d^3*e^7 + 6*(a^2*b
^7*c - 8*a^3*b^5*c^2 + 16*a^4*b^3*c^3)*d^2*e^8 - (4*a^3*b^6*c - 33*a^4*b^4*c^2 +
72*a^5*b^2*c^3 - 16*a^6*c^4)*d*e^9 + (a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)
*e^10)*x^5 + ((b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^10 - 3*(3*b^6*c^4 - 26*a*b^
4*c^5 + 64*a^2*b^2*c^6 - 32*a^3*c^7)*d^8*e^2 + 4*(4*b^7*c^3 - 33*a*b^5*c^4 + 72*
a^2*b^3*c^5 - 16*a^3*b*c^6)*d^7*e^3 - (9*b^8*c^2 - 52*a*b^6*c^3 - 30*a^2*b^4*c^4
+ 432*a^3*b^2*c^5 - 224*a^4*c^6)*d^6*e^4 + 12*(2*a*b^7*c^2 - 17*a^2*b^5*c^3 + 4
0*a^3*b^3*c^4 - 16*a^4*b*c^5)*d^5*e^5 + (b^10 - 10*a*b^8*c + 20*a^2*b^6*c^2 + 80
*a^3*b^4*c^3 - 320*a^4*b^2*c^4 + 256*a^5*c^5)*d^4*e^6 - 4*(a*b^9 - 9*a^2*b^7*c +
27*a^3*b^5*c^2 - 40*a^4*b^3*c^3 + 48*a^5*b*c^4)*d^3*e^7 + 3*(2*a^2*b^8 - 16*a^3
*b^6*c + 35*a^4*b^4*c^2 - 24*a^5*b^2*c^3 + 48*a^6*c^4)*d^2*e^8 - 4*(a^3*b^7 - 7*
a^4*b^5*c + 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*d*e^9 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^
7*c^3)*e^10)*x^4 + 2*((b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^10 - (3*b^6*c^4 -
26*a*b^4*c^5 + 64*a^2*b^2*c^6 - 32*a^3*c^7)*d^9*e + (2*b^7*c^3 - 19*a*b^5*c^4 +
56*a^2*b^3*c^5 - 48*a^3*b*c^6)*d^8*e^2 + 2*(b^8*c^2 - 8*a*b^6*c^3 + 20*a^2*b^4*
c^4 - 32*a^3*b^2*c^5 + 64*a^4*c^6)*d^7*e^3 - (3*b^9*c - 22*a*b^7*c^2 + 46*a^2*b^
5*c^3 - 80*a^3*b^3*c^4 + 224*a^4*b*c^5)*d^6*e^4 + (b^10 - 2*a*b^8*c - 26*a^2*b^6
*c^2 + 60*a^3*b^4*c^3 + 192*a^5*c^5)*d^5*e^5 - (3*a*b^9 - 22*a^2*b^7*c + 46*a^3*
b^5*c^2 - 80*a^4*b^3*c^3 + 224*a^5*b*c^4)*d^4*e^6 + 2*(a^2*b^8 - 8*a^3*b^6*c + 2
0*a^4*b^4*c^2 - 32*a^5*b^2*c^3 + 64*a^6*c^4)*d^3*e^7 + (2*a^3*b^7 - 19*a^4*b^5*c
+ 56*a^5*b^3*c^2 - 48*a^6*b*c^3)*d^2*e^8 - (3*a^4*b^6 - 26*a^5*b^4*c + 64*a^6*b
^2*c^2 - 32*a^7*c^3)*d*e^9 + (a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*e^10)*x^3 +
((b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*d^10 - 4*(b^7*c^3 - 7*a*b^5*c^4 + 8*a^2*b^
3*c^5 + 16*a^3*b*c^6)*d^9*e + 3*(2*b^8*c^2 - 16*a*b^6*c^3 + 35*a^2*b^4*c^4 - 24*
a^3*b^2*c^5 + 48*a^4*c^6)*d^8*e^2 - 4*(b^9*c - 9*a*b^7*c^2 + 27*a^2*b^5*c^3 - 40
*a^3*b^3*c^4 + 48*a^4*b*c^5)*d^7*e^3 + (b^10 - 10*a*b^8*c + 20*a^2*b^6*c^2 + 80*
a^3*b^4*c^3 - 320*a^4*b^2*c^4 + 256*a^5*c^5)*d^6*e^4 + 12*(2*a^2*b^7*c - 17*a^3*
b^5*c^2 + 40*a^4*b^3*c^3 - 16*a^5*b*c^4)*d^5*e^5 - (9*a^2*b^8 - 52*a^3*b^6*c - 3
0*a^4*b^4*c^2 + 432*a^5*b^2*c^3 - 224*a^6*c^4)*d^4*e^6 + 4*(4*a^3*b^7 - 33*a^4*b
^5*c + 72*a^5*b^3*c^2 - 16*a^6*b*c^3)*d^3*e^7 - 3*(3*a^4*b^6 - 26*a^5*b^4*c + 64
*a^6*b^2*c^2 - 32*a^7*c^3)*d^2*e^8 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*e^10)*
x^2 + 2*((a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*d^10 - (4*a*b^6*c^3 - 33*a^2
*b^4*c^4 + 72*a^3*b^2*c^5 - 16*a^4*c^6)*d^9*e + 6*(a*b^7*c^2 - 8*a^2*b^5*c^3 + 1
6*a^3*b^3*c^4)*d^8*e^2 - 2*(2*a*b^8*c - 13*a^2*b^6*c^2 + 6*a^3*b^4*c^3 + 64*a^4*
b^2*c^4 - 32*a^5*c^5)*d^7*e^3 + (a*b^9 - 54*a^3*b^5*c^2 + 176*a^4*b^3*c^3 - 96*a
^5*b*c^4)*d^6*e^4 - 3*(a^2*b^8 - 8*a^3*b^6*c + 14*a^4*b^4*c^2 + 16*a^5*b^2*c^3 -
32*a^6*c^4)*d^5*e^5 + 2*(a^3*b^7 - 12*a^4*b^5*c + 48*a^5*b^3*c^2 - 64*a^6*b*c^3
)*d^4*e^6 + 2*(a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*d^3*e^7 - 3*(a^5*b^5 - 8*a^6*
b^3*c + 16*a^7*b*c^2)*d^2*e^8 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*d*e^9)*x)*s
qrt(-c*d^2 + b*d*e - a*e^2))]

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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(1/(e*x+d)**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)^3),x, algorithm="giac")

[Out]

Exception raised: TypeError