3.2378 \(\int \frac{1}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=371 \[ -\frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{4 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{3 e^2 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)*(d + e*x)^2*Sqrt[a + b*x + c*x^2]) - (e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e
*(2*b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2
)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*
a*e))*Sqrt[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x
)) + (3*e^2*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*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)^(7/2))

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Rubi [A]  time = 1.28739, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{4 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{3 e^2 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)*(d + e*x)^2*Sqrt[a + b*x + c*x^2]) - (e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e
*(2*b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2
)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*
a*e))*Sqrt[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x
)) + (3*e^2*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*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)^(7/2))

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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)**(3/2),x)

[Out]

Timed out

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Mathematica [A]  time = 5.35495, size = 389, normalized size = 1.05 \[ \frac{1}{8} \left (\frac{2 \sqrt{a+x (b+c x)} \left (\frac{8 \left (2 c^2 \left (a^2 e^3-3 a c d e (d-e x)-c^2 d^3 x\right )+b^2 c e \left (3 c d (d-e x)-4 a e^2\right )+b c^2 \left (-3 a e^2 (e x-3 d)-c d^2 (d-3 e x)\right )+b^4 e^3+b^3 c e^2 (e x-3 d)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{2 e^3 \left (e (a e-b d)+c d^2\right )}{(d+e x)^2}+\frac{7 e^3 (b e-2 c d)}{d+e x}\right )}{\left (e (a e-b d)+c d^2\right )^3}+\frac{3 e^2 \log (d+e x) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^{7/2}}+\frac{3 e^2 \left (4 c e (a e+4 b d)-5 b^2 e^2-16 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{7/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a + x*(b + c*x)]*((-2*e^3*(c*d^2 + e*(-(b*d) + a*e)))/(d + e*x)^2 + (7*
e^3*(-2*c*d + b*e))/(d + e*x) + (8*(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)))))/(c*d^2 + e*(-(b*d) + a*e))^3 + (3*e^2*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*
e*(4*b*d + a*e))*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^(7/2) + (3*e^2*(-16*c^
2*d^2 - 5*b^2*e^2 + 4*c*e*(4*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)]])/(c*d^2 + e*(-(b*d) + a*
e))^(7/2))/8

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Maple [B]  time = 0.027, size = 2380, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*e*c/(a*e^2-b*d*e+c*d^2)^2/((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))+15/8*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)^(1/2)
*b^2+15/2*e^2/(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))*b*c*d+13*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)^(1/2)*x*b-26/(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)^(1/2)*x*d-13/(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)
^(1/2)*b*d-45*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)^(1/2)*x*b*c^3*d^2+5/4*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)^(1/2)*b+15/2*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)
^(1/2)*c^2*d^2-15/8*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)^(1/2)*b^4-15/8*e^3/(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))*b^2-15/2*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)^(1/2)*b*c*d-5/2/(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)^(1/2)*
c*d-15/2*e/(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))*c^2*d^2+13/2*
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)^(1/2)*b^2+45/4*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)^(1/2)*b^3*c*d-45/2*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)^(1/2)*b^2*c^2*d^2+15/(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*c^3*d^3-3/2*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)^(1/2)-15/4*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)^(1/2)*x*b^3*c+30/(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*c^4*d^3+45/2*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)^(1/2)*x*b^2*c^2*d-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)^(1/2)

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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)^(3/2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.1984, 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)^(3/2)*(e*x + d)^3),x, algorithm="fricas")

[Out]

[-1/16*(4*(8*b*c^3*d^5 - 24*(b^2*c^2 - 2*a*c^3)*d^4*e + 24*(b^3*c - 3*a*b*c^2)*d
^3*e^2 - 8*(b^4 - 6*a*b^2*c + 10*a^2*c^2)*d^2*e^3 - 9*(a*b^3 - 4*a^2*b*c)*d*e^4
+ 2*(a^2*b^2 - 4*a^3*c)*e^5 + (16*c^4*d^3*e^2 - 24*b*c^3*d^2*e^3 + 2*(19*b^2*c^2
- 52*a*c^3)*d*e^4 - (15*b^3*c - 52*a*b*c^2)*e^5)*x^3 + (32*c^4*d^4*e - 40*b*c^3
*d^3*e^2 + 8*(5*b^2*c^2 - 14*a*c^3)*d^2*e^3 + (13*b^3*c - 44*a*b*c^2)*d*e^4 - (1
5*b^4 - 62*a*b^2*c + 24*a^2*c^2)*e^5)*x^2 + (16*c^4*d^5 - 8*b*c^3*d^4*e - 24*(b^
2*c^2 - 2*a*c^3)*d^3*e^2 + 8*(7*b^3*c - 23*a*b*c^2)*d^2*e^3 - (25*b^4 - 114*a*b^
2*c + 88*a^2*c^2)*d*e^4 - 5*(a*b^3 - 4*a^2*b*c)*e^5)*x)*sqrt(c*d^2 - b*d*e + a*e
^2)*sqrt(c*x^2 + b*x + a) + 3*(16*(a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - 16*(a*b^3*c
- 4*a^2*b*c^2)*d^3*e^3 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*d^2*e^4 + (16*(b^
2*c^3 - 4*a*c^4)*d^2*e^4 - 16*(b^3*c^2 - 4*a*b*c^3)*d*e^5 + (5*b^4*c - 24*a*b^2*
c^2 + 16*a^2*c^3)*e^6)*x^4 + (32*(b^2*c^3 - 4*a*c^4)*d^3*e^3 - 16*(b^3*c^2 - 4*a
*b*c^3)*d^2*e^4 - 2*(3*b^4*c - 8*a*b^2*c^2 - 16*a^2*c^3)*d*e^5 + (5*b^5 - 24*a*b
^3*c + 16*a^2*b*c^2)*e^6)*x^3 + (16*(b^2*c^3 - 4*a*c^4)*d^4*e^2 + 16*(b^3*c^2 -
4*a*b*c^3)*d^3*e^3 - 3*(9*b^4*c - 40*a*b^2*c^2 + 16*a^2*c^3)*d^2*e^4 + 2*(5*b^5
- 32*a*b^3*c + 48*a^2*b*c^2)*d*e^5 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*e^6)*
x^2 + (16*(b^3*c^2 - 4*a*b*c^3)*d^4*e^2 - 16*(b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d
^3*e^3 + (5*b^5 - 56*a*b^3*c + 144*a^2*b*c^2)*d^2*e^4 + 2*(5*a*b^4 - 24*a^2*b^2*
c + 16*a^3*c^2)*d*e^5)*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*b^2*c^3 - 4*a^2
*c^4)*d^8 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^7*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a
^3*c^3)*d^6*e^2 - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^5*e^3 + 3*(a^2*b^4 - 3*
a^3*b^2*c - 4*a^4*c^2)*d^4*e^4 - 3*(a^3*b^3 - 4*a^4*b*c)*d^3*e^5 + (a^4*b^2 - 4*
a^5*c)*d^2*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e^2 - 3*(b^3*c^3 - 4*a*b*c^4)*d^5*e^3
+ 3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^4 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*
b*c^3)*d^3*e^5 + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^6 - 3*(a^2*b^3*c
- 4*a^3*b*c^2)*d*e^7 + (a^3*b^2*c - 4*a^4*c^2)*e^8)*x^4 + (2*(b^2*c^4 - 4*a*c^5)
*d^7*e - 5*(b^3*c^3 - 4*a*b*c^4)*d^6*e^2 + 3*(b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)
*d^5*e^3 + (b^5*c - 13*a*b^3*c^2 + 36*a^2*b*c^3)*d^4*e^4 - (b^6 - 4*a*b^4*c - 6*
a^2*b^2*c^2 + 24*a^3*c^3)*d^3*e^5 + 3*(a*b^5 - 5*a^2*b^3*c + 4*a^3*b*c^2)*d^2*e^
6 - (3*a^2*b^4 - 14*a^3*b^2*c + 8*a^4*c^2)*d*e^7 + (a^3*b^3 - 4*a^4*b*c)*e^8)*x^
3 + ((b^2*c^4 - 4*a*c^5)*d^8 - (b^3*c^3 - 4*a*b*c^4)*d^7*e - (3*b^4*c^2 - 16*a*b
^2*c^3 + 16*a^2*c^4)*d^6*e^2 + (5*b^5*c - 23*a*b^3*c^2 + 12*a^2*b*c^3)*d^5*e^3 -
2*(b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^4 + (5*a*b^5 - 23*a^2*b^3
*c + 12*a^3*b*c^2)*d^3*e^5 - (3*a^2*b^4 - 16*a^3*b^2*c + 16*a^4*c^2)*d^2*e^6 - (
a^3*b^3 - 4*a^4*b*c)*d*e^7 + (a^4*b^2 - 4*a^5*c)*e^8)*x^2 + ((b^3*c^3 - 4*a*b*c^
4)*d^8 - (3*b^4*c^2 - 14*a*b^2*c^3 + 8*a^2*c^4)*d^7*e + 3*(b^5*c - 5*a*b^3*c^2 +
4*a^2*b*c^3)*d^6*e^2 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 + 24*a^3*c^3)*d^5*e^3 +
(a*b^5 - 13*a^2*b^3*c + 36*a^3*b*c^2)*d^4*e^4 + 3*(a^2*b^4 - 2*a^3*b^2*c - 8*a^
4*c^2)*d^3*e^5 - 5*(a^3*b^3 - 4*a^4*b*c)*d^2*e^6 + 2*(a^4*b^2 - 4*a^5*c)*d*e^7)*
x)*sqrt(c*d^2 - b*d*e + a*e^2)), -1/8*(2*(8*b*c^3*d^5 - 24*(b^2*c^2 - 2*a*c^3)*d
^4*e + 24*(b^3*c - 3*a*b*c^2)*d^3*e^2 - 8*(b^4 - 6*a*b^2*c + 10*a^2*c^2)*d^2*e^3
- 9*(a*b^3 - 4*a^2*b*c)*d*e^4 + 2*(a^2*b^2 - 4*a^3*c)*e^5 + (16*c^4*d^3*e^2 - 2
4*b*c^3*d^2*e^3 + 2*(19*b^2*c^2 - 52*a*c^3)*d*e^4 - (15*b^3*c - 52*a*b*c^2)*e^5)
*x^3 + (32*c^4*d^4*e - 40*b*c^3*d^3*e^2 + 8*(5*b^2*c^2 - 14*a*c^3)*d^2*e^3 + (13
*b^3*c - 44*a*b*c^2)*d*e^4 - (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)*e^5)*x^2 + (16*c
^4*d^5 - 8*b*c^3*d^4*e - 24*(b^2*c^2 - 2*a*c^3)*d^3*e^2 + 8*(7*b^3*c - 23*a*b*c^
2)*d^2*e^3 - (25*b^4 - 114*a*b^2*c + 88*a^2*c^2)*d*e^4 - 5*(a*b^3 - 4*a^2*b*c)*e
^5)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 3*(16*(a*b^2*c^2 - 4
*a^2*c^3)*d^4*e^2 - 16*(a*b^3*c - 4*a^2*b*c^2)*d^3*e^3 + (5*a*b^4 - 24*a^2*b^2*c
+ 16*a^3*c^2)*d^2*e^4 + (16*(b^2*c^3 - 4*a*c^4)*d^2*e^4 - 16*(b^3*c^2 - 4*a*b*c
^3)*d*e^5 + (5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*e^6)*x^4 + (32*(b^2*c^3 - 4*a*
c^4)*d^3*e^3 - 16*(b^3*c^2 - 4*a*b*c^3)*d^2*e^4 - 2*(3*b^4*c - 8*a*b^2*c^2 - 16*
a^2*c^3)*d*e^5 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^3 + (16*(b^2*c^3 - 4
*a*c^4)*d^4*e^2 + 16*(b^3*c^2 - 4*a*b*c^3)*d^3*e^3 - 3*(9*b^4*c - 40*a*b^2*c^2 +
16*a^2*c^3)*d^2*e^4 + 2*(5*b^5 - 32*a*b^3*c + 48*a^2*b*c^2)*d*e^5 + (5*a*b^4 -
24*a^2*b^2*c + 16*a^3*c^2)*e^6)*x^2 + (16*(b^3*c^2 - 4*a*b*c^3)*d^4*e^2 - 16*(b^
4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d^3*e^3 + (5*b^5 - 56*a*b^3*c + 144*a^2*b*c^2)*d^
2*e^4 + 2*(5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*d*e^5)*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*b^2*c^3 - 4*a^2*c^4)*d^8 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)
*d^7*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^6*e^2 - (a*b^5 + 2*a^2*b^3*c
- 24*a^3*b*c^2)*d^5*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^4*e^4 - 3*(a^3
*b^3 - 4*a^4*b*c)*d^3*e^5 + (a^4*b^2 - 4*a^5*c)*d^2*e^6 + ((b^2*c^4 - 4*a*c^5)*d
^6*e^2 - 3*(b^3*c^3 - 4*a*b*c^4)*d^5*e^3 + 3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)
*d^4*e^4 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^5 + 3*(a*b^4*c - 3*a^2*b^2
*c^2 - 4*a^3*c^3)*d^2*e^6 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^7 + (a^3*b^2*c - 4*a
^4*c^2)*e^8)*x^4 + (2*(b^2*c^4 - 4*a*c^5)*d^7*e - 5*(b^3*c^3 - 4*a*b*c^4)*d^6*e^
2 + 3*(b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^5*e^3 + (b^5*c - 13*a*b^3*c^2 + 36*a
^2*b*c^3)*d^4*e^4 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 + 24*a^3*c^3)*d^3*e^5 + 3*(
a*b^5 - 5*a^2*b^3*c + 4*a^3*b*c^2)*d^2*e^6 - (3*a^2*b^4 - 14*a^3*b^2*c + 8*a^4*c
^2)*d*e^7 + (a^3*b^3 - 4*a^4*b*c)*e^8)*x^3 + ((b^2*c^4 - 4*a*c^5)*d^8 - (b^3*c^3
- 4*a*b*c^4)*d^7*e - (3*b^4*c^2 - 16*a*b^2*c^3 + 16*a^2*c^4)*d^6*e^2 + (5*b^5*c
- 23*a*b^3*c^2 + 12*a^2*b*c^3)*d^5*e^3 - 2*(b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12
*a^3*c^3)*d^4*e^4 + (5*a*b^5 - 23*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^5 - (3*a^2*b^4
- 16*a^3*b^2*c + 16*a^4*c^2)*d^2*e^6 - (a^3*b^3 - 4*a^4*b*c)*d*e^7 + (a^4*b^2 -
4*a^5*c)*e^8)*x^2 + ((b^3*c^3 - 4*a*b*c^4)*d^8 - (3*b^4*c^2 - 14*a*b^2*c^3 + 8*
a^2*c^4)*d^7*e + 3*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d^6*e^2 - (b^6 - 4*a*b^4*
c - 6*a^2*b^2*c^2 + 24*a^3*c^3)*d^5*e^3 + (a*b^5 - 13*a^2*b^3*c + 36*a^3*b*c^2)*
d^4*e^4 + 3*(a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d^3*e^5 - 5*(a^3*b^3 - 4*a^4*b*c
)*d^2*e^6 + 2*(a^4*b^2 - 4*a^5*c)*d*e^7)*x)*sqrt(-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)**(3/2),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)^(3/2)*(e*x + d)^3),x, algorithm="giac")

[Out]

Exception raised: TypeError