3.2212 \(\int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^5} \, dx\)

Optimal. Leaf size=378 \[ \frac{5 (b+2 c x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{10 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^2 \left (-16 a c e-3 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{2 x (2 c d-b e) \left (-c e (13 a e+35 b d)+12 b^2 e^2+35 c^2 d^2\right )-b^2 \left (27 a e^3+49 c d^2 e\right )+2 b c d \left (99 a e^2+35 c d^2\right )-32 a c e \left (a e^2+7 c d^2\right )+3 b^3 d e^2}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2} \]

[Out]

-((b + 2*c*x)*(d + e*x)^3)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^2*
(14*b*c*d - 3*b^2*e - 16*a*c*e + 14*c*(2*c*d - b*e)*x))/(12*(b^2 - 4*a*c)^2*(a +
b*x + c*x^2)^3) - (3*b^3*d*e^2 - 32*a*c*e*(7*c*d^2 + a*e^2) + 2*b*c*d*(35*c*d^2
+ 99*a*e^2) - b^2*(49*c*d^2*e + 27*a*e^3) + 2*(2*c*d - b*e)*(35*c^2*d^2 + 12*b^
2*e^2 - c*e*(35*b*d + 13*a*e))*x)/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (5*
(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(b + 2*c*x))/(2*(b^2 -
4*a*c)^4*(a + b*x + c*x^2)) - (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7
*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

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Rubi [A]  time = 1.15831, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{5 (b+2 c x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{10 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^2 \left (-16 a c e-3 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{2 x (2 c d-b e) \left (-c e (13 a e+35 b d)+12 b^2 e^2+35 c^2 d^2\right )-b^2 \left (27 a e^3+49 c d^2 e\right )+2 b c d \left (99 a e^2+35 c d^2\right )-32 a c e \left (a e^2+7 c d^2\right )+3 b^3 d e^2}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b + 2*c*x)*(d + e*x)^3)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^2*
(14*b*c*d - 3*b^2*e - 16*a*c*e + 14*c*(2*c*d - b*e)*x))/(12*(b^2 - 4*a*c)^2*(a +
b*x + c*x^2)^3) - (3*b^3*d*e^2 - 32*a*c*e*(7*c*d^2 + a*e^2) + 2*b*c*d*(35*c*d^2
+ 99*a*e^2) - b^2*(49*c*d^2*e + 27*a*e^3) + 2*(2*c*d - b*e)*(35*c^2*d^2 + 12*b^
2*e^2 - c*e*(35*b*d + 13*a*e))*x)/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (5*
(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(b + 2*c*x))/(2*(b^2 -
4*a*c)^4*(a + b*x + c*x^2)) - (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7
*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

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Rubi in Sympy [A]  time = 177.25, size = 391, normalized size = 1.03 \[ \frac{10 c \left (b e - 2 c d\right ) \left (3 a c e^{2} + b^{2} e^{2} - 7 b c d e + 7 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{9}{2}}} - \frac{\left (b + 2 c x\right ) \left (d + e x\right )^{3}}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{4}} - \frac{5 \left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (3 a c e^{2} + b^{2} e^{2} - 7 b c d e + 7 c^{2} d^{2}\right )}{2 \left (- 4 a c + b^{2}\right )^{4} \left (a + b x + c x^{2}\right )} - \frac{\left (d + e x\right )^{2} \left (16 a c e + b \left (3 b e - 14 c d\right ) + 14 c x \left (b e - 2 c d\right )\right )}{12 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{3}} + \frac{64 a^{2} c e^{3} + 54 a b^{2} e^{3} - 396 a b c d e^{2} + 448 a c^{2} d^{2} e - 6 b^{3} d e^{2} + 98 b^{2} c d^{2} e - 140 b c^{2} d^{3} + 4 x \left (b e - 2 c d\right ) \left (- 13 a c e^{2} + 12 b^{2} e^{2} - 35 b c d e + 35 c^{2} d^{2}\right )}{24 \left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*c*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2)*atanh((b +
2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(9/2) - (b + 2*c*x)*(d + e*x)**3/(
4*(-4*a*c + b**2)*(a + b*x + c*x**2)**4) - 5*(b + 2*c*x)*(b*e - 2*c*d)*(3*a*c*e*
*2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2)/(2*(-4*a*c + b**2)**4*(a + b*x + c*x**
2)) - (d + e*x)**2*(16*a*c*e + b*(3*b*e - 14*c*d) + 14*c*x*(b*e - 2*c*d))/(12*(-
4*a*c + b**2)**2*(a + b*x + c*x**2)**3) + (64*a**2*c*e**3 + 54*a*b**2*e**3 - 396
*a*b*c*d*e**2 + 448*a*c**2*d**2*e - 6*b**3*d*e**2 + 98*b**2*c*d**2*e - 140*b*c**
2*d**3 + 4*x*(b*e - 2*c*d)*(-13*a*c*e**2 + 12*b**2*e**2 - 35*b*c*d*e + 35*c**2*d
**2))/(24*(-4*a*c + b**2)**3*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 2.43254, size = 467, normalized size = 1.24 \[ \frac{1}{12} \left (\frac{3 \left (2 c \left (a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right )+b^2 e^2 (3 c d x-a e)+b c \left (3 a e^2 (d+e x)+c d^2 (d-3 e x)\right )-b^3 e^3 x\right )}{c^2 \left (4 a c-b^2\right ) (a+x (b+c x))^4}+\frac{4 c^2 \left (-8 a^2 e^3+3 a c d e^2 x+7 c^2 d^3 x\right )+b^2 c e \left (13 a e^2-3 c d (7 d-6 e x)\right )+2 b c^2 \left (3 a e^2 (d-e x)+7 c d^2 (d-3 e x)\right )-3 b^4 e^3+b^3 c e^2 (9 d-2 e x)}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}+\frac{30 (b+2 c x) (2 c d-b e) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right )}{\left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac{5 (b+2 c x) (2 c d-b e) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right )}{c \left (4 a c-b^2\right )^3 (a+x (b+c x))^2}+\frac{120 c (2 c d-b e) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{9/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*(b + 2*c*x))/(c*(
-b^2 + 4*a*c)^3*(a + x*(b + c*x))^2) + (30*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 +
c*e*(-7*b*d + 3*a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (-3*b^4
*e^3 + b^3*c*e^2*(9*d - 2*e*x) + 4*c^2*(-8*a^2*e^3 + 7*c^2*d^3*x + 3*a*c*d*e^2*x
) + b^2*c*e*(13*a*e^2 - 3*c*d*(7*d - 6*e*x)) + 2*b*c^2*(7*c*d^2*(d - 3*e*x) + 3*
a*e^2*(d - e*x)))/(c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) + (3*(-(b^3*e^3*x) +
b^2*e^2*(-(a*e) + 3*c*d*x) + 2*c*(a^2*e^3 + c^2*d^3*x - 3*a*c*d*e*(d + e*x)) +
b*c*(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x))))/(c^2*(-b^2 + 4*a*c)*(a + x*(b + c*
x))^4) + (120*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*ArcTa
n[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(9/2))/12

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Maple [B]  time = 0.028, size = 1742, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-5*c^4*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d
^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^7-35/2*b*c^3*(
3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/(256*
a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^6-5/3*c^2*(11*a*c+13*b^
2)*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/(
256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^5-25/12*b*(22*a*c+5
*b^2)*c*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d
^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^4-1/3*(73*a^2*
c^2+101*a*b^2*c+3*b^4)*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2
*d^2*e-14*c^3*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x
^3-1/6*(256*a^4*c^3*e^3+401*a^3*b^2*c^2*e^3-1314*a^3*b*c^3*d*e^2+399*a^2*b^4*c*e
^3-2139*a^2*b^3*c^2*d*e^2+4599*a^2*b^2*c^3*d^2*e-3066*a^2*b*c^4*d^3+9*a*b^6*e^3-
246*a*b^5*c*d*e^2+588*a*b^4*c^2*d^2*e-392*a*b^3*c^3*d^3+9*b^7*d*e^2-21*b^6*c*d^2
*e+14*b^5*c^2*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x
^2-1/3*(83*a^4*b*c^2*e^3+90*a^4*c^3*d*e^2+151*a^3*b^3*c*e^3-837*a^3*b^2*c^2*d*e^
2+837*a^3*b*c^3*d^2*e-558*a^3*c^4*d^3+3*a^2*b^5*e^3-84*a^2*b^4*c*d*e^2+522*a^2*b
^3*c^2*d^2*e-348*a^2*b^2*c^3*d^3+3*a*b^6*d*e^2-57*a*b^5*c*d^2*e+38*a*b^4*c^2*d^3
+3*b^7*d^2*e-2*b^6*c*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c
+b^8)*x-1/12*(128*a^5*c^2*e^3+166*a^4*b^2*c*e^3-972*a^4*b*c^2*d*e^2+1152*a^4*c^3
*d^2*e+3*a^3*b^4*e^3-84*a^3*b^3*c*d*e^2+522*a^3*b^2*c^2*d^2*e-1116*a^3*b*c^3*d^3
+3*a^2*b^5*d*e^2-57*a^2*b^4*c*d^2*e+326*a^2*b^3*c^2*d^3+3*a*b^6*d^2*e-50*a*b^5*c
*d^3+3*b^7*d^3)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8))/(c*
x^2+b*x+a)^4-30*c^2/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/
(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*e^3+60*c^3/(256*a^4*c^
4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x
+b)/(4*a*c-b^2)^(1/2))*a*e^2*d-10*c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-
16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*e^3+90
*c^2/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/
2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d*e^2-210*c^3/(256*a^4*c^4-256*a^3*b^
2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b
^2)^(1/2))*b*d^2*e+140*c^4/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*
c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250015, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(60*(14*a^4*c^4*d^3 - 21*a^4*b*c^3*d^2*e + (14*c^8*d^3 - 21*b*c^7*d^2*e +
3*(3*b^2*c^6 + 2*a*c^7)*d*e^2 - (b^3*c^5 + 3*a*b*c^6)*e^3)*x^8 + 4*(14*b*c^7*d^
3 - 21*b^2*c^6*d^2*e + 3*(3*b^3*c^5 + 2*a*b*c^6)*d*e^2 - (b^4*c^4 + 3*a*b^2*c^5)
*e^3)*x^7 + 2*(14*(3*b^2*c^6 + 2*a*c^7)*d^3 - 21*(3*b^3*c^5 + 2*a*b*c^6)*d^2*e +
3*(9*b^4*c^4 + 12*a*b^2*c^5 + 4*a^2*c^6)*d*e^2 - (3*b^5*c^3 + 11*a*b^3*c^4 + 6*
a^2*b*c^5)*e^3)*x^6 + 4*(14*(b^3*c^5 + 3*a*b*c^6)*d^3 - 21*(b^4*c^4 + 3*a*b^2*c^
5)*d^2*e + 3*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d*e^2 - (b^6*c^2 + 6*a*b^4
*c^3 + 9*a^2*b^2*c^4)*e^3)*x^5 + (14*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^3 -
21*(b^5*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*d^2*e + 3*(3*b^6*c^2 + 38*a*b^4*c^3 +
42*a^2*b^2*c^4 + 12*a^3*c^5)*d*e^2 - (b^7*c + 15*a*b^5*c^2 + 42*a^2*b^3*c^3 + 18
*a^3*b*c^4)*e^3)*x^4 + 3*(3*a^4*b^2*c^2 + 2*a^5*c^3)*d*e^2 - (a^4*b^3*c + 3*a^5*
b*c^2)*e^3 + 4*(14*(a*b^3*c^4 + 3*a^2*b*c^5)*d^3 - 21*(a*b^4*c^3 + 3*a^2*b^2*c^4
)*d^2*e + 3*(3*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*d*e^2 - (a*b^6*c + 6*a^
2*b^4*c^2 + 9*a^3*b^2*c^3)*e^3)*x^3 + 2*(14*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d^3 - 21
*(3*a^2*b^3*c^3 + 2*a^3*b*c^4)*d^2*e + 3*(9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4
*c^4)*d*e^2 - (3*a^2*b^5*c + 11*a^3*b^3*c^2 + 6*a^4*b*c^3)*e^3)*x^2 + 4*(14*a^3*
b*c^4*d^3 - 21*a^3*b^2*c^3*d^2*e + 3*(3*a^3*b^3*c^2 + 2*a^4*b*c^3)*d*e^2 - (a^3*
b^4*c + 3*a^4*b^2*c^2)*e^3)*x)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x + (2*c
^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a)) - (60*(14*
c^7*d^3 - 21*b*c^6*d^2*e + 3*(3*b^2*c^5 + 2*a*c^6)*d*e^2 - (b^3*c^4 + 3*a*b*c^5)
*e^3)*x^7 + 210*(14*b*c^6*d^3 - 21*b^2*c^5*d^2*e + 3*(3*b^3*c^4 + 2*a*b*c^5)*d*e
^2 - (b^4*c^3 + 3*a*b^2*c^4)*e^3)*x^6 + 20*(14*(13*b^2*c^5 + 11*a*c^6)*d^3 - 21*
(13*b^3*c^4 + 11*a*b*c^5)*d^2*e + 3*(39*b^4*c^3 + 59*a*b^2*c^4 + 22*a^2*c^5)*d*e
^2 - (13*b^5*c^2 + 50*a*b^3*c^3 + 33*a^2*b*c^4)*e^3)*x^5 + 25*(14*(5*b^3*c^4 + 2
2*a*b*c^5)*d^3 - 21*(5*b^4*c^3 + 22*a*b^2*c^4)*d^2*e + 3*(15*b^5*c^2 + 76*a*b^3*
c^3 + 44*a^2*b*c^4)*d*e^2 - (5*b^6*c + 37*a*b^4*c^2 + 66*a^2*b^2*c^3)*e^3)*x^4 -
(3*b^7 - 50*a*b^5*c + 326*a^2*b^3*c^2 - 1116*a^3*b*c^3)*d^3 - 3*(a*b^6 - 19*a^2
*b^4*c + 174*a^3*b^2*c^2 + 384*a^4*c^3)*d^2*e - 3*(a^2*b^5 - 28*a^3*b^3*c - 324*
a^4*b*c^2)*d*e^2 - (3*a^3*b^4 + 166*a^4*b^2*c + 128*a^5*c^2)*e^3 + 4*(14*(3*b^4*
c^3 + 101*a*b^2*c^4 + 73*a^2*c^5)*d^3 - 21*(3*b^5*c^2 + 101*a*b^3*c^3 + 73*a^2*b
*c^4)*d^2*e + 3*(9*b^6*c + 309*a*b^4*c^2 + 421*a^2*b^2*c^3 + 146*a^3*c^4)*d*e^2
- (3*b^7 + 110*a*b^5*c + 376*a^2*b^3*c^2 + 219*a^3*b*c^3)*e^3)*x^3 - 2*(14*(b^5*
c^2 - 28*a*b^3*c^3 - 219*a^2*b*c^4)*d^3 - 21*(b^6*c - 28*a*b^4*c^2 - 219*a^2*b^2
*c^3)*d^2*e + 3*(3*b^7 - 82*a*b^5*c - 713*a^2*b^3*c^2 - 438*a^3*b*c^3)*d*e^2 + (
9*a*b^6 + 399*a^2*b^4*c + 401*a^3*b^2*c^2 + 256*a^4*c^3)*e^3)*x^2 + 4*(2*(b^6*c
- 19*a*b^4*c^2 + 174*a^2*b^2*c^3 + 279*a^3*c^4)*d^3 - 3*(b^7 - 19*a*b^5*c + 174*
a^2*b^3*c^2 + 279*a^3*b*c^3)*d^2*e - 3*(a*b^6 - 28*a^2*b^4*c - 279*a^3*b^2*c^2 +
30*a^4*c^3)*d*e^2 - (3*a^2*b^5 + 151*a^3*b^3*c + 83*a^4*b*c^2)*e^3)*x)*sqrt(b^2
- 4*a*c))/((a^4*b^8 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^3 + 256*a^8
*c^4 + (b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8)
*x^8 + 4*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*
c^7)*x^7 + 2*(3*b^10*c^2 - 46*a*b^8*c^3 + 256*a^2*b^6*c^4 - 576*a^3*b^4*c^5 + 25
6*a^4*b^2*c^6 + 512*a^5*c^7)*x^6 + 4*(b^11*c - 13*a*b^9*c^2 + 48*a^2*b^7*c^3 + 3
2*a^3*b^5*c^4 - 512*a^4*b^3*c^5 + 768*a^5*b*c^6)*x^5 + (b^12 - 4*a*b^10*c - 90*a
^2*b^8*c^2 + 800*a^3*b^6*c^3 - 2240*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 + 1536*a^6*c^
6)*x^4 + 4*(a*b^11 - 13*a^2*b^9*c + 48*a^3*b^7*c^2 + 32*a^4*b^5*c^3 - 512*a^5*b^
3*c^4 + 768*a^6*b*c^5)*x^3 + 2*(3*a^2*b^10 - 46*a^3*b^8*c + 256*a^4*b^6*c^2 - 57
6*a^5*b^4*c^3 + 256*a^6*b^2*c^4 + 512*a^7*c^5)*x^2 + 4*(a^3*b^9 - 16*a^4*b^7*c +
96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4)*x)*sqrt(b^2 - 4*a*c)), 1/12*(
120*(14*a^4*c^4*d^3 - 21*a^4*b*c^3*d^2*e + (14*c^8*d^3 - 21*b*c^7*d^2*e + 3*(3*b
^2*c^6 + 2*a*c^7)*d*e^2 - (b^3*c^5 + 3*a*b*c^6)*e^3)*x^8 + 4*(14*b*c^7*d^3 - 21*
b^2*c^6*d^2*e + 3*(3*b^3*c^5 + 2*a*b*c^6)*d*e^2 - (b^4*c^4 + 3*a*b^2*c^5)*e^3)*x
^7 + 2*(14*(3*b^2*c^6 + 2*a*c^7)*d^3 - 21*(3*b^3*c^5 + 2*a*b*c^6)*d^2*e + 3*(9*b
^4*c^4 + 12*a*b^2*c^5 + 4*a^2*c^6)*d*e^2 - (3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c
^5)*e^3)*x^6 + 4*(14*(b^3*c^5 + 3*a*b*c^6)*d^3 - 21*(b^4*c^4 + 3*a*b^2*c^5)*d^2*
e + 3*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d*e^2 - (b^6*c^2 + 6*a*b^4*c^3 +
9*a^2*b^2*c^4)*e^3)*x^5 + (14*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^3 - 21*(b^5
*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*d^2*e + 3*(3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*
b^2*c^4 + 12*a^3*c^5)*d*e^2 - (b^7*c + 15*a*b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*
c^4)*e^3)*x^4 + 3*(3*a^4*b^2*c^2 + 2*a^5*c^3)*d*e^2 - (a^4*b^3*c + 3*a^5*b*c^2)*
e^3 + 4*(14*(a*b^3*c^4 + 3*a^2*b*c^5)*d^3 - 21*(a*b^4*c^3 + 3*a^2*b^2*c^4)*d^2*e
+ 3*(3*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*d*e^2 - (a*b^6*c + 6*a^2*b^4*c
^2 + 9*a^3*b^2*c^3)*e^3)*x^3 + 2*(14*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d^3 - 21*(3*a^2
*b^3*c^3 + 2*a^3*b*c^4)*d^2*e + 3*(9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c^4)*d
*e^2 - (3*a^2*b^5*c + 11*a^3*b^3*c^2 + 6*a^4*b*c^3)*e^3)*x^2 + 4*(14*a^3*b*c^4*d
^3 - 21*a^3*b^2*c^3*d^2*e + 3*(3*a^3*b^3*c^2 + 2*a^4*b*c^3)*d*e^2 - (a^3*b^4*c +
3*a^4*b^2*c^2)*e^3)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) +
(60*(14*c^7*d^3 - 21*b*c^6*d^2*e + 3*(3*b^2*c^5 + 2*a*c^6)*d*e^2 - (b^3*c^4 + 3*
a*b*c^5)*e^3)*x^7 + 210*(14*b*c^6*d^3 - 21*b^2*c^5*d^2*e + 3*(3*b^3*c^4 + 2*a*b*
c^5)*d*e^2 - (b^4*c^3 + 3*a*b^2*c^4)*e^3)*x^6 + 20*(14*(13*b^2*c^5 + 11*a*c^6)*d
^3 - 21*(13*b^3*c^4 + 11*a*b*c^5)*d^2*e + 3*(39*b^4*c^3 + 59*a*b^2*c^4 + 22*a^2*
c^5)*d*e^2 - (13*b^5*c^2 + 50*a*b^3*c^3 + 33*a^2*b*c^4)*e^3)*x^5 + 25*(14*(5*b^3
*c^4 + 22*a*b*c^5)*d^3 - 21*(5*b^4*c^3 + 22*a*b^2*c^4)*d^2*e + 3*(15*b^5*c^2 + 7
6*a*b^3*c^3 + 44*a^2*b*c^4)*d*e^2 - (5*b^6*c + 37*a*b^4*c^2 + 66*a^2*b^2*c^3)*e^
3)*x^4 - (3*b^7 - 50*a*b^5*c + 326*a^2*b^3*c^2 - 1116*a^3*b*c^3)*d^3 - 3*(a*b^6
- 19*a^2*b^4*c + 174*a^3*b^2*c^2 + 384*a^4*c^3)*d^2*e - 3*(a^2*b^5 - 28*a^3*b^3*
c - 324*a^4*b*c^2)*d*e^2 - (3*a^3*b^4 + 166*a^4*b^2*c + 128*a^5*c^2)*e^3 + 4*(14
*(3*b^4*c^3 + 101*a*b^2*c^4 + 73*a^2*c^5)*d^3 - 21*(3*b^5*c^2 + 101*a*b^3*c^3 +
73*a^2*b*c^4)*d^2*e + 3*(9*b^6*c + 309*a*b^4*c^2 + 421*a^2*b^2*c^3 + 146*a^3*c^4
)*d*e^2 - (3*b^7 + 110*a*b^5*c + 376*a^2*b^3*c^2 + 219*a^3*b*c^3)*e^3)*x^3 - 2*(
14*(b^5*c^2 - 28*a*b^3*c^3 - 219*a^2*b*c^4)*d^3 - 21*(b^6*c - 28*a*b^4*c^2 - 219
*a^2*b^2*c^3)*d^2*e + 3*(3*b^7 - 82*a*b^5*c - 713*a^2*b^3*c^2 - 438*a^3*b*c^3)*d
*e^2 + (9*a*b^6 + 399*a^2*b^4*c + 401*a^3*b^2*c^2 + 256*a^4*c^3)*e^3)*x^2 + 4*(2
*(b^6*c - 19*a*b^4*c^2 + 174*a^2*b^2*c^3 + 279*a^3*c^4)*d^3 - 3*(b^7 - 19*a*b^5*
c + 174*a^2*b^3*c^2 + 279*a^3*b*c^3)*d^2*e - 3*(a*b^6 - 28*a^2*b^4*c - 279*a^3*b
^2*c^2 + 30*a^4*c^3)*d*e^2 - (3*a^2*b^5 + 151*a^3*b^3*c + 83*a^4*b*c^2)*e^3)*x)*
sqrt(-b^2 + 4*a*c))/((a^4*b^8 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^3
+ 256*a^8*c^4 + (b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256
*a^4*c^8)*x^8 + 4*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 2
56*a^4*b*c^7)*x^7 + 2*(3*b^10*c^2 - 46*a*b^8*c^3 + 256*a^2*b^6*c^4 - 576*a^3*b^4
*c^5 + 256*a^4*b^2*c^6 + 512*a^5*c^7)*x^6 + 4*(b^11*c - 13*a*b^9*c^2 + 48*a^2*b^
7*c^3 + 32*a^3*b^5*c^4 - 512*a^4*b^3*c^5 + 768*a^5*b*c^6)*x^5 + (b^12 - 4*a*b^10
*c - 90*a^2*b^8*c^2 + 800*a^3*b^6*c^3 - 2240*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 + 15
36*a^6*c^6)*x^4 + 4*(a*b^11 - 13*a^2*b^9*c + 48*a^3*b^7*c^2 + 32*a^4*b^5*c^3 - 5
12*a^5*b^3*c^4 + 768*a^6*b*c^5)*x^3 + 2*(3*a^2*b^10 - 46*a^3*b^8*c + 256*a^4*b^6
*c^2 - 576*a^5*b^4*c^3 + 256*a^6*b^2*c^4 + 512*a^7*c^5)*x^2 + 4*(a^3*b^9 - 16*a^
4*b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4)*x)*sqrt(-b^2 + 4*a*c
))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.210748, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done