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3.2204 \(\int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=306 \[ \frac{2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{2 x (2 c d-b e) \left (-c e (a e+15 b d)+4 b^2 e^2+15 c^2 d^2\right )-b^2 \left (11 a e^3+25 c d^2 e\right )+6 b c d \left (13 a e^2+5 c d^2\right )-16 a c e \left (a e^2+5 c d^2\right )+3 b^3 d e^2}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )} \]

[Out]

-((b + 2*c*x)*(d + e*x)^3)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + ((d + e*x)^2*
(10*b*c*d - 3*b^2*e - 8*a*c*e + 10*c*(2*c*d - b*e)*x))/(6*(b^2 - 4*a*c)^2*(a + b
*x + c*x^2)^2) - (3*b^3*d*e^2 - 16*a*c*e*(5*c*d^2 + a*e^2) + 6*b*c*d*(5*c*d^2 +
13*a*e^2) - b^2*(25*c*d^2*e + 11*a*e^3) + 2*(2*c*d - b*e)*(15*c^2*d^2 + 4*b^2*e^
2 - c*e*(15*b*d + a*e))*x)/(3*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (2*(2*c*d - b
*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2
- 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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Rubi [A]  time = 0.977771, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{2 x (2 c d-b e) \left (-c e (a e+15 b d)+4 b^2 e^2+15 c^2 d^2\right )-b^2 \left (11 a e^3+25 c d^2 e\right )+6 b c d \left (13 a e^2+5 c d^2\right )-16 a c e \left (a e^2+5 c d^2\right )+3 b^3 d e^2}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(a + b*x + c*x^2)^4,x]

[Out]

-((b + 2*c*x)*(d + e*x)^3)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + ((d + e*x)^2*
(10*b*c*d - 3*b^2*e - 8*a*c*e + 10*c*(2*c*d - b*e)*x))/(6*(b^2 - 4*a*c)^2*(a + b
*x + c*x^2)^2) - (3*b^3*d*e^2 - 16*a*c*e*(5*c*d^2 + a*e^2) + 6*b*c*d*(5*c*d^2 +
13*a*e^2) - b^2*(25*c*d^2*e + 11*a*e^3) + 2*(2*c*d - b*e)*(15*c^2*d^2 + 4*b^2*e^
2 - c*e*(15*b*d + a*e))*x)/(3*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (2*(2*c*d - b
*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2
- 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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Rubi in Sympy [A]  time = 133.918, size = 289, normalized size = 0.94 \[ - \frac{\left (b + 2 c x\right ) \left (d + e x\right )^{3}}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{3}} - \frac{\left (d + e x\right )^{2} \left (8 a c e + b \left (3 b e - 10 c d\right ) + 10 c x \left (b e - 2 c d\right )\right )}{6 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{2}} - \frac{\left (d + e x\right ) \left (36 a b c e^{2} - 40 a c^{2} d e + 6 b^{3} e^{2} - 50 b^{2} c d e + 60 b c^{2} d^{2} + 2 c x \left (16 a c e^{2} + 11 b^{2} e^{2} - 60 b c d e + 60 c^{2} d^{2}\right )\right )}{6 \left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )} - \frac{2 \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(c*x**2+b*x+a)**4,x)

[Out]

-(b + 2*c*x)*(d + e*x)**3/(3*(-4*a*c + b**2)*(a + b*x + c*x**2)**3) - (d + e*x)*
*2*(8*a*c*e + b*(3*b*e - 10*c*d) + 10*c*x*(b*e - 2*c*d))/(6*(-4*a*c + b**2)**2*(
a + b*x + c*x**2)**2) - (d + e*x)*(36*a*b*c*e**2 - 40*a*c**2*d*e + 6*b**3*e**2 -
 50*b**2*c*d*e + 60*b*c**2*d**2 + 2*c*x*(16*a*c*e**2 + 11*b**2*e**2 - 60*b*c*d*e
 + 60*c**2*d**2))/(6*(-4*a*c + b**2)**3*(a + b*x + c*x**2)) - 2*(b*e - 2*c*d)*(6
*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)*atanh((b + 2*c*x)/sqrt(-4*a*c
 + b**2))/(-4*a*c + b**2)**(7/2)

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Mathematica [A]  time = 1.61679, size = 401, normalized size = 1.31 \[ \frac{1}{6} \left (\frac{2 \left (2 c \left (a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right )+b^2 e^2 (3 c d x-a e)+b c \left (3 a e^2 (d+e x)+c d^2 (d-3 e x)\right )-b^3 e^3 x\right )}{c^2 \left (4 a c-b^2\right ) (a+x (b+c x))^3}+\frac{4 c^2 \left (-6 a^2 e^3+3 a c d e^2 x+5 c^2 d^3 x\right )+3 b^2 c e \left (3 a e^2+c d (4 e x-5 d)\right )+2 b c^2 \left (3 a e^2 (d-e x)+5 c d^2 (d-3 e x)\right )-2 b^4 e^3+b^3 c e^2 (6 d-e x)}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac{3 (b+2 c x) (2 c d-b e) \left (2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right )}{c \left (4 a c-b^2\right )^3 (a+x (b+c x))}-\frac{12 (b e-2 c d) \left (2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(a + b*x + c*x^2)^4,x]

[Out]

((3*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e))*(b + 2*c*x))/(
c*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))) + (2*(-(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))^3) + (-2*b^4*e^3 +
 b^3*c*e^2*(6*d - e*x) + 4*c^2*(-6*a^2*e^3 + 5*c^2*d^3*x + 3*a*c*d*e^2*x) + 2*b*
c^2*(5*c*d^2*(d - 3*e*x) + 3*a*e^2*(d - e*x)) + 3*b^2*c*e*(3*a*e^2 + c*d*(-5*d +
 4*e*x)))/(c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (12*(-2*c*d + b*e)*(10*c^2
*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])
/(-b^2 + 4*a*c)^(7/2))/6

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Maple [B]  time = 0.021, size = 1213, normalized size = 4. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^2+b*x+a)^4,x)

[Out]

(-(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/
(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^2*x^5-5/2*b*c*(6*a*b*c*e^3-12*a*c^2
*d*e^2+b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/(64*a^3*c^3-48*a^2*b^2*
c^2+12*a*b^4*c-b^6)*x^4-1/6*(16*a*c+11*b^2)*(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-
12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-
b^6)*x^3-1/2*(32*a^3*c^2*e^3+24*a^2*b^2*c*e^3-96*a^2*b*c^2*d*e^2+17*a*b^4*e^3-10
2*a*b^3*c*d*e^2+240*a*b^2*c^2*d^2*e-160*a*b*c^3*d^3-6*b^5*d*e^2+15*b^4*c*d^2*e-1
0*b^3*c^2*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-1/2*(20*a^3*b*c*e^
3+24*a^3*c^2*d*e^2+20*a^2*b^3*e^3-132*a^2*b^2*c*d*e^2+132*a^2*b*c^2*d^2*e-88*a^2
*c^3*d^3-6*a*b^4*d*e^2+54*a*b^3*c*d^2*e-36*a*b^2*c^2*d^3-3*b^5*d^2*e+2*b^4*c*d^3
)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x-1/6*(32*a^4*c*e^3+22*a^3*b^2*e^3-
156*a^3*b*c*d*e^2+192*a^3*c^2*d^2*e-6*a^2*b^3*d*e^2+54*a^2*b^2*c*d^2*e-132*a^2*b
*c^2*d^3-3*a*b^4*d^2*e+26*a*b^3*c*d^3-2*b^5*d^3)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a
*b^4*c-b^6))/(c*x^2+b*x+a)^3-12/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*
c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c*e^3+24/(64*a^3*c^3-48*a^2
*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a
*c^2*e^2*d-2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan
((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*e^3+24/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b
^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c*d*e^2-60/(64*a^3
*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^
2)^(1/2))*b*c^2*d^2*e+40/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^
(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^3*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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228575, 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)^4,x, algorithm="fricas")

[Out]

[1/6*(6*(20*a^3*c^3*d^3 - 30*a^3*b*c^2*d^2*e + (20*c^6*d^3 - 30*b*c^5*d^2*e + 12
*(b^2*c^4 + a*c^5)*d*e^2 - (b^3*c^3 + 6*a*b*c^4)*e^3)*x^6 + 3*(20*b*c^5*d^3 - 30
*b^2*c^4*d^2*e + 12*(b^3*c^3 + a*b*c^4)*d*e^2 - (b^4*c^2 + 6*a*b^2*c^3)*e^3)*x^5
 + 3*(20*(b^2*c^4 + a*c^5)*d^3 - 30*(b^3*c^3 + a*b*c^4)*d^2*e + 12*(b^4*c^2 + 2*
a*b^2*c^3 + a^2*c^4)*d*e^2 - (b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*e^3)*x^4 + 12*(
a^3*b^2*c + a^4*c^2)*d*e^2 - (a^3*b^3 + 6*a^4*b*c)*e^3 + (20*(b^3*c^3 + 6*a*b*c^
4)*d^3 - 30*(b^4*c^2 + 6*a*b^2*c^3)*d^2*e + 12*(b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^
3)*d*e^2 - (b^6 + 12*a*b^4*c + 36*a^2*b^2*c^2)*e^3)*x^3 + 3*(20*(a*b^2*c^3 + a^2
*c^4)*d^3 - 30*(a*b^3*c^2 + a^2*b*c^3)*d^2*e + 12*(a*b^4*c + 2*a^2*b^2*c^2 + a^3
*c^3)*d*e^2 - (a*b^5 + 7*a^2*b^3*c + 6*a^3*b*c^2)*e^3)*x^2 + 3*(20*a^2*b*c^3*d^3
 - 30*a^2*b^2*c^2*d^2*e + 12*(a^2*b^3*c + a^3*b*c^2)*d*e^2 - (a^2*b^4 + 6*a^3*b^
2*c)*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)) - (6*(20*c^5*d^3 - 30*b*c^4*
d^2*e + 12*(b^2*c^3 + a*c^4)*d*e^2 - (b^3*c^2 + 6*a*b*c^3)*e^3)*x^5 + 15*(20*b*c
^4*d^3 - 30*b^2*c^3*d^2*e + 12*(b^3*c^2 + a*b*c^3)*d*e^2 - (b^4*c + 6*a*b^2*c^2)
*e^3)*x^4 + 2*(b^5 - 13*a*b^3*c + 66*a^2*b*c^2)*d^3 + 3*(a*b^4 - 18*a^2*b^2*c -
64*a^3*c^2)*d^2*e + 6*(a^2*b^3 + 26*a^3*b*c)*d*e^2 - 2*(11*a^3*b^2 + 16*a^4*c)*e
^3 + (20*(11*b^2*c^3 + 16*a*c^4)*d^3 - 30*(11*b^3*c^2 + 16*a*b*c^3)*d^2*e + 12*(
11*b^4*c + 27*a*b^2*c^2 + 16*a^2*c^3)*d*e^2 - (11*b^5 + 82*a*b^3*c + 96*a^2*b*c^
2)*e^3)*x^3 + 3*(10*(b^3*c^2 + 16*a*b*c^3)*d^3 - 15*(b^4*c + 16*a*b^2*c^2)*d^2*e
 + 6*(b^5 + 17*a*b^3*c + 16*a^2*b*c^2)*d*e^2 - (17*a*b^4 + 24*a^2*b^2*c + 32*a^3
*c^2)*e^3)*x^2 - 3*(2*(b^4*c - 18*a*b^2*c^2 - 44*a^2*c^3)*d^3 - 3*(b^5 - 18*a*b^
3*c - 44*a^2*b*c^2)*d^2*e - 6*(a*b^4 + 22*a^2*b^2*c - 4*a^3*c^2)*d*e^2 + 20*(a^2
*b^3 + a^3*b*c)*e^3)*x)*sqrt(b^2 - 4*a*c))/((a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2
*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^6 +
 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 3*(b^8*c - 11*
a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*x^4 + (b^9 - 6*a*b^7*c
 - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*x^3 + 3*(a*b^8 - 11*a^2*b^6
*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*x^2 + 3*(a^2*b^7 - 12*a^3*b^5
*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*sqrt(b^2 - 4*a*c)), -1/6*(12*(20*a^3*c^3*
d^3 - 30*a^3*b*c^2*d^2*e + (20*c^6*d^3 - 30*b*c^5*d^2*e + 12*(b^2*c^4 + a*c^5)*d
*e^2 - (b^3*c^3 + 6*a*b*c^4)*e^3)*x^6 + 3*(20*b*c^5*d^3 - 30*b^2*c^4*d^2*e + 12*
(b^3*c^3 + a*b*c^4)*d*e^2 - (b^4*c^2 + 6*a*b^2*c^3)*e^3)*x^5 + 3*(20*(b^2*c^4 +
a*c^5)*d^3 - 30*(b^3*c^3 + a*b*c^4)*d^2*e + 12*(b^4*c^2 + 2*a*b^2*c^3 + a^2*c^4)
*d*e^2 - (b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*e^3)*x^4 + 12*(a^3*b^2*c + a^4*c^2)
*d*e^2 - (a^3*b^3 + 6*a^4*b*c)*e^3 + (20*(b^3*c^3 + 6*a*b*c^4)*d^3 - 30*(b^4*c^2
 + 6*a*b^2*c^3)*d^2*e + 12*(b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*d*e^2 - (b^6 + 12
*a*b^4*c + 36*a^2*b^2*c^2)*e^3)*x^3 + 3*(20*(a*b^2*c^3 + a^2*c^4)*d^3 - 30*(a*b^
3*c^2 + a^2*b*c^3)*d^2*e + 12*(a*b^4*c + 2*a^2*b^2*c^2 + a^3*c^3)*d*e^2 - (a*b^5
 + 7*a^2*b^3*c + 6*a^3*b*c^2)*e^3)*x^2 + 3*(20*a^2*b*c^3*d^3 - 30*a^2*b^2*c^2*d^
2*e + 12*(a^2*b^3*c + a^3*b*c^2)*d*e^2 - (a^2*b^4 + 6*a^3*b^2*c)*e^3)*x)*arctan(
-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (6*(20*c^5*d^3 - 30*b*c^4*d^2*e
 + 12*(b^2*c^3 + a*c^4)*d*e^2 - (b^3*c^2 + 6*a*b*c^3)*e^3)*x^5 + 15*(20*b*c^4*d^
3 - 30*b^2*c^3*d^2*e + 12*(b^3*c^2 + a*b*c^3)*d*e^2 - (b^4*c + 6*a*b^2*c^2)*e^3)
*x^4 + 2*(b^5 - 13*a*b^3*c + 66*a^2*b*c^2)*d^3 + 3*(a*b^4 - 18*a^2*b^2*c - 64*a^
3*c^2)*d^2*e + 6*(a^2*b^3 + 26*a^3*b*c)*d*e^2 - 2*(11*a^3*b^2 + 16*a^4*c)*e^3 +
(20*(11*b^2*c^3 + 16*a*c^4)*d^3 - 30*(11*b^3*c^2 + 16*a*b*c^3)*d^2*e + 12*(11*b^
4*c + 27*a*b^2*c^2 + 16*a^2*c^3)*d*e^2 - (11*b^5 + 82*a*b^3*c + 96*a^2*b*c^2)*e^
3)*x^3 + 3*(10*(b^3*c^2 + 16*a*b*c^3)*d^3 - 15*(b^4*c + 16*a*b^2*c^2)*d^2*e + 6*
(b^5 + 17*a*b^3*c + 16*a^2*b*c^2)*d*e^2 - (17*a*b^4 + 24*a^2*b^2*c + 32*a^3*c^2)
*e^3)*x^2 - 3*(2*(b^4*c - 18*a*b^2*c^2 - 44*a^2*c^3)*d^3 - 3*(b^5 - 18*a*b^3*c -
 44*a^2*b*c^2)*d^2*e - 6*(a*b^4 + 22*a^2*b^2*c - 4*a^3*c^2)*d*e^2 + 20*(a^2*b^3
+ a^3*b*c)*e^3)*x)*sqrt(-b^2 + 4*a*c))/((a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2
 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^6 + 3*(
b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 3*(b^8*c - 11*a*b^
6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*x^4 + (b^9 - 6*a*b^7*c - 2
4*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*x^3 + 3*(a*b^8 - 11*a^2*b^6*c +
 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*x^2 + 3*(a^2*b^7 - 12*a^3*b^5*c +
 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 90.8387, size = 2057, normalized size = 6.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(c*x**2+b*x+a)**4,x)

[Out]

sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e +
10*c**2*d**2)*log(x + (-256*a**4*c**4*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(
6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + 256*a**3*b**2*c**3*sqrt(-1
/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2
*d**2) - 96*a**2*b**4*c**2*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2
+ b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + 16*a*b**6*c*sqrt(-1/(4*a*c - b**2)**7
)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + 6*a*b**2*
c*e**3 - 12*a*b*c**2*d*e**2 - b**8*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a
*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + b**4*e**3 - 12*b**3*c*d*e**2
+ 30*b**2*c**2*d**2*e - 20*b*c**3*d**3)/(12*a*b*c**2*e**3 - 24*a*c**3*d*e**2 + 2
*b**3*c*e**3 - 24*b**2*c**2*d*e**2 + 60*b*c**3*d**2*e - 40*c**4*d**3)) - sqrt(-1
/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2
*d**2)*log(x + (256*a**4*c**4*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e*
*2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) - 256*a**3*b**2*c**3*sqrt(-1/(4*a*c
- b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) +
 96*a**2*b**4*c**2*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2 + b**2*e
**2 - 10*b*c*d*e + 10*c**2*d**2) - 16*a*b**6*c*sqrt(-1/(4*a*c - b**2)**7)*(b*e -
 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + 6*a*b**2*c*e**3 -
 12*a*b*c**2*d*e**2 + b**8*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*(6*a*c*e**2
+ b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2) + b**4*e**3 - 12*b**3*c*d*e**2 + 30*b**
2*c**2*d**2*e - 20*b*c**3*d**3)/(12*a*b*c**2*e**3 - 24*a*c**3*d*e**2 + 2*b**3*c*
e**3 - 24*b**2*c**2*d*e**2 + 60*b*c**3*d**2*e - 40*c**4*d**3)) - (32*a**4*c*e**3
 + 22*a**3*b**2*e**3 - 156*a**3*b*c*d*e**2 + 192*a**3*c**2*d**2*e - 6*a**2*b**3*
d*e**2 + 54*a**2*b**2*c*d**2*e - 132*a**2*b*c**2*d**3 - 3*a*b**4*d**2*e + 26*a*b
**3*c*d**3 - 2*b**5*d**3 + x**5*(36*a*b*c**3*e**3 - 72*a*c**4*d*e**2 + 6*b**3*c*
*2*e**3 - 72*b**2*c**3*d*e**2 + 180*b*c**4*d**2*e - 120*c**5*d**3) + x**4*(90*a*
b**2*c**2*e**3 - 180*a*b*c**3*d*e**2 + 15*b**4*c*e**3 - 180*b**3*c**2*d*e**2 + 4
50*b**2*c**3*d**2*e - 300*b*c**4*d**3) + x**3*(96*a**2*b*c**2*e**3 - 192*a**2*c*
*3*d*e**2 + 82*a*b**3*c*e**3 - 324*a*b**2*c**2*d*e**2 + 480*a*b*c**3*d**2*e - 32
0*a*c**4*d**3 + 11*b**5*e**3 - 132*b**4*c*d*e**2 + 330*b**3*c**2*d**2*e - 220*b*
*2*c**3*d**3) + x**2*(96*a**3*c**2*e**3 + 72*a**2*b**2*c*e**3 - 288*a**2*b*c**2*
d*e**2 + 51*a*b**4*e**3 - 306*a*b**3*c*d*e**2 + 720*a*b**2*c**2*d**2*e - 480*a*b
*c**3*d**3 - 18*b**5*d*e**2 + 45*b**4*c*d**2*e - 30*b**3*c**2*d**3) + x*(60*a**3
*b*c*e**3 + 72*a**3*c**2*d*e**2 + 60*a**2*b**3*e**3 - 396*a**2*b**2*c*d*e**2 + 3
96*a**2*b*c**2*d**2*e - 264*a**2*c**3*d**3 - 18*a*b**4*d*e**2 + 162*a*b**3*c*d**
2*e - 108*a*b**2*c**2*d**3 - 9*b**5*d**2*e + 6*b**4*c*d**3))/(384*a**6*c**3 - 28
8*a**5*b**2*c**2 + 72*a**4*b**4*c - 6*a**3*b**6 + x**6*(384*a**3*c**6 - 288*a**2
*b**2*c**5 + 72*a*b**4*c**4 - 6*b**6*c**3) + x**5*(1152*a**3*b*c**5 - 864*a**2*b
**3*c**4 + 216*a*b**5*c**3 - 18*b**7*c**2) + x**4*(1152*a**4*c**5 + 288*a**3*b**
2*c**4 - 648*a**2*b**4*c**3 + 198*a*b**6*c**2 - 18*b**8*c) + x**3*(2304*a**4*b*c
**4 - 1344*a**3*b**3*c**3 + 144*a**2*b**5*c**2 + 36*a*b**7*c - 6*b**9) + x**2*(1
152*a**5*c**4 + 288*a**4*b**2*c**3 - 648*a**3*b**4*c**2 + 198*a**2*b**6*c - 18*a
*b**8) + x*(1152*a**5*b*c**3 - 864*a**4*b**3*c**2 + 216*a**3*b**5*c - 18*a**2*b*
*7))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211295, size = 1118, normalized size = 3.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*x^2 + b*x + a)^4,x, algorithm="giac")

[Out]

-2*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 12*a*c^2*d*e^2 - b^3*e^3 - 6*
a*b*c*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4*c + 48*a^2*b^
2*c^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)) - 1/6*(120*c^5*d^3*x^5 - 180*b*c^4*d^2*x
^5*e + 300*b*c^4*d^3*x^4 + 72*b^2*c^3*d*x^5*e^2 + 72*a*c^4*d*x^5*e^2 - 450*b^2*c
^3*d^2*x^4*e + 220*b^2*c^3*d^3*x^3 + 320*a*c^4*d^3*x^3 - 6*b^3*c^2*x^5*e^3 - 36*
a*b*c^3*x^5*e^3 + 180*b^3*c^2*d*x^4*e^2 + 180*a*b*c^3*d*x^4*e^2 - 330*b^3*c^2*d^
2*x^3*e - 480*a*b*c^3*d^2*x^3*e + 30*b^3*c^2*d^3*x^2 + 480*a*b*c^3*d^3*x^2 - 15*
b^4*c*x^4*e^3 - 90*a*b^2*c^2*x^4*e^3 + 132*b^4*c*d*x^3*e^2 + 324*a*b^2*c^2*d*x^3
*e^2 + 192*a^2*c^3*d*x^3*e^2 - 45*b^4*c*d^2*x^2*e - 720*a*b^2*c^2*d^2*x^2*e - 6*
b^4*c*d^3*x + 108*a*b^2*c^2*d^3*x + 264*a^2*c^3*d^3*x - 11*b^5*x^3*e^3 - 82*a*b^
3*c*x^3*e^3 - 96*a^2*b*c^2*x^3*e^3 + 18*b^5*d*x^2*e^2 + 306*a*b^3*c*d*x^2*e^2 +
288*a^2*b*c^2*d*x^2*e^2 + 9*b^5*d^2*x*e - 162*a*b^3*c*d^2*x*e - 396*a^2*b*c^2*d^
2*x*e + 2*b^5*d^3 - 26*a*b^3*c*d^3 + 132*a^2*b*c^2*d^3 - 51*a*b^4*x^2*e^3 - 72*a
^2*b^2*c*x^2*e^3 - 96*a^3*c^2*x^2*e^3 + 18*a*b^4*d*x*e^2 + 396*a^2*b^2*c*d*x*e^2
 - 72*a^3*c^2*d*x*e^2 + 3*a*b^4*d^2*e - 54*a^2*b^2*c*d^2*e - 192*a^3*c^2*d^2*e -
 60*a^2*b^3*x*e^3 - 60*a^3*b*c*x*e^3 + 6*a^2*b^3*d*e^2 + 156*a^3*b*c*d*e^2 - 22*
a^3*b^2*e^3 - 32*a^4*c*e^3)/((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(c
*x^2 + b*x + a)^3)

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