3.2203 \(\int \frac{(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=259 \[ -\frac{2 (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{8 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^3 \left (-2 a c e-2 b^2 e+5 c x (2 c d-b e)+5 b c d\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]

[Out]

-((b + 2*c*x)*(d + e*x)^4)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + ((d + e*x)^3*
(5*b*c*d - 2*b^2*e - 2*a*c*e + 5*c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)^2*(a + b*x
+ c*x^2)^2) - (2*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)*(b*d - 2*a
*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (8*(c*d^2 - b*d*e +
a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - 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 = 0.979925, antiderivative size = 259, 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 \[ -\frac{2 (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{8 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^3 \left (-2 a c e-2 b^2 e+5 c x (2 c d-b e)+5 b c d\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b + 2*c*x)*(d + e*x)^4)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + ((d + e*x)^3*
(5*b*c*d - 2*b^2*e - 2*a*c*e + 5*c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)^2*(a + b*x
+ c*x^2)^2) - (2*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)*(b*d - 2*a
*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (8*(c*d^2 - b*d*e +
a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - 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 = 91.8287, size = 255, normalized size = 0.98 \[ - \frac{\left (b + 2 c x\right ) \left (d + e x\right )^{4}}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{3}} - \frac{\left (d + e x\right )^{3} \left (4 a c e + 2 b \left (2 b e - 5 c d\right ) + 10 c x \left (b e - 2 c d\right )\right )}{6 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{2}} + \frac{2 \left (d + e x\right ) \left (2 a e - b d + x \left (b e - 2 c d\right )\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{\left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )} + \frac{8 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b + 2*c*x)*(d + e*x)**4/(3*(-4*a*c + b**2)*(a + b*x + c*x**2)**3) - (d + e*x)*
*3*(4*a*c*e + 2*b*(2*b*e - 5*c*d) + 10*c*x*(b*e - 2*c*d))/(6*(-4*a*c + b**2)**2*
(a + b*x + c*x**2)**2) + 2*(d + e*x)*(2*a*e - b*d + x*(b*e - 2*c*d))*(a*c*e**2 +
b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/((-4*a*c + b**2)**3*(a + b*x + c*x**2)) +
8*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*ata
nh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(7/2)

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Mathematica [B]  time = 2.33727, size = 572, normalized size = 2.21 \[ \frac{1}{3} \left (\frac{6 (b+2 c x) \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right )}{c \left (4 a c-b^2\right )^3 (a+x (b+c x))}+\frac{24 \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}+\frac{b c \left (-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right )+2 c^2 \left (a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right )+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left (3 c d^2 x-2 a e (d+e x)\right )+b^4 e^4 x}{c^3 \left (4 a c-b^2\right ) (a+x (b+c x))^3}+\frac{b c^2 \left (17 a^2 e^4+6 a c d e^2 (d-2 e x)+5 c^2 d^3 (d-4 e x)\right )+2 c^3 \left (-a^2 e^3 (24 d+7 e x)+6 a c d^2 e^2 x+5 c^2 d^4 x\right )+b^3 c e^2 \left (2 c d (3 d-e x)-7 a e^2\right )+2 b^2 c^2 e \left (a e^2 (9 d+5 e x)+c d^2 (6 e x-5 d)\right )+b^5 e^4-b^4 c e^3 (4 d+e x)}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((6*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(
6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*(b + 2*c*x))/(c*(-b^2 + 4*a*c)^3*(a + x*(b + c
*x))) + (b^4*e^4*x + b^3*e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3*c*d^2*x - 2*a*e*(d
+ e*x)) + b*c*(-3*a^2*e^4 + c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) + 2*
c^2*(c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x)))/(c^3*(-b^2 +
4*a*c)*(a + x*(b + c*x))^3) + (b^5*e^4 - b^4*c*e^3*(4*d + e*x) + b*c^2*(17*a^2*e
^4 + 5*c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d - 2*e*x)) + b^3*c*e^2*(-7*a*e^2 + 2*
c*d*(3*d - e*x)) + 2*b^2*c^2*e*(a*e^2*(9*d + 5*e*x) + c*d^2*(-5*d + 6*e*x)) + 2*
c^3*(5*c^2*d^4*x + 6*a*c*d^2*e^2*x - a^2*e^3*(24*d + 7*e*x)))/(c^3*(b^2 - 4*a*c)
^2*(a + x*(b + c*x))^2) + (24*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*
(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*ArcTan[(b + 2*c*x)/S
qrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/3

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Maple [B]  time = 0.022, size = 1666, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4*(a^2*c*e^4+a*b^2*e^4-6*a*b*c*d*e^3+6*a*c^2*d^2*e^2-b^3*d*e^3+6*b^2*c*d^2*e^2-
10*b*c^2*d^3*e+5*c^3*d^4)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^2*x^5+10*
b*c*(a^2*c*e^4+a*b^2*e^4-6*a*b*c*d*e^3+6*a*c^2*d^2*e^2-b^3*d*e^3+6*b^2*c*d^2*e^2
-10*b*c^2*d^3*e+5*c^3*d^4)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^4-1/3*(3
2*a^3*c^3*e^4-102*a^2*b^2*c^2*e^4+192*a^2*b*c^3*d*e^3-192*a^2*c^4*d^2*e^2-10*a*b
^4*c*e^4+164*a*b^3*c^2*d*e^3-324*a*b^2*c^3*d^2*e^2+320*a*b*c^4*d^3*e-160*a*c^5*d
^4-b^6*e^4+22*b^5*c*d*e^3-132*b^4*c^2*d^2*e^2+220*b^3*c^3*d^3*e-110*b^2*c^4*d^4)
/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3+(16*a^3*b*c^2*e^4-64*a^3*c^3*d
*e^3+17*a^2*b^3*c*e^4-48*a^2*b^2*c^2*d*e^3+96*a^2*b*c^3*d^2*e^2+a*b^5*e^4-34*a*b
^4*c*d*e^3+102*a*b^3*c^2*d^2*e^2-160*a*b^2*c^3*d^3*e+80*a*b*c^4*d^4+6*b^5*c*d^2*
e^2-10*b^4*c^2*d^3*e+5*b^3*c^3*d^4)/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)
*x^2-(4*a^4*c^2*e^4-22*a^3*b^2*c*e^4+40*a^3*b*c^2*d*e^3+24*a^3*c^3*d^2*e^2-a^2*b
^4*e^4+40*a^2*b^3*c*d*e^3-132*a^2*b^2*c^2*d^2*e^2+88*a^2*b*c^3*d^3*e-44*a^2*c^4*
d^4-6*a*b^4*c*d^2*e^2+36*a*b^3*c^2*d^3*e-18*a*b^2*c^3*d^4-2*b^5*c*d^3*e+b^4*c^2*
d^4)/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x+1/3*(26*a^4*b*c*e^4-64*a^4*c
^2*d*e^3+a^3*b^3*e^4-44*a^3*b^2*c*d*e^3+156*a^3*b*c^2*d^2*e^2-128*a^3*c^3*d^3*e+
6*a^2*b^3*c*d^2*e^2-36*a^2*b^2*c^2*d^3*e+66*a^2*b*c^3*d^4+2*a*b^4*c*d^3*e-13*a*b
^3*c^2*d^4+b^5*c*d^4)/c/(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+8/(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^2*c*e^4+8/(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^2*e^4-48/(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*d*e^3+48/(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)*ar
ctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c^2*d^2*e^2-8/(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*d*e^3+48/
(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^2*e^2-80/(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))*d^3*e*b*c^2+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^4

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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)^4/(c*x^2 + b*x + a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238726, 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)^4/(c*x^2 + b*x + a)^4,x, algorithm="fricas")

[Out]

[-1/3*(12*(5*a^3*c^4*d^4 - 10*a^3*b*c^3*d^3*e + (5*c^7*d^4 - 10*b*c^6*d^3*e + 6*
(b^2*c^5 + a*c^6)*d^2*e^2 - (b^3*c^4 + 6*a*b*c^5)*d*e^3 + (a*b^2*c^4 + a^2*c^5)*
e^4)*x^6 + 3*(5*b*c^6*d^4 - 10*b^2*c^5*d^3*e + 6*(b^3*c^4 + a*b*c^5)*d^2*e^2 - (
b^4*c^3 + 6*a*b^2*c^4)*d*e^3 + (a*b^3*c^3 + a^2*b*c^4)*e^4)*x^5 + 6*(a^3*b^2*c^2
+ a^4*c^3)*d^2*e^2 - (a^3*b^3*c + 6*a^4*b*c^2)*d*e^3 + (a^4*b^2*c + a^5*c^2)*e^
4 + 3*(5*(b^2*c^5 + a*c^6)*d^4 - 10*(b^3*c^4 + a*b*c^5)*d^3*e + 6*(b^4*c^3 + 2*a
*b^2*c^4 + a^2*c^5)*d^2*e^2 - (b^5*c^2 + 7*a*b^3*c^3 + 6*a^2*b*c^4)*d*e^3 + (a*b
^4*c^2 + 2*a^2*b^2*c^3 + a^3*c^4)*e^4)*x^4 + (5*(b^3*c^4 + 6*a*b*c^5)*d^4 - 10*(
b^4*c^3 + 6*a*b^2*c^4)*d^3*e + 6*(b^5*c^2 + 7*a*b^3*c^3 + 6*a^2*b*c^4)*d^2*e^2 -
(b^6*c + 12*a*b^4*c^2 + 36*a^2*b^2*c^3)*d*e^3 + (a*b^5*c + 7*a^2*b^3*c^2 + 6*a^
3*b*c^3)*e^4)*x^3 + 3*(5*(a*b^2*c^4 + a^2*c^5)*d^4 - 10*(a*b^3*c^3 + a^2*b*c^4)*
d^3*e + 6*(a*b^4*c^2 + 2*a^2*b^2*c^3 + a^3*c^4)*d^2*e^2 - (a*b^5*c + 7*a^2*b^3*c
^2 + 6*a^3*b*c^3)*d*e^3 + (a^2*b^4*c + 2*a^3*b^2*c^2 + a^4*c^3)*e^4)*x^2 + 3*(5*
a^2*b*c^4*d^4 - 10*a^2*b^2*c^3*d^3*e + 6*(a^2*b^3*c^2 + a^3*b*c^3)*d^2*e^2 - (a^
2*b^4*c + 6*a^3*b^2*c^2)*d*e^3 + (a^3*b^3*c + a^4*b*c^2)*e^4)*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)) + (12*(5*c^6*d^4 - 10*b*c^5*d^3*e + 6*(b^2*c^4 + a*c^5)
*d^2*e^2 - (b^3*c^3 + 6*a*b*c^4)*d*e^3 + (a*b^2*c^3 + a^2*c^4)*e^4)*x^5 + (b^5*c
- 13*a*b^3*c^2 + 66*a^2*b*c^3)*d^4 + 2*(a*b^4*c - 18*a^2*b^2*c^2 - 64*a^3*c^3)*
d^3*e + 6*(a^2*b^3*c + 26*a^3*b*c^2)*d^2*e^2 - 4*(11*a^3*b^2*c + 16*a^4*c^2)*d*e
^3 + (a^3*b^3 + 26*a^4*b*c)*e^4 + 30*(5*b*c^5*d^4 - 10*b^2*c^4*d^3*e + 6*(b^3*c^
3 + a*b*c^4)*d^2*e^2 - (b^4*c^2 + 6*a*b^2*c^3)*d*e^3 + (a*b^3*c^2 + a^2*b*c^3)*e
^4)*x^4 + (10*(11*b^2*c^4 + 16*a*c^5)*d^4 - 20*(11*b^3*c^3 + 16*a*b*c^4)*d^3*e +
12*(11*b^4*c^2 + 27*a*b^2*c^3 + 16*a^2*c^4)*d^2*e^2 - 2*(11*b^5*c + 82*a*b^3*c^
2 + 96*a^2*b*c^3)*d*e^3 + (b^6 + 10*a*b^4*c + 102*a^2*b^2*c^2 - 32*a^3*c^3)*e^4)
*x^3 + 3*(5*(b^3*c^3 + 16*a*b*c^4)*d^4 - 10*(b^4*c^2 + 16*a*b^2*c^3)*d^3*e + 6*(
b^5*c + 17*a*b^3*c^2 + 16*a^2*b*c^3)*d^2*e^2 - 2*(17*a*b^4*c + 24*a^2*b^2*c^2 +
32*a^3*c^3)*d*e^3 + (a*b^5 + 17*a^2*b^3*c + 16*a^3*b*c^2)*e^4)*x^2 - 3*((b^4*c^2
- 18*a*b^2*c^3 - 44*a^2*c^4)*d^4 - 2*(b^5*c - 18*a*b^3*c^2 - 44*a^2*b*c^3)*d^3*
e - 6*(a*b^4*c + 22*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^2 + 40*(a^2*b^3*c + a^3*b*c^2
)*d*e^3 - (a^2*b^4 + 22*a^3*b^2*c - 4*a^4*c^2)*e^4)*x)*sqrt(b^2 - 4*a*c))/((a^3*
b^6*c - 12*a^4*b^4*c^2 + 48*a^5*b^2*c^3 - 64*a^6*c^4 + (b^6*c^4 - 12*a*b^4*c^5 +
48*a^2*b^2*c^6 - 64*a^3*c^7)*x^6 + 3*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 -
64*a^3*b*c^6)*x^5 + 3*(b^8*c^2 - 11*a*b^6*c^3 + 36*a^2*b^4*c^4 - 16*a^3*b^2*c^5
- 64*a^4*c^6)*x^4 + (b^9*c - 6*a*b^7*c^2 - 24*a^2*b^5*c^3 + 224*a^3*b^3*c^4 - 3
84*a^4*b*c^5)*x^3 + 3*(a*b^8*c - 11*a^2*b^6*c^2 + 36*a^3*b^4*c^3 - 16*a^4*b^2*c^
4 - 64*a^5*c^5)*x^2 + 3*(a^2*b^7*c - 12*a^3*b^5*c^2 + 48*a^4*b^3*c^3 - 64*a^5*b*
c^4)*x)*sqrt(b^2 - 4*a*c)), -1/3*(24*(5*a^3*c^4*d^4 - 10*a^3*b*c^3*d^3*e + (5*c^
7*d^4 - 10*b*c^6*d^3*e + 6*(b^2*c^5 + a*c^6)*d^2*e^2 - (b^3*c^4 + 6*a*b*c^5)*d*e
^3 + (a*b^2*c^4 + a^2*c^5)*e^4)*x^6 + 3*(5*b*c^6*d^4 - 10*b^2*c^5*d^3*e + 6*(b^3
*c^4 + a*b*c^5)*d^2*e^2 - (b^4*c^3 + 6*a*b^2*c^4)*d*e^3 + (a*b^3*c^3 + a^2*b*c^4
)*e^4)*x^5 + 6*(a^3*b^2*c^2 + a^4*c^3)*d^2*e^2 - (a^3*b^3*c + 6*a^4*b*c^2)*d*e^3
+ (a^4*b^2*c + a^5*c^2)*e^4 + 3*(5*(b^2*c^5 + a*c^6)*d^4 - 10*(b^3*c^4 + a*b*c^
5)*d^3*e + 6*(b^4*c^3 + 2*a*b^2*c^4 + a^2*c^5)*d^2*e^2 - (b^5*c^2 + 7*a*b^3*c^3
+ 6*a^2*b*c^4)*d*e^3 + (a*b^4*c^2 + 2*a^2*b^2*c^3 + a^3*c^4)*e^4)*x^4 + (5*(b^3*
c^4 + 6*a*b*c^5)*d^4 - 10*(b^4*c^3 + 6*a*b^2*c^4)*d^3*e + 6*(b^5*c^2 + 7*a*b^3*c
^3 + 6*a^2*b*c^4)*d^2*e^2 - (b^6*c + 12*a*b^4*c^2 + 36*a^2*b^2*c^3)*d*e^3 + (a*b
^5*c + 7*a^2*b^3*c^2 + 6*a^3*b*c^3)*e^4)*x^3 + 3*(5*(a*b^2*c^4 + a^2*c^5)*d^4 -
10*(a*b^3*c^3 + a^2*b*c^4)*d^3*e + 6*(a*b^4*c^2 + 2*a^2*b^2*c^3 + a^3*c^4)*d^2*e
^2 - (a*b^5*c + 7*a^2*b^3*c^2 + 6*a^3*b*c^3)*d*e^3 + (a^2*b^4*c + 2*a^3*b^2*c^2
+ a^4*c^3)*e^4)*x^2 + 3*(5*a^2*b*c^4*d^4 - 10*a^2*b^2*c^3*d^3*e + 6*(a^2*b^3*c^2
+ a^3*b*c^3)*d^2*e^2 - (a^2*b^4*c + 6*a^3*b^2*c^2)*d*e^3 + (a^3*b^3*c + a^4*b*c
^2)*e^4)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (12*(5*c^6*d
^4 - 10*b*c^5*d^3*e + 6*(b^2*c^4 + a*c^5)*d^2*e^2 - (b^3*c^3 + 6*a*b*c^4)*d*e^3
+ (a*b^2*c^3 + a^2*c^4)*e^4)*x^5 + (b^5*c - 13*a*b^3*c^2 + 66*a^2*b*c^3)*d^4 + 2
*(a*b^4*c - 18*a^2*b^2*c^2 - 64*a^3*c^3)*d^3*e + 6*(a^2*b^3*c + 26*a^3*b*c^2)*d^
2*e^2 - 4*(11*a^3*b^2*c + 16*a^4*c^2)*d*e^3 + (a^3*b^3 + 26*a^4*b*c)*e^4 + 30*(5
*b*c^5*d^4 - 10*b^2*c^4*d^3*e + 6*(b^3*c^3 + a*b*c^4)*d^2*e^2 - (b^4*c^2 + 6*a*b
^2*c^3)*d*e^3 + (a*b^3*c^2 + a^2*b*c^3)*e^4)*x^4 + (10*(11*b^2*c^4 + 16*a*c^5)*d
^4 - 20*(11*b^3*c^3 + 16*a*b*c^4)*d^3*e + 12*(11*b^4*c^2 + 27*a*b^2*c^3 + 16*a^2
*c^4)*d^2*e^2 - 2*(11*b^5*c + 82*a*b^3*c^2 + 96*a^2*b*c^3)*d*e^3 + (b^6 + 10*a*b
^4*c + 102*a^2*b^2*c^2 - 32*a^3*c^3)*e^4)*x^3 + 3*(5*(b^3*c^3 + 16*a*b*c^4)*d^4
- 10*(b^4*c^2 + 16*a*b^2*c^3)*d^3*e + 6*(b^5*c + 17*a*b^3*c^2 + 16*a^2*b*c^3)*d^
2*e^2 - 2*(17*a*b^4*c + 24*a^2*b^2*c^2 + 32*a^3*c^3)*d*e^3 + (a*b^5 + 17*a^2*b^3
*c + 16*a^3*b*c^2)*e^4)*x^2 - 3*((b^4*c^2 - 18*a*b^2*c^3 - 44*a^2*c^4)*d^4 - 2*(
b^5*c - 18*a*b^3*c^2 - 44*a^2*b*c^3)*d^3*e - 6*(a*b^4*c + 22*a^2*b^2*c^2 - 4*a^3
*c^3)*d^2*e^2 + 40*(a^2*b^3*c + a^3*b*c^2)*d*e^3 - (a^2*b^4 + 22*a^3*b^2*c - 4*a
^4*c^2)*e^4)*x)*sqrt(-b^2 + 4*a*c))/((a^3*b^6*c - 12*a^4*b^4*c^2 + 48*a^5*b^2*c^
3 - 64*a^6*c^4 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^6 + 3*
(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^5 + 3*(b^8*c^2 - 11*a
*b^6*c^3 + 36*a^2*b^4*c^4 - 16*a^3*b^2*c^5 - 64*a^4*c^6)*x^4 + (b^9*c - 6*a*b^7*
c^2 - 24*a^2*b^5*c^3 + 224*a^3*b^3*c^4 - 384*a^4*b*c^5)*x^3 + 3*(a*b^8*c - 11*a^
2*b^6*c^2 + 36*a^3*b^4*c^3 - 16*a^4*b^2*c^4 - 64*a^5*c^5)*x^2 + 3*(a^2*b^7*c - 1
2*a^3*b^5*c^2 + 48*a^4*b^3*c^3 - 64*a^5*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)**4/(c*x**2+b*x+a)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.210467, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done