3.2171 \(\int \frac{(d+e x)^4}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=243 \[ -\frac{\left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}+\frac{e (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{e^2 x \left (-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )}{c^3}+\frac{e^3 x^2 (4 c d-b e)}{2 c^2}+\frac{e^4 x^3}{3 c} \]

[Out]

(e^2*(6*c^2*d^2 + b^2*e^2 - c*e*(4*b*d + a*e))*x)/c^3 + (e^3*(4*c*d - b*e)*x^2)/
(2*c^2) + (e^4*x^3)/(3*c) - ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*
c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*ArcTanh[(
b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*Sqrt[b^2 - 4*a*c]) + (e*(2*c*d - b*e)*(2*c^2
*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^4)

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Rubi [A]  time = 0.939809, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}+\frac{e (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{e^2 x \left (-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )}{c^3}+\frac{e^3 x^2 (4 c d-b e)}{2 c^2}+\frac{e^4 x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(6*c^2*d^2 + b^2*e^2 - c*e*(4*b*d + a*e))*x)/c^3 + (e^3*(4*c*d - b*e)*x^2)/
(2*c^2) + (e^4*x^3)/(3*c) - ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*
c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*ArcTanh[(
b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*Sqrt[b^2 - 4*a*c]) + (e*(2*c*d - b*e)*(2*c^2
*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^4)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{2} \left (- a c e^{2} + b^{2} e^{2} - 4 b c d e + 6 c^{2} d^{2}\right ) \int \frac{1}{c^{3}}\, dx + \frac{e^{4} x^{3}}{3 c} - \frac{e^{3} \left (b e - 4 c d\right ) \int x\, dx}{c^{2}} - \frac{e \left (b e - 2 c d\right ) \left (- 2 a c e^{2} + b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{4}} - \frac{\left (b e \left (b e - 2 c d\right ) \left (- 2 a c e^{2} + b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right ) + 2 c \left (a^{2} c e^{4} - a b^{2} e^{4} + 4 a b c d e^{3} - 6 a c^{2} d^{2} e^{2} + c^{3} d^{4}\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{4} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**2*(-a*c*e**2 + b**2*e**2 - 4*b*c*d*e + 6*c**2*d**2)*Integral(c**(-3), x) + e*
*4*x**3/(3*c) - e**3*(b*e - 4*c*d)*Integral(x, x)/c**2 - e*(b*e - 2*c*d)*(-2*a*c
*e**2 + b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)*log(a + b*x + c*x**2)/(2*c**4) - (b
*e*(b*e - 2*c*d)*(-2*a*c*e**2 + b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2) + 2*c*(a**2
*c*e**4 - a*b**2*e**4 + 4*a*b*c*d*e**3 - 6*a*c**2*d**2*e**2 + c**3*d**4))*atanh(
(b + 2*c*x)/sqrt(-4*a*c + b**2))/(c**4*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.36421, size = 240, normalized size = 0.99 \[ \frac{\frac{6 \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+6 c e^2 x \left (-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )+3 e (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log (a+x (b+c x))+3 c^2 e^3 x^2 (4 c d-b e)+2 c^3 e^4 x^3}{6 c^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.009, size = 595, normalized size = 2.5 \[{\frac{{e}^{4}{x}^{3}}{3\,c}}-{\frac{{e}^{4}{x}^{2}b}{2\,{c}^{2}}}+2\,{\frac{d{e}^{3}{x}^{2}}{c}}-{\frac{{e}^{4}ax}{{c}^{2}}}+{\frac{{b}^{2}{e}^{4}x}{{c}^{3}}}-4\,{\frac{bd{e}^{3}x}{{c}^{2}}}+6\,{\frac{{d}^{2}{e}^{2}x}{c}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) ab{e}^{4}}{{c}^{3}}}-2\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ) ad{e}^{3}}{{c}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}{e}^{4}}{2\,{c}^{4}}}+2\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}d{e}^{3}}{{c}^{3}}}-3\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ){d}^{2}{e}^{2}b}{{c}^{2}}}+2\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ){d}^{3}e}{c}}+2\,{\frac{{a}^{2}{e}^{4}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{a{b}^{2}{e}^{4}}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{d{e}^{3}ab}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-12\,{\frac{a{d}^{2}{e}^{2}}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{d}^{4}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}{e}^{4}}{{c}^{4}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-4\,{\frac{{b}^{3}d{e}^{3}}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+6\,{\frac{{b}^{2}{d}^{2}{e}^{2}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{d}^{3}eb}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*x^2+b*x+a),x)

[Out]

1/3*e^4*x^3/c-1/2*e^4/c^2*x^2*b+2*d*e^3*x^2/c-e^4/c^2*a*x+e^4/c^3*b^2*x-4*e^3/c^
2*b*d*x+6*e^2/c*d^2*x+1/c^3*ln(c*x^2+b*x+a)*a*b*e^4-2/c^2*ln(c*x^2+b*x+a)*a*d*e^
3-1/2/c^4*ln(c*x^2+b*x+a)*b^3*e^4+2/c^3*ln(c*x^2+b*x+a)*b^2*d*e^3-3/c^2*ln(c*x^2
+b*x+a)*d^2*e^2*b+2/c*ln(c*x^2+b*x+a)*d^3*e+2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*
x+b)/(4*a*c-b^2)^(1/2))*a^2*e^4-4/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-
b^2)^(1/2))*a*b^2*e^4+12/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2
))*a*b*d*e^3-12/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d^2*e^
2+2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^4+1/c^4/(4*a*c-b^2)^
(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*e^4-4/c^3/(4*a*c-b^2)^(1/2)*arctan
((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*d*e^3+6/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)
/(4*a*c-b^2)^(1/2))*b^2*d^2*e^2-4/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^
2)^(1/2))*d^3*e*b

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270608, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2} e^{2} - 4 \,{\left (b^{3} c - 3 \, a b c^{2}\right )} d e^{3} +{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e^{4}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (2 \, c^{3} e^{4} x^{3} + 3 \,{\left (4 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} x^{2} + 6 \,{\left (6 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} +{\left (b^{2} c - a c^{2}\right )} e^{4}\right )} x + 3 \,{\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \,{\left (b^{2} c - a c^{2}\right )} d e^{3} -{\left (b^{3} - 2 \, a b c\right )} e^{4}\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{6 \, \sqrt{b^{2} - 4 \, a c} c^{4}}, \frac{6 \,{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2} e^{2} - 4 \,{\left (b^{3} c - 3 \, a b c^{2}\right )} d e^{3} +{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e^{4}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, c^{3} e^{4} x^{3} + 3 \,{\left (4 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} x^{2} + 6 \,{\left (6 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} +{\left (b^{2} c - a c^{2}\right )} e^{4}\right )} x + 3 \,{\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \,{\left (b^{2} c - a c^{2}\right )} d e^{3} -{\left (b^{3} - 2 \, a b c\right )} e^{4}\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{6 \, \sqrt{-b^{2} + 4 \, a c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*(2*c^4*d^4 - 4*b*c^3*d^3*e + 6*(b^2*c^2 - 2*a*c^3)*d^2*e^2 - 4*(b^3*c -
3*a*b*c^2)*d*e^3 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^4)*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)) + (2*c^3*e^4*x^3 + 3*(4*c^3*d*e^3 - b*c^2*e^4)*x^2 + 6*(6*c^3*d^2*
e^2 - 4*b*c^2*d*e^3 + (b^2*c - a*c^2)*e^4)*x + 3*(4*c^3*d^3*e - 6*b*c^2*d^2*e^2
+ 4*(b^2*c - a*c^2)*d*e^3 - (b^3 - 2*a*b*c)*e^4)*log(c*x^2 + b*x + a))*sqrt(b^2
- 4*a*c))/(sqrt(b^2 - 4*a*c)*c^4), 1/6*(6*(2*c^4*d^4 - 4*b*c^3*d^3*e + 6*(b^2*c^
2 - 2*a*c^3)*d^2*e^2 - 4*(b^3*c - 3*a*b*c^2)*d*e^3 + (b^4 - 4*a*b^2*c + 2*a^2*c^
2)*e^4)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*c^3*e^4*x^3 +
3*(4*c^3*d*e^3 - b*c^2*e^4)*x^2 + 6*(6*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + (b^2*c - a
*c^2)*e^4)*x + 3*(4*c^3*d^3*e - 6*b*c^2*d^2*e^2 + 4*(b^2*c - a*c^2)*d*e^3 - (b^3
- 2*a*b*c)*e^4)*log(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c
^4)]

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Sympy [A]  time = 19.9173, size = 1554, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*(b*e - 2*c*d)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**4) - s
qrt(-4*a*c + b**2)*(2*a**2*c**2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3 - 12
*a*c**3*d**2*e**2 + b**4*e**4 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*b*c*
*3*d**3*e + 2*c**4*d**4)/(2*c**4*(4*a*c - b**2)))*log(x + (-3*a**2*b*c*e**4 + 8*
a**2*c**2*d*e**3 + a*b**3*e**4 - 4*a*b**2*c*d*e**3 + 6*a*b*c**2*d**2*e**2 + 4*a*
c**4*(e*(b*e - 2*c*d)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**4
) - sqrt(-4*a*c + b**2)*(2*a**2*c**2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3
- 12*a*c**3*d**2*e**2 + b**4*e**4 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4
*b*c**3*d**3*e + 2*c**4*d**4)/(2*c**4*(4*a*c - b**2))) - 8*a*c**3*d**3*e - b**2*
c**3*(e*(b*e - 2*c*d)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**4
) - sqrt(-4*a*c + b**2)*(2*a**2*c**2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3
- 12*a*c**3*d**2*e**2 + b**4*e**4 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4
*b*c**3*d**3*e + 2*c**4*d**4)/(2*c**4*(4*a*c - b**2))) + b*c**3*d**4)/(2*a**2*c*
*2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3 - 12*a*c**3*d**2*e**2 + b**4*e**4
- 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*b*c**3*d**3*e + 2*c**4*d**4)) + (
e*(b*e - 2*c*d)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**4) + sq
rt(-4*a*c + b**2)*(2*a**2*c**2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3 - 12*
a*c**3*d**2*e**2 + b**4*e**4 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*b*c**
3*d**3*e + 2*c**4*d**4)/(2*c**4*(4*a*c - b**2)))*log(x + (-3*a**2*b*c*e**4 + 8*a
**2*c**2*d*e**3 + a*b**3*e**4 - 4*a*b**2*c*d*e**3 + 6*a*b*c**2*d**2*e**2 + 4*a*c
**4*(e*(b*e - 2*c*d)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**4)
+ sqrt(-4*a*c + b**2)*(2*a**2*c**2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3
- 12*a*c**3*d**2*e**2 + b**4*e**4 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*
b*c**3*d**3*e + 2*c**4*d**4)/(2*c**4*(4*a*c - b**2))) - 8*a*c**3*d**3*e - b**2*c
**3*(e*(b*e - 2*c*d)*(2*a*c*e**2 - b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)/(2*c**4)
+ sqrt(-4*a*c + b**2)*(2*a**2*c**2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3
- 12*a*c**3*d**2*e**2 + b**4*e**4 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*
b*c**3*d**3*e + 2*c**4*d**4)/(2*c**4*(4*a*c - b**2))) + b*c**3*d**4)/(2*a**2*c**
2*e**4 - 4*a*b**2*c*e**4 + 12*a*b*c**2*d*e**3 - 12*a*c**3*d**2*e**2 + b**4*e**4
- 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*b*c**3*d**3*e + 2*c**4*d**4)) + e*
*4*x**3/(3*c) - x**2*(b*e**4 - 4*c*d*e**3)/(2*c**2) - x*(a*c*e**4 - b**2*e**4 +
4*b*c*d*e**3 - 6*c**2*d**2*e**2)/c**3

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GIAC/XCAS [A]  time = 0.206986, size = 358, normalized size = 1.47 \[ \frac{2 \, c^{2} x^{3} e^{4} + 12 \, c^{2} d x^{2} e^{3} + 36 \, c^{2} d^{2} x e^{2} - 3 \, b c x^{2} e^{4} - 24 \, b c d x e^{3} + 6 \, b^{2} x e^{4} - 6 \, a c x e^{4}}{6 \, c^{3}} + \frac{{\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \, b^{2} c d e^{3} - 4 \, a c^{2} d e^{3} - b^{3} e^{4} + 2 \, a b c e^{4}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 12 \, a c^{3} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + 12 \, a b c^{2} d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*c^2*x^3*e^4 + 12*c^2*d*x^2*e^3 + 36*c^2*d^2*x*e^2 - 3*b*c*x^2*e^4 - 24*b*
c*d*x*e^3 + 6*b^2*x*e^4 - 6*a*c*x*e^4)/c^3 + 1/2*(4*c^3*d^3*e - 6*b*c^2*d^2*e^2
+ 4*b^2*c*d*e^3 - 4*a*c^2*d*e^3 - b^3*e^4 + 2*a*b*c*e^4)*ln(c*x^2 + b*x + a)/c^4
+ (2*c^4*d^4 - 4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 12*a*c^3*d^2*e^2 - 4*b^3*c*d
*e^3 + 12*a*b*c^2*d*e^3 + b^4*e^4 - 4*a*b^2*c*e^4 + 2*a^2*c^2*e^4)*arctan((2*c*x
+ b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^4)