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

Optimal. Leaf size=280 \[ -\frac{x^2 (a d+b e)}{2 a^2 e^2}+\frac{\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e+b^4 d-b^3 c e\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}+\frac{\left (5 a^2 b c^2 d-2 a^2 c^3 e-5 a b^3 c d+4 a b^2 c^2 e+b^5 d-b^4 c e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{x \left (a^2 d^2+a e (b d-c e)+b^2 e^2\right )}{a^3 e^3}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac{x^3}{3 a e} \]

[Out]

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

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Rubi [A]  time = 1.20744, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{x^2 (a d+b e)}{2 a^2 e^2}+\frac{\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e+b^4 d-b^3 c e\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-e (b d-c e)\right )}+\frac{\left (5 a^2 b c^2 d-2 a^2 c^3 e-5 a b^3 c d+4 a b^2 c^2 e+b^5 d-b^4 c e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{x \left (a^2 d^2+a e (b d-c e)+b^2 e^2\right )}{a^3 e^3}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )}+\frac{x^3}{3 a e} \]

Antiderivative was successfully verified.

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

[Out]

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

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

[Out]

Timed out

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Mathematica [A]  time = 0.404622, size = 283, normalized size = 1.01 \[ -\frac{x^2 (a d+b e)}{2 a^2 e^2}+\frac{\left (a^2 c^2 d-3 a b^2 c d+2 a b c^2 e+b^4 d-b^3 c e\right ) \log \left (a x^2+b x+c\right )}{2 a^4 \left (a d^2-b d e+c e^2\right )}+\frac{\left (5 a^2 b c^2 d-2 a^2 c^3 e-5 a b^3 c d+4 a b^2 c^2 e+b^5 d-b^4 c e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a^4 \sqrt{4 a c-b^2} \left (-a d^2+b d e-c e^2\right )}+\frac{x \left (a^2 d^2+a b d e-a c e^2+b^2 e^2\right )}{a^3 e^3}-\frac{d^5 \log (d+e x)}{e^4 \left (a d^2-b d e+c e^2\right )}+\frac{x^3}{3 a e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.018, size = 662, normalized size = 2.4 \[{\frac{{x}^{3}}{3\,ae}}-{\frac{{x}^{2}d}{2\,a{e}^{2}}}-{\frac{b{x}^{2}}{2\,{a}^{2}e}}+{\frac{{d}^{2}x}{{e}^{3}a}}+{\frac{bdx}{{a}^{2}{e}^{2}}}-{\frac{cx}{{a}^{2}e}}+{\frac{{b}^{2}x}{{a}^{3}e}}-{\frac{{d}^{5}\ln \left ( ex+d \right ) }{{e}^{4} \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){c}^{2}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{2}}}-{\frac{3\,\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}cd}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{3}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) b{c}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{4}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{4}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{3}ce}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{4}}}-5\,{\frac{{c}^{2}db}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{c}^{3}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+5\,{\frac{{b}^{3}cd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{2}{c}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{5}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{4}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}ce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{4}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3/a/e-1/2/a/e^2*x^2*d-1/2/a^2/e*x^2*b+1/a/e^3*d^2*x+1/a^2/e^2*b*d*x-1/a^2/
e*c*x+1/a^3/e*b^2*x-1/e^4*d^5/(a*d^2-b*d*e+c*e^2)*ln(e*x+d)+1/2/(a*d^2-b*d*e+c*e
^2)/a^2*ln(a*x^2+b*x+c)*c^2*d-3/2/(a*d^2-b*d*e+c*e^2)/a^3*ln(a*x^2+b*x+c)*b^2*c*
d+1/(a*d^2-b*d*e+c*e^2)/a^3*ln(a*x^2+b*x+c)*b*c^2*e+1/2/(a*d^2-b*d*e+c*e^2)/a^4*
ln(a*x^2+b*x+c)*b^4*d-1/2/(a*d^2-b*d*e+c*e^2)/a^4*ln(a*x^2+b*x+c)*b^3*c*e-5/(a*d
^2-b*d*e+c*e^2)/a^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*b*c^2*
d+2/(a*d^2-b*d*e+c*e^2)/a^2/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2)
)*c^3*e+5/(a*d^2-b*d*e+c*e^2)/a^3/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)
^(1/2))*b^3*c*d-4/(a*d^2-b*d*e+c*e^2)/a^3/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*
a*c-b^2)^(1/2))*b^2*c^2*e-1/(a*d^2-b*d*e+c*e^2)/a^4/(4*a*c-b^2)^(1/2)*arctan((2*
a*x+b)/(4*a*c-b^2)^(1/2))*b^5*d+1/(a*d^2-b*d*e+c*e^2)/a^4/(4*a*c-b^2)^(1/2)*arct
an((2*a*x+b)/(4*a*c-b^2)^(1/2))*b^4*c*e

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 28.9781, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left ({\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} d e^{4} -{\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} e^{5}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} x -{\left (2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{a x^{2} + b x + c}\right ) +{\left (6 \, a^{4} d^{5} \log \left (e x + d\right ) - 2 \,{\left (a^{4} d^{2} e^{3} - a^{3} b d e^{4} + a^{3} c e^{5}\right )} x^{3} + 3 \,{\left (a^{4} d^{3} e^{2} + a^{2} b c e^{5} -{\left (a^{2} b^{2} - a^{3} c\right )} d e^{4}\right )} x^{2} - 6 \,{\left (a^{4} d^{4} e -{\left (a b^{3} - 2 \, a^{2} b c\right )} d e^{4} +{\left (a b^{2} c - a^{2} c^{2}\right )} e^{5}\right )} x - 3 \,{\left ({\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} d e^{4} -{\left (b^{3} c - 2 \, a b c^{2}\right )} e^{5}\right )} \log \left (a x^{2} + b x + c\right )\right )} \sqrt{b^{2} - 4 \, a c}}{6 \,{\left (a^{5} d^{2} e^{4} - a^{4} b d e^{5} + a^{4} c e^{6}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{6 \,{\left ({\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} d e^{4} -{\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} e^{5}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (6 \, a^{4} d^{5} \log \left (e x + d\right ) - 2 \,{\left (a^{4} d^{2} e^{3} - a^{3} b d e^{4} + a^{3} c e^{5}\right )} x^{3} + 3 \,{\left (a^{4} d^{3} e^{2} + a^{2} b c e^{5} -{\left (a^{2} b^{2} - a^{3} c\right )} d e^{4}\right )} x^{2} - 6 \,{\left (a^{4} d^{4} e -{\left (a b^{3} - 2 \, a^{2} b c\right )} d e^{4} +{\left (a b^{2} c - a^{2} c^{2}\right )} e^{5}\right )} x - 3 \,{\left ({\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} d e^{4} -{\left (b^{3} c - 2 \, a b c^{2}\right )} e^{5}\right )} \log \left (a x^{2} + b x + c\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{6 \,{\left (a^{5} d^{2} e^{4} - a^{4} b d e^{5} + a^{4} c e^{6}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(3*((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d*e^4 - (b^4*c - 4*a*b^2*c^2 + 2*a^2*c
^3)*e^5)*log(-(b^3 - 4*a*b*c + 2*(a*b^2 - 4*a^2*c)*x - (2*a^2*x^2 + 2*a*b*x + b^
2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(a*x^2 + b*x + c)) + (6*a^4*d^5*log(e*x + d) - 2*(
a^4*d^2*e^3 - a^3*b*d*e^4 + a^3*c*e^5)*x^3 + 3*(a^4*d^3*e^2 + a^2*b*c*e^5 - (a^2
*b^2 - a^3*c)*d*e^4)*x^2 - 6*(a^4*d^4*e - (a*b^3 - 2*a^2*b*c)*d*e^4 + (a*b^2*c -
a^2*c^2)*e^5)*x - 3*((b^4 - 3*a*b^2*c + a^2*c^2)*d*e^4 - (b^3*c - 2*a*b*c^2)*e^
5)*log(a*x^2 + b*x + c))*sqrt(b^2 - 4*a*c))/((a^5*d^2*e^4 - a^4*b*d*e^5 + a^4*c*
e^6)*sqrt(b^2 - 4*a*c)), -1/6*(6*((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d*e^4 - (b^4*c
- 4*a*b^2*c^2 + 2*a^2*c^3)*e^5)*arctan(-sqrt(-b^2 + 4*a*c)*(2*a*x + b)/(b^2 - 4
*a*c)) + (6*a^4*d^5*log(e*x + d) - 2*(a^4*d^2*e^3 - a^3*b*d*e^4 + a^3*c*e^5)*x^3
+ 3*(a^4*d^3*e^2 + a^2*b*c*e^5 - (a^2*b^2 - a^3*c)*d*e^4)*x^2 - 6*(a^4*d^4*e -
(a*b^3 - 2*a^2*b*c)*d*e^4 + (a*b^2*c - a^2*c^2)*e^5)*x - 3*((b^4 - 3*a*b^2*c + a
^2*c^2)*d*e^4 - (b^3*c - 2*a*b*c^2)*e^5)*log(a*x^2 + b*x + c))*sqrt(-b^2 + 4*a*c
))/((a^5*d^2*e^4 - a^4*b*d*e^5 + a^4*c*e^6)*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(x**3/(a+c/x**2+b/x)/(e*x+d),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.291317, size = 398, normalized size = 1.42 \[ -\frac{d^{5}{\rm ln}\left ({\left | x e + d \right |}\right )}{a d^{2} e^{4} - b d e^{5} + c e^{6}} + \frac{{\left (b^{4} d - 3 \, a b^{2} c d + a^{2} c^{2} d - b^{3} c e + 2 \, a b c^{2} e\right )}{\rm ln}\left (a x^{2} + b x + c\right )}{2 \,{\left (a^{5} d^{2} - a^{4} b d e + a^{4} c e^{2}\right )}} - \frac{{\left (b^{5} d - 5 \, a b^{3} c d + 5 \, a^{2} b c^{2} d - b^{4} c e + 4 \, a b^{2} c^{2} e - 2 \, a^{2} c^{3} e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} d^{2} - a^{4} b d e + a^{4} c e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (2 \, a^{2} x^{3} e^{2} - 3 \, a^{2} d x^{2} e + 6 \, a^{2} d^{2} x - 3 \, a b x^{2} e^{2} + 6 \, a b d x e + 6 \, b^{2} x e^{2} - 6 \, a c x e^{2}\right )} e^{\left (-3\right )}}{6 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d^5*ln(abs(x*e + d))/(a*d^2*e^4 - b*d*e^5 + c*e^6) + 1/2*(b^4*d - 3*a*b^2*c*d +
a^2*c^2*d - b^3*c*e + 2*a*b*c^2*e)*ln(a*x^2 + b*x + c)/(a^5*d^2 - a^4*b*d*e + a
^4*c*e^2) - (b^5*d - 5*a*b^3*c*d + 5*a^2*b*c^2*d - b^4*c*e + 4*a*b^2*c^2*e - 2*a
^2*c^3*e)*arctan((2*a*x + b)/sqrt(-b^2 + 4*a*c))/((a^5*d^2 - a^4*b*d*e + a^4*c*e
^2)*sqrt(-b^2 + 4*a*c)) + 1/6*(2*a^2*x^3*e^2 - 3*a^2*d*x^2*e + 6*a^2*d^2*x - 3*a
*b*x^2*e^2 + 6*a*b*d*x*e + 6*b^2*x*e^2 - 6*a*c*x*e^2)*e^(-3)/a^3