3.1606 \(\int \frac{x^3}{a+\frac{b}{x}} \, dx\)

Optimal. Leaf size=57 \[ \frac{b^4 \log (a x+b)}{a^5}-\frac{b^3 x}{a^4}+\frac{b^2 x^2}{2 a^3}-\frac{b x^3}{3 a^2}+\frac{x^4}{4 a} \]

[Out]

-((b^3*x)/a^4) + (b^2*x^2)/(2*a^3) - (b*x^3)/(3*a^2) + x^4/(4*a) + (b^4*Log[b +
a*x])/a^5

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Rubi [A]  time = 0.0770509, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{b^4 \log (a x+b)}{a^5}-\frac{b^3 x}{a^4}+\frac{b^2 x^2}{2 a^3}-\frac{b x^3}{3 a^2}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x),x]

[Out]

-((b^3*x)/a^4) + (b^2*x^2)/(2*a^3) - (b*x^3)/(3*a^2) + x^4/(4*a) + (b^4*Log[b +
a*x])/a^5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - b^{3} \int \frac{1}{a^{4}}\, dx + \frac{x^{4}}{4 a} - \frac{b x^{3}}{3 a^{2}} + \frac{b^{2} \int x\, dx}{a^{3}} + \frac{b^{4} \log{\left (a x + b \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a+b/x),x)

[Out]

-b**3*Integral(a**(-4), x) + x**4/(4*a) - b*x**3/(3*a**2) + b**2*Integral(x, x)/
a**3 + b**4*log(a*x + b)/a**5

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Mathematica [A]  time = 0.00749848, size = 57, normalized size = 1. \[ \frac{b^4 \log (a x+b)}{a^5}-\frac{b^3 x}{a^4}+\frac{b^2 x^2}{2 a^3}-\frac{b x^3}{3 a^2}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b/x),x]

[Out]

-((b^3*x)/a^4) + (b^2*x^2)/(2*a^3) - (b*x^3)/(3*a^2) + x^4/(4*a) + (b^4*Log[b +
a*x])/a^5

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Maple [A]  time = 0.004, size = 52, normalized size = 0.9 \[ -{\frac{{b}^{3}x}{{a}^{4}}}+{\frac{{b}^{2}{x}^{2}}{2\,{a}^{3}}}-{\frac{b{x}^{3}}{3\,{a}^{2}}}+{\frac{{x}^{4}}{4\,a}}+{\frac{{b}^{4}\ln \left ( ax+b \right ) }{{a}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x),x)

[Out]

-b^3*x/a^4+1/2*b^2*x^2/a^3-1/3*b*x^3/a^2+1/4*x^4/a+b^4*ln(a*x+b)/a^5

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Maxima [A]  time = 1.43825, size = 70, normalized size = 1.23 \[ \frac{b^{4} \log \left (a x + b\right )}{a^{5}} + \frac{3 \, a^{3} x^{4} - 4 \, a^{2} b x^{3} + 6 \, a b^{2} x^{2} - 12 \, b^{3} x}{12 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x),x, algorithm="maxima")

[Out]

b^4*log(a*x + b)/a^5 + 1/12*(3*a^3*x^4 - 4*a^2*b*x^3 + 6*a*b^2*x^2 - 12*b^3*x)/a
^4

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Fricas [A]  time = 0.219245, size = 70, normalized size = 1.23 \[ \frac{3 \, a^{4} x^{4} - 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a b^{3} x + 12 \, b^{4} \log \left (a x + b\right )}{12 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x),x, algorithm="fricas")

[Out]

1/12*(3*a^4*x^4 - 4*a^3*b*x^3 + 6*a^2*b^2*x^2 - 12*a*b^3*x + 12*b^4*log(a*x + b)
)/a^5

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Sympy [A]  time = 1.15064, size = 49, normalized size = 0.86 \[ \frac{x^{4}}{4 a} - \frac{b x^{3}}{3 a^{2}} + \frac{b^{2} x^{2}}{2 a^{3}} - \frac{b^{3} x}{a^{4}} + \frac{b^{4} \log{\left (a x + b \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x),x)

[Out]

x**4/(4*a) - b*x**3/(3*a**2) + b**2*x**2/(2*a**3) - b**3*x/a**4 + b**4*log(a*x +
b)/a**5

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GIAC/XCAS [A]  time = 0.228316, size = 72, normalized size = 1.26 \[ \frac{b^{4}{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{5}} + \frac{3 \, a^{3} x^{4} - 4 \, a^{2} b x^{3} + 6 \, a b^{2} x^{2} - 12 \, b^{3} x}{12 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x),x, algorithm="giac")

[Out]

b^4*ln(abs(a*x + b))/a^5 + 1/12*(3*a^3*x^4 - 4*a^2*b*x^3 + 6*a*b^2*x^2 - 12*b^3*
x)/a^4