3.1607 \(\int \frac{x^2}{a+\frac{b}{x}} \, dx\)

Optimal. Leaf size=44 \[ -\frac{b^3 \log (a x+b)}{a^4}+\frac{b^2 x}{a^3}-\frac{b x^2}{2 a^2}+\frac{x^3}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.0618754, antiderivative size = 44, 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 \[ -\frac{b^3 \log (a x+b)}{a^4}+\frac{b^2 x}{a^3}-\frac{b x^2}{2 a^2}+\frac{x^3}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(a+b/x),x)

[Out]

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

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Mathematica [A]  time = 0.00632414, size = 44, normalized size = 1. \[ -\frac{b^3 \log (a x+b)}{a^4}+\frac{b^2 x}{a^3}-\frac{b x^2}{2 a^2}+\frac{x^3}{3 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b/x),x]

[Out]

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

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Maple [A]  time = 0.004, size = 41, normalized size = 0.9 \[{\frac{{b}^{2}x}{{a}^{3}}}-{\frac{b{x}^{2}}{2\,{a}^{2}}}+{\frac{{x}^{3}}{3\,a}}-{\frac{{b}^{3}\ln \left ( ax+b \right ) }{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x),x)

[Out]

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

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Maxima [A]  time = 1.43947, size = 57, normalized size = 1.3 \[ -\frac{b^{3} \log \left (a x + b\right )}{a^{4}} + \frac{2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{6 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a + b/x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.22032, size = 55, normalized size = 1.25 \[ \frac{2 \, a^{3} x^{3} - 3 \, a^{2} b x^{2} + 6 \, a b^{2} x - 6 \, b^{3} \log \left (a x + b\right )}{6 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a + b/x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.12002, size = 37, normalized size = 0.84 \[ \frac{x^{3}}{3 a} - \frac{b x^{2}}{2 a^{2}} + \frac{b^{2} x}{a^{3}} - \frac{b^{3} \log{\left (a x + b \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b/x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223908, size = 58, normalized size = 1.32 \[ -\frac{b^{3}{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{4}} + \frac{2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{6 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a + b/x),x, algorithm="giac")

[Out]

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