3.330 \(\int \frac{x^5}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{a}{3 b^2 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right )}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0566092, antiderivative size = 33, 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 \[ \frac{a}{3 b^2 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^3)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.29711, size = 26, normalized size = 0.79 \[ \frac{a}{3 b^{2} \left (a + b x^{3}\right )} + \frac{\log{\left (a + b x^{3} \right )}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**3+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0159, size = 27, normalized size = 0.82 \[ \frac{\frac{a}{a+b x^3}+\log \left (a+b x^3\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^3)^2,x]

[Out]

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

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Maple [A]  time = 0.007, size = 30, normalized size = 0.9 \[{\frac{a}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^3+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.44014, size = 43, normalized size = 1.3 \[ \frac{a}{3 \,{\left (b^{3} x^{3} + a b^{2}\right )}} + \frac{\log \left (b x^{3} + a\right )}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.210681, size = 47, normalized size = 1.42 \[ \frac{{\left (b x^{3} + a\right )} \log \left (b x^{3} + a\right ) + a}{3 \,{\left (b^{3} x^{3} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.59763, size = 29, normalized size = 0.88 \[ \frac{a}{3 a b^{2} + 3 b^{3} x^{3}} + \frac{\log{\left (a + b x^{3} \right )}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**3+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215381, size = 65, normalized size = 1.97 \[ -\frac{\frac{{\rm ln}\left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x^{3} + a\right )} b}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

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