3.1292 \(\int \frac{1}{x^6 \left (3+b x^5\right )} \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{45} b \log \left (b x^5+3\right )-\frac{1}{9} b \log (x)-\frac{1}{15 x^5} \]

[Out]

-1/(15*x^5) - (b*Log[x])/9 + (b*Log[3 + b*x^5])/45

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Rubi [A]  time = 0.0447055, antiderivative size = 28, 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 \[ \frac{1}{45} b \log \left (b x^5+3\right )-\frac{1}{9} b \log (x)-\frac{1}{15 x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(3 + b*x^5)),x]

[Out]

-1/(15*x^5) - (b*Log[x])/9 + (b*Log[3 + b*x^5])/45

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Rubi in Sympy [A]  time = 6.23248, size = 26, normalized size = 0.93 \[ - \frac{b \log{\left (x^{5} \right )}}{45} + \frac{b \log{\left (b x^{5} + 3 \right )}}{45} - \frac{1}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**5+3),x)

[Out]

-b*log(x**5)/45 + b*log(b*x**5 + 3)/45 - 1/(15*x**5)

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Mathematica [A]  time = 0.0083346, size = 28, normalized size = 1. \[ \frac{1}{45} b \log \left (b x^5+3\right )-\frac{1}{9} b \log (x)-\frac{1}{15 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(3 + b*x^5)),x]

[Out]

-1/(15*x^5) - (b*Log[x])/9 + (b*Log[3 + b*x^5])/45

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Maple [A]  time = 0.009, size = 23, normalized size = 0.8 \[ -{\frac{1}{15\,{x}^{5}}}-{\frac{b\ln \left ( x \right ) }{9}}+{\frac{b\ln \left ( b{x}^{5}+3 \right ) }{45}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^5+3),x)

[Out]

-1/15/x^5-1/9*b*ln(x)+1/45*b*ln(b*x^5+3)

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Maxima [A]  time = 1.43884, size = 32, normalized size = 1.14 \[ \frac{1}{45} \, b \log \left (b x^{5} + 3\right ) - \frac{1}{45} \, b \log \left (x^{5}\right ) - \frac{1}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 3)*x^6),x, algorithm="maxima")

[Out]

1/45*b*log(b*x^5 + 3) - 1/45*b*log(x^5) - 1/15/x^5

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Fricas [A]  time = 0.214911, size = 38, normalized size = 1.36 \[ \frac{b x^{5} \log \left (b x^{5} + 3\right ) - 5 \, b x^{5} \log \left (x\right ) - 3}{45 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 3)*x^6),x, algorithm="fricas")

[Out]

1/45*(b*x^5*log(b*x^5 + 3) - 5*b*x^5*log(x) - 3)/x^5

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Sympy [A]  time = 2.72954, size = 24, normalized size = 0.86 \[ - \frac{b \log{\left (x \right )}}{9} + \frac{b \log{\left (x^{5} + \frac{3}{b} \right )}}{45} - \frac{1}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**5+3),x)

[Out]

-b*log(x)/9 + b*log(x**5 + 3/b)/45 - 1/(15*x**5)

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GIAC/XCAS [A]  time = 0.284526, size = 42, normalized size = 1.5 \[ \frac{1}{45} \, b{\rm ln}\left ({\left | b x^{5} + 3 \right |}\right ) - \frac{1}{9} \, b{\rm ln}\left ({\left | x \right |}\right ) + \frac{b x^{5} - 3}{45 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 3)*x^6),x, algorithm="giac")

[Out]

1/45*b*ln(abs(b*x^5 + 3)) - 1/9*b*ln(abs(x)) + 1/45*(b*x^5 - 3)/x^5