3.1271 \(\int \frac{1}{x^6 \left (a+b x^5\right )} \, dx\)

Optimal. Leaf size=35 \[ \frac{b \log \left (a+b x^5\right )}{5 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{5 a x^5} \]

[Out]

-1/(5*a*x^5) - (b*Log[x])/a^2 + (b*Log[a + b*x^5])/(5*a^2)

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Rubi [A]  time = 0.0538295, antiderivative size = 35, 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 \[ \frac{b \log \left (a+b x^5\right )}{5 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^5)),x]

[Out]

-1/(5*a*x^5) - (b*Log[x])/a^2 + (b*Log[a + b*x^5])/(5*a^2)

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Rubi in Sympy [A]  time = 8.0728, size = 34, normalized size = 0.97 \[ - \frac{1}{5 a x^{5}} - \frac{b \log{\left (x^{5} \right )}}{5 a^{2}} + \frac{b \log{\left (a + b x^{5} \right )}}{5 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(5*a*x**5) - b*log(x**5)/(5*a**2) + b*log(a + b*x**5)/(5*a**2)

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Mathematica [A]  time = 0.0113012, size = 35, normalized size = 1. \[ \frac{b \log \left (a+b x^5\right )}{5 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{5 a x^5} \]

Antiderivative was successfully verified.

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

[Out]

-1/(5*a*x^5) - (b*Log[x])/a^2 + (b*Log[a + b*x^5])/(5*a^2)

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Maple [A]  time = 0.009, size = 32, normalized size = 0.9 \[ -{\frac{1}{5\,a{x}^{5}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}}+{\frac{b\ln \left ( b{x}^{5}+a \right ) }{5\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^5+a),x)

[Out]

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

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Maxima [A]  time = 1.43319, size = 45, normalized size = 1.29 \[ \frac{b \log \left (b x^{5} + a\right )}{5 \, a^{2}} - \frac{b \log \left (x^{5}\right )}{5 \, a^{2}} - \frac{1}{5 \, a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*log(b*x^5 + a)/a^2 - 1/5*b*log(x^5)/a^2 - 1/5/(a*x^5)

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Fricas [A]  time = 0.219342, size = 45, normalized size = 1.29 \[ \frac{b x^{5} \log \left (b x^{5} + a\right ) - 5 \, b x^{5} \log \left (x\right ) - a}{5 \, a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(b*x^5*log(b*x^5 + a) - 5*b*x^5*log(x) - a)/(a^2*x^5)

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Sympy [A]  time = 2.9688, size = 31, normalized size = 0.89 \[ - \frac{1}{5 a x^{5}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{5} \right )}}{5 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**5+a),x)

[Out]

-1/(5*a*x**5) - b*log(x)/a**2 + b*log(a/b + x**5)/(5*a**2)

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GIAC/XCAS [A]  time = 0.231607, size = 57, normalized size = 1.63 \[ \frac{b{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{2}} - \frac{b{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{b x^{5} - a}{5 \, a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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