3.1458 \(\int \frac{1}{x^9 \left (a+b x^8\right )} \, dx\)

Optimal. Leaf size=35 \[ \frac{b \log \left (a+b x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{8 a x^8} \]

[Out]

-1/(8*a*x^8) - (b*Log[x])/a^2 + (b*Log[a + b*x^8])/(8*a^2)

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Rubi [A]  time = 0.0611305, 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 \[ \frac{b \log \left (a+b x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^9*(a + b*x^8)),x]

[Out]

-1/(8*a*x^8) - (b*Log[x])/a^2 + (b*Log[a + b*x^8])/(8*a^2)

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Rubi in Sympy [A]  time = 8.37455, size = 34, normalized size = 0.97 \[ - \frac{1}{8 a x^{8}} - \frac{b \log{\left (x^{8} \right )}}{8 a^{2}} + \frac{b \log{\left (a + b x^{8} \right )}}{8 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**9/(b*x**8+a),x)

[Out]

-1/(8*a*x**8) - b*log(x**8)/(8*a**2) + b*log(a + b*x**8)/(8*a**2)

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Mathematica [A]  time = 0.0123321, size = 35, normalized size = 1. \[ \frac{b \log \left (a+b x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^9*(a + b*x^8)),x]

[Out]

-1/(8*a*x^8) - (b*Log[x])/a^2 + (b*Log[a + b*x^8])/(8*a^2)

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Maple [A]  time = 0.008, size = 32, normalized size = 0.9 \[ -{\frac{1}{8\,a{x}^{8}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}}+{\frac{b\ln \left ( b{x}^{8}+a \right ) }{8\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^9/(b*x^8+a),x)

[Out]

-1/8/a/x^8-b*ln(x)/a^2+1/8*b*ln(b*x^8+a)/a^2

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Maxima [A]  time = 1.43521, size = 45, normalized size = 1.29 \[ \frac{b \log \left (b x^{8} + a\right )}{8 \, a^{2}} - \frac{b \log \left (x^{8}\right )}{8 \, a^{2}} - \frac{1}{8 \, a x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^9),x, algorithm="maxima")

[Out]

1/8*b*log(b*x^8 + a)/a^2 - 1/8*b*log(x^8)/a^2 - 1/8/(a*x^8)

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Fricas [A]  time = 0.224552, size = 45, normalized size = 1.29 \[ \frac{b x^{8} \log \left (b x^{8} + a\right ) - 8 \, b x^{8} \log \left (x\right ) - a}{8 \, a^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^9),x, algorithm="fricas")

[Out]

1/8*(b*x^8*log(b*x^8 + a) - 8*b*x^8*log(x) - a)/(a^2*x^8)

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Sympy [A]  time = 26.8054, size = 31, normalized size = 0.89 \[ - \frac{1}{8 a x^{8}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{8} \right )}}{8 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**9/(b*x**8+a),x)

[Out]

-1/(8*a*x**8) - b*log(x)/a**2 + b*log(a/b + x**8)/(8*a**2)

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GIAC/XCAS [A]  time = 0.228609, size = 58, normalized size = 1.66 \[ -\frac{b{\rm ln}\left (x^{8}\right )}{8 \, a^{2}} + \frac{b{\rm ln}\left ({\left | b x^{8} + a \right |}\right )}{8 \, a^{2}} + \frac{b x^{8} - a}{8 \, a^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^9),x, algorithm="giac")

[Out]

-1/8*b*ln(x^8)/a^2 + 1/8*b*ln(abs(b*x^8 + a))/a^2 + 1/8*(b*x^8 - a)/(a^2*x^8)