3.1272 \(\int \frac{1}{x^{11} \left (a+b x^5\right )} \, dx\)

Optimal. Leaf size=49 \[ -\frac{b^2 \log \left (a+b x^5\right )}{5 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{5 a^2 x^5}-\frac{1}{10 a x^{10}} \]

[Out]

-1/(10*a*x^10) + b/(5*a^2*x^5) + (b^2*Log[x])/a^3 - (b^2*Log[a + b*x^5])/(5*a^3)

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Rubi [A]  time = 0.0654487, antiderivative size = 49, 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 \[ -\frac{b^2 \log \left (a+b x^5\right )}{5 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{5 a^2 x^5}-\frac{1}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^11*(a + b*x^5)),x]

[Out]

-1/(10*a*x^10) + b/(5*a^2*x^5) + (b^2*Log[x])/a^3 - (b^2*Log[a + b*x^5])/(5*a^3)

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Rubi in Sympy [A]  time = 10.5353, size = 48, normalized size = 0.98 \[ - \frac{1}{10 a x^{10}} + \frac{b}{5 a^{2} x^{5}} + \frac{b^{2} \log{\left (x^{5} \right )}}{5 a^{3}} - \frac{b^{2} \log{\left (a + b x^{5} \right )}}{5 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**11/(b*x**5+a),x)

[Out]

-1/(10*a*x**10) + b/(5*a**2*x**5) + b**2*log(x**5)/(5*a**3) - b**2*log(a + b*x**
5)/(5*a**3)

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Mathematica [A]  time = 0.0119805, size = 49, normalized size = 1. \[ -\frac{b^2 \log \left (a+b x^5\right )}{5 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{5 a^2 x^5}-\frac{1}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^11*(a + b*x^5)),x]

[Out]

-1/(10*a*x^10) + b/(5*a^2*x^5) + (b^2*Log[x])/a^3 - (b^2*Log[a + b*x^5])/(5*a^3)

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Maple [A]  time = 0.011, size = 44, normalized size = 0.9 \[ -{\frac{1}{10\,a{x}^{10}}}+{\frac{b}{5\,{x}^{5}{a}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{3}}}-{\frac{{b}^{2}\ln \left ( b{x}^{5}+a \right ) }{5\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^11/(b*x^5+a),x)

[Out]

-1/10/a/x^10+1/5*b/x^5/a^2+b^2*ln(x)/a^3-1/5*b^2*ln(b*x^5+a)/a^3

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Maxima [A]  time = 1.44414, size = 63, normalized size = 1.29 \[ -\frac{b^{2} \log \left (b x^{5} + a\right )}{5 \, a^{3}} + \frac{b^{2} \log \left (x^{5}\right )}{5 \, a^{3}} + \frac{2 \, b x^{5} - a}{10 \, a^{2} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)*x^11),x, algorithm="maxima")

[Out]

-1/5*b^2*log(b*x^5 + a)/a^3 + 1/5*b^2*log(x^5)/a^3 + 1/10*(2*b*x^5 - a)/(a^2*x^1
0)

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Fricas [A]  time = 0.22277, size = 61, normalized size = 1.24 \[ -\frac{2 \, b^{2} x^{10} \log \left (b x^{5} + a\right ) - 10 \, b^{2} x^{10} \log \left (x\right ) - 2 \, a b x^{5} + a^{2}}{10 \, a^{3} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)*x^11),x, algorithm="fricas")

[Out]

-1/10*(2*b^2*x^10*log(b*x^5 + a) - 10*b^2*x^10*log(x) - 2*a*b*x^5 + a^2)/(a^3*x^
10)

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Sympy [A]  time = 18.8188, size = 42, normalized size = 0.86 \[ \frac{- a + 2 b x^{5}}{10 a^{2} x^{10}} + \frac{b^{2} \log{\left (x \right )}}{a^{3}} - \frac{b^{2} \log{\left (\frac{a}{b} + x^{5} \right )}}{5 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**11/(b*x**5+a),x)

[Out]

(-a + 2*b*x**5)/(10*a**2*x**10) + b**2*log(x)/a**3 - b**2*log(a/b + x**5)/(5*a**
3)

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GIAC/XCAS [A]  time = 0.238084, size = 74, normalized size = 1.51 \[ -\frac{b^{2}{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{3}} + \frac{b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{3 \, b^{2} x^{10} - 2 \, a b x^{5} + a^{2}}{10 \, a^{3} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)*x^11),x, algorithm="giac")

[Out]

-1/5*b^2*ln(abs(b*x^5 + a))/a^3 + b^2*ln(abs(x))/a^3 - 1/10*(3*b^2*x^10 - 2*a*b*
x^5 + a^2)/(a^3*x^10)