Optimal. Leaf size=69 \[ -\frac {3 b^2 \log \left (a+b x^5\right )}{5 a^4}+\frac {3 b^2 \log (x)}{a^4}+\frac {b^2}{5 a^3 \left (a+b x^5\right )}+\frac {2 b}{5 a^3 x^5}-\frac {1}{10 a^2 x^{10}} \]
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Rubi [A] time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac {b^2}{5 a^3 \left (a+b x^5\right )}-\frac {3 b^2 \log \left (a+b x^5\right )}{5 a^4}+\frac {3 b^2 \log (x)}{a^4}+\frac {2 b}{5 a^3 x^5}-\frac {1}{10 a^2 x^{10}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^{11} \left (a+b x^5\right )^2} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)^2} \, dx,x,x^5\right )\\ &=\frac {1}{5} \operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^3}-\frac {2 b}{a^3 x^2}+\frac {3 b^2}{a^4 x}-\frac {b^3}{a^3 (a+b x)^2}-\frac {3 b^3}{a^4 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac {1}{10 a^2 x^{10}}+\frac {2 b}{5 a^3 x^5}+\frac {b^2}{5 a^3 \left (a+b x^5\right )}+\frac {3 b^2 \log (x)}{a^4}-\frac {3 b^2 \log \left (a+b x^5\right )}{5 a^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 57, normalized size = 0.83 \[ \frac {-6 b^2 \log \left (a+b x^5\right )+a \left (\frac {2 b^2}{a+b x^5}-\frac {a}{x^{10}}+\frac {4 b}{x^5}\right )+30 b^2 \log (x)}{10 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 90, normalized size = 1.30 \[ \frac {6 \, a b^{2} x^{10} + 3 \, a^{2} b x^{5} - a^{3} - 6 \, {\left (b^{3} x^{15} + a b^{2} x^{10}\right )} \log \left (b x^{5} + a\right ) + 30 \, {\left (b^{3} x^{15} + a b^{2} x^{10}\right )} \log \relax (x)}{10 \, {\left (a^{4} b x^{15} + a^{5} x^{10}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 85, normalized size = 1.23 \[ -\frac {3 \, b^{2} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{4}} + \frac {3 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {3 \, b^{3} x^{5} + 4 \, a b^{2}}{5 \, {\left (b x^{5} + a\right )} a^{4}} - \frac {9 \, b^{2} x^{10} - 4 \, a b x^{5} + a^{2}}{10 \, a^{4} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 62, normalized size = 0.90 \[ \frac {b^{2}}{5 \left (b \,x^{5}+a \right ) a^{3}}+\frac {3 b^{2} \ln \relax (x )}{a^{4}}-\frac {3 b^{2} \ln \left (b \,x^{5}+a \right )}{5 a^{4}}+\frac {2 b}{5 a^{3} x^{5}}-\frac {1}{10 a^{2} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 70, normalized size = 1.01 \[ \frac {6 \, b^{2} x^{10} + 3 \, a b x^{5} - a^{2}}{10 \, {\left (a^{3} b x^{15} + a^{4} x^{10}\right )}} - \frac {3 \, b^{2} \log \left (b x^{5} + a\right )}{5 \, a^{4}} + \frac {3 \, b^{2} \log \left (x^{5}\right )}{5 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 67, normalized size = 0.97 \[ \frac {\frac {3\,b\,x^5}{10\,a^2}-\frac {1}{10\,a}+\frac {3\,b^2\,x^{10}}{5\,a^3}}{b\,x^{15}+a\,x^{10}}-\frac {3\,b^2\,\ln \left (b\,x^5+a\right )}{5\,a^4}+\frac {3\,b^2\,\ln \relax (x)}{a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.60, size = 68, normalized size = 0.99 \[ \frac {- a^{2} + 3 a b x^{5} + 6 b^{2} x^{10}}{10 a^{4} x^{10} + 10 a^{3} b x^{15}} + \frac {3 b^{2} \log {\relax (x )}}{a^{4}} - \frac {3 b^{2} \log {\left (\frac {a}{b} + x^{5} \right )}}{5 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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