Optimal. Leaf size=78 \[ -\frac {3 b^2 \log \left (a+b x^n\right )}{a^4 n}+\frac {3 b^2 \log (x)}{a^4}+\frac {b^2}{a^3 n \left (a+b x^n\right )}+\frac {2 b x^{-n}}{a^3 n}-\frac {x^{-2 n}}{2 a^2 n} \]
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Rubi [A] time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 44} \[ \frac {b^2}{a^3 n \left (a+b x^n\right )}-\frac {3 b^2 \log \left (a+b x^n\right )}{a^4 n}+\frac {3 b^2 \log (x)}{a^4}+\frac {2 b x^{-n}}{a^3 n}-\frac {x^{-2 n}}{2 a^2 n} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1-2 n}}{\left (a+b x^n\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac {\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^n\right )}{n}\\ &=-\frac {x^{-2 n}}{2 a^2 n}+\frac {2 b x^{-n}}{a^3 n}+\frac {b^2}{a^3 n \left (a+b x^n\right )}+\frac {3 b^2 \log (x)}{a^4}-\frac {3 b^2 \log \left (a+b x^n\right )}{a^4 n}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 65, normalized size = 0.83 \[ \frac {-6 b^2 \log \left (a+b x^n\right )+a \left (\frac {2 b^2}{a+b x^n}-a x^{-2 n}+4 b x^{-n}\right )+6 b^2 n \log (x)}{2 a^4 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 105, normalized size = 1.35 \[ \frac {6 \, b^{3} n x^{3 \, n} \log \relax (x) + 3 \, a^{2} b x^{n} - a^{3} + 6 \, {\left (a b^{2} n \log \relax (x) + a b^{2}\right )} x^{2 \, n} - 6 \, {\left (b^{3} x^{3 \, n} + a b^{2} x^{2 \, n}\right )} \log \left (b x^{n} + a\right )}{2 \, {\left (a^{4} b n x^{3 \, n} + a^{5} n x^{2 \, n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{-2 \, n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 117, normalized size = 1.50 \[ \frac {\left (\frac {3 b^{2} {\mathrm e}^{2 n \ln \relax (x )} \ln \relax (x )}{a^{3}}+\frac {3 b^{3} {\mathrm e}^{3 n \ln \relax (x )} \ln \relax (x )}{a^{4}}+\frac {3 b \,{\mathrm e}^{n \ln \relax (x )}}{2 a^{2} n}-\frac {3 b^{3} {\mathrm e}^{3 n \ln \relax (x )}}{a^{4} n}-\frac {1}{2 a n}\right ) {\mathrm e}^{-2 n \ln \relax (x )}}{b \,{\mathrm e}^{n \ln \relax (x )}+a}-\frac {3 b^{2} \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{a^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{2\,n+1}\,{\left (a+b\,x^n\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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