Optimal. Leaf size=101 \[ -\frac {6 b^2 \log \left (a+b x^n\right )}{a^5 n}+\frac {6 b^2 \log (x)}{a^5}+\frac {3 b^2}{a^4 n \left (a+b x^n\right )}+\frac {3 b x^{-n}}{a^4 n}+\frac {b^2}{2 a^3 n \left (a+b x^n\right )^2}-\frac {x^{-2 n}}{2 a^3 n} \]
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Rubi [A] time = 0.06, antiderivative size = 101, 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} \[ \frac {3 b^2}{a^4 n \left (a+b x^n\right )}+\frac {b^2}{2 a^3 n \left (a+b x^n\right )^2}-\frac {6 b^2 \log \left (a+b x^n\right )}{a^5 n}+\frac {6 b^2 \log (x)}{a^5}+\frac {3 b x^{-n}}{a^4 n}-\frac {x^{-2 n}}{2 a^3 n} \]
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 )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)^3} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^3}-\frac {3 b}{a^4 x^2}+\frac {6 b^2}{a^5 x}-\frac {b^3}{a^3 (a+b x)^3}-\frac {3 b^3}{a^4 (a+b x)^2}-\frac {6 b^3}{a^5 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-2 n}}{2 a^3 n}+\frac {3 b x^{-n}}{a^4 n}+\frac {b^2}{2 a^3 n \left (a+b x^n\right )^2}+\frac {3 b^2}{a^4 n \left (a+b x^n\right )}+\frac {6 b^2 \log (x)}{a^5}-\frac {6 b^2 \log \left (a+b x^n\right )}{a^5 n}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 79, normalized size = 0.78 \[ \frac {\frac {a x^{-2 n} \left (a+2 b x^n\right ) \left (-a^2+6 a b x^n+6 b^2 x^{2 n}\right )}{\left (a+b x^n\right )^2}-12 b^2 \log \left (a+b x^n\right )+12 b^2 n \log (x)}{2 a^5 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 160, normalized size = 1.58 \[ \frac {12 \, b^{4} n x^{4 \, n} \log \relax (x) + 4 \, a^{3} b x^{n} - a^{4} + 12 \, {\left (2 \, a b^{3} n \log \relax (x) + a b^{3}\right )} x^{3 \, n} + 6 \, {\left (2 \, a^{2} b^{2} n \log \relax (x) + 3 \, a^{2} b^{2}\right )} x^{2 \, n} - 12 \, {\left (b^{4} x^{4 \, n} + 2 \, a b^{3} x^{3 \, n} + a^{2} b^{2} x^{2 \, n}\right )} \log \left (b x^{n} + a\right )}{2 \, {\left (a^{5} b^{2} n x^{4 \, n} + 2 \, a^{6} b n x^{3 \, n} + a^{7} n x^{2 \, n}\right )}} \]
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 \[ \int \frac {x^{-2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 152, normalized size = 1.50 \[ \frac {\left (\frac {6 b^{2} {\mathrm e}^{2 n \ln \relax (x )} \ln \relax (x )}{a^{3}}+\frac {12 b^{3} {\mathrm e}^{3 n \ln \relax (x )} \ln \relax (x )}{a^{4}}+\frac {6 b^{4} {\mathrm e}^{4 n \ln \relax (x )} \ln \relax (x )}{a^{5}}+\frac {2 b \,{\mathrm e}^{n \ln \relax (x )}}{a^{2} n}+\frac {9 b^{2} {\mathrm e}^{2 n \ln \relax (x )}}{a^{3} n}+\frac {6 b^{3} {\mathrm e}^{3 n \ln \relax (x )}}{a^{4} n}-\frac {1}{2 a n}\right ) {\mathrm e}^{-2 n \ln \relax (x )}}{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )^{2}}-\frac {6 b^{2} \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{a^{5} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 110, normalized size = 1.09 \[ \frac {12 \, b^{3} x^{3 \, n} + 18 \, a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} - a^{3}}{2 \, {\left (a^{4} b^{2} n x^{4 \, n} + 2 \, a^{5} b n x^{3 \, n} + a^{6} n x^{2 \, n}\right )}} + \frac {6 \, b^{2} \log \relax (x)}{a^{5}} - \frac {6 \, b^{2} \log \left (\frac {b x^{n} + a}{b}\right )}{a^{5} n} \]
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 \[ \int \frac {1}{x^{2\,n+1}\,{\left (a+b\,x^n\right )}^3} \,d x \]
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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