Optimal. Leaf size=77 \[ \frac {3 b \log \left (a+b x^n\right )}{a^4 n}-\frac {3 b \log (x)}{a^4}-\frac {2 b}{a^3 n \left (a+b x^n\right )}-\frac {x^{-n}}{a^3 n}-\frac {b}{2 a^2 n \left (a+b x^n\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 77, 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} \[ -\frac {2 b}{a^3 n \left (a+b x^n\right )}-\frac {b}{2 a^2 n \left (a+b x^n\right )^2}+\frac {3 b \log \left (a+b x^n\right )}{a^4 n}-\frac {3 b \log (x)}{a^4}-\frac {x^{-n}}{a^3 n} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1-n}}{\left (a+b x^n\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^3} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^2}-\frac {3 b}{a^4 x}+\frac {b^2}{a^2 (a+b x)^3}+\frac {2 b^2}{a^3 (a+b x)^2}+\frac {3 b^2}{a^4 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n}}{a^3 n}-\frac {b}{2 a^2 n \left (a+b x^n\right )^2}-\frac {2 b}{a^3 n \left (a+b x^n\right )}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log \left (a+b x^n\right )}{a^4 n}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 58, normalized size = 0.75 \[ -\frac {\frac {a b \left (5 a+4 b x^n\right )}{\left (a+b x^n\right )^2}-6 b \log \left (a+b x^n\right )+2 a x^{-n}+6 b n \log (x)}{2 a^4 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 139, normalized size = 1.81 \[ -\frac {6 \, b^{3} n x^{3 \, n} \log \relax (x) + 2 \, a^{3} + 6 \, {\left (2 \, a b^{2} n \log \relax (x) + a b^{2}\right )} x^{2 \, n} + 3 \, {\left (2 \, a^{2} b n \log \relax (x) + 3 \, a^{2} b\right )} x^{n} - 6 \, {\left (b^{3} x^{3 \, n} + 2 \, a b^{2} x^{2 \, n} + a^{2} b x^{n}\right )} \log \left (b x^{n} + a\right )}{2 \, {\left (a^{4} b^{2} n x^{3 \, n} + 2 \, a^{5} b n x^{2 \, n} + a^{6} n x^{n}\right )}} \]
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 \[ \int \frac {x^{-n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 132, normalized size = 1.71 \[ \frac {\left (-\frac {3 b \,{\mathrm e}^{n \ln \relax (x )} \ln \relax (x )}{a^{2}}-\frac {6 b^{2} {\mathrm e}^{2 n \ln \relax (x )} \ln \relax (x )}{a^{3}}-\frac {3 b^{3} {\mathrm e}^{3 n \ln \relax (x )} \ln \relax (x )}{a^{4}}+\frac {6 b^{2} {\mathrm e}^{2 n \ln \relax (x )}}{a^{3} n}+\frac {9 b^{3} {\mathrm e}^{3 n \ln \relax (x )}}{2 a^{4} n}-\frac {1}{a n}\right ) {\mathrm e}^{-n \ln \relax (x )}}{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )^{2}}+\frac {3 b \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{a^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 91, normalized size = 1.18 \[ -\frac {6 \, b^{2} x^{2 \, n} + 9 \, a b x^{n} + 2 \, a^{2}}{2 \, {\left (a^{3} b^{2} n x^{3 \, n} + 2 \, a^{4} b n x^{2 \, n} + a^{5} n x^{n}\right )}} - \frac {3 \, b \log \relax (x)}{a^{4}} + \frac {3 \, b \log \left (\frac {b x^{n} + a}{b}\right )}{a^{4} n} \]
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 \[ \int \frac {1}{x^{n+1}\,{\left (a+b\,x^n\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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