Optimal. Leaf size=56 \[ -\frac {a^2}{2 b^3 n \left (a+b x^n\right )^2}+\frac {2 a}{b^3 n \left (a+b x^n\right )}+\frac {\log \left (a+b x^n\right )}{b^3 n} \]
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Rubi [A] time = 0.03, antiderivative size = 56, 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, 43} \[ -\frac {a^2}{2 b^3 n \left (a+b x^n\right )^2}+\frac {2 a}{b^3 n \left (a+b x^n\right )}+\frac {\log \left (a+b x^n\right )}{b^3 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1+3 n}}{\left (a+b x^n\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^3} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^3}-\frac {2 a}{b^2 (a+b x)^2}+\frac {1}{b^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^2}{2 b^3 n \left (a+b x^n\right )^2}+\frac {2 a}{b^3 n \left (a+b x^n\right )}+\frac {\log \left (a+b x^n\right )}{b^3 n}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 42, normalized size = 0.75 \[ \frac {\frac {a \left (3 a+4 b x^n\right )}{\left (a+b x^n\right )^2}+2 \log \left (a+b x^n\right )}{2 b^3 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 76, normalized size = 1.36 \[ \frac {4 \, a b x^{n} + 3 \, a^{2} + 2 \, {\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{2 \, {\left (b^{5} n x^{2 \, n} + 2 \, a b^{4} n x^{n} + a^{2} b^{3} n\right )}} \]
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 \[ \int \frac {x^{3 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 57, normalized size = 1.02 \[ \frac {\ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{b^{3} n}+\frac {\frac {2 a \,{\mathrm e}^{n \ln \relax (x )}}{b^{2} n}+\frac {3 a^{2}}{2 b^{3} n}}{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 66, normalized size = 1.18 \[ \frac {4 \, a b x^{n} + 3 \, a^{2}}{2 \, {\left (b^{5} n x^{2 \, n} + 2 \, a b^{4} n x^{n} + a^{2} b^{3} n\right )}} + \frac {\log \left (\frac {b x^{n} + a}{b}\right )}{b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^{3\,n-1}}{{\left (a+b\,x^n\right )}^3} \,d x \]
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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