Optimal. Leaf size=52 \[ -\frac {a^3 x^{-3 n}}{3 n}-\frac {3 a^2 b x^{-2 n}}{2 n}-\frac {3 a b^2 x^{-n}}{n}+b^3 \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 52, 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} \[ -\frac {3 a^2 b x^{-2 n}}{2 n}-\frac {a^3 x^{-3 n}}{3 n}-\frac {3 a b^2 x^{-n}}{n}+b^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-3 n} \left (a+b x^n\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^3}{x^4} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^3}{x^4}+\frac {3 a^2 b}{x^3}+\frac {3 a b^2}{x^2}+\frac {b^3}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^3 x^{-3 n}}{3 n}-\frac {3 a^2 b x^{-2 n}}{2 n}-\frac {3 a b^2 x^{-n}}{n}+b^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 43, normalized size = 0.83 \[ b^3 \log (x)-\frac {a x^{-3 n} \left (2 a^2+9 a b x^n+18 b^2 x^{2 n}\right )}{6 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 51, normalized size = 0.98 \[ \frac {6 \, b^{3} n x^{3 \, n} \log \relax (x) - 18 \, a b^{2} x^{2 \, n} - 9 \, a^{2} b x^{n} - 2 \, a^{3}}{6 \, n x^{3 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 51, normalized size = 0.98 \[ \frac {6 \, b^{3} n x^{3 \, n} \log \relax (x) - 18 \, a b^{2} x^{2 \, n} - 9 \, a^{2} b x^{n} - 2 \, a^{3}}{6 \, n x^{3 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 61, normalized size = 1.17 \[ \left (b^{3} {\mathrm e}^{3 n \ln \relax (x )} \ln \relax (x )-\frac {3 a^{2} b \,{\mathrm e}^{n \ln \relax (x )}}{2 n}-\frac {3 a \,b^{2} {\mathrm e}^{2 n \ln \relax (x )}}{n}-\frac {a^{3}}{3 n}\right ) {\mathrm e}^{-3 n \ln \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 52, normalized size = 1.00 \[ b^{3} \log \relax (x) - \frac {a^{3}}{3 \, n x^{3 \, n}} - \frac {3 \, a^{2} b}{2 \, n x^{2 \, n}} - \frac {3 \, a b^{2}}{n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 52, normalized size = 1.00 \[ b^3\,\ln \relax (x)-\frac {a^3}{3\,n\,x^{3\,n}}-\frac {3\,a\,b^2}{n\,x^n}-\frac {3\,a^2\,b}{2\,n\,x^{2\,n}} \]
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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