Optimal. Leaf size=85 \[ -\frac{10 a^3 b^2 x^{-n}}{n}+10 a^2 b^3 \log (x)-\frac{5 a^4 b x^{-2 n}}{2 n}-\frac{a^5 x^{-3 n}}{3 n}+\frac{5 a b^4 x^n}{n}+\frac{b^5 x^{2 n}}{2 n} \]
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Rubi [A] time = 0.036219, antiderivative size = 85, normalized size of antiderivative = 1., 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} \[ -\frac{10 a^3 b^2 x^{-n}}{n}+10 a^2 b^3 \log (x)-\frac{5 a^4 b x^{-2 n}}{2 n}-\frac{a^5 x^{-3 n}}{3 n}+\frac{5 a b^4 x^n}{n}+\frac{b^5 x^{2 n}}{2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-3 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^4} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (5 a b^4+\frac{a^5}{x^4}+\frac{5 a^4 b}{x^3}+\frac{10 a^3 b^2}{x^2}+\frac{10 a^2 b^3}{x}+b^5 x\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-3 n}}{3 n}-\frac{5 a^4 b x^{-2 n}}{2 n}-\frac{10 a^3 b^2 x^{-n}}{n}+\frac{5 a b^4 x^n}{n}+\frac{b^5 x^{2 n}}{2 n}+10 a^2 b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0491566, size = 72, normalized size = 0.85 \[ \frac{x^{-3 n} \left (-60 a^3 b^2 x^{2 n}-15 a^4 b x^n-2 a^5+30 a b^4 x^{4 n}+3 b^5 x^{5 n}\right )}{6 n}+10 a^2 b^3 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 98, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( 10\,{a}^{2}{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}-{\frac{{a}^{5}}{3\,n}}+{\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{2\,n}}+5\,{\frac{a{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{n}}-10\,{\frac{{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{5\,{a}^{4}b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,n}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34595, size = 170, normalized size = 2. \begin{align*} \frac{60 \, a^{2} b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, b^{5} x^{5 \, n} + 30 \, a b^{4} x^{4 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 15 \, a^{4} b x^{n} - 2 \, a^{5}}{6 \, n x^{3 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25381, size = 104, normalized size = 1.22 \begin{align*} \frac{60 \, a^{2} b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, b^{5} x^{5 \, n} + 30 \, a b^{4} x^{4 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 15 \, a^{4} b x^{n} - 2 \, a^{5}}{6 \, n x^{3 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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