Optimal. Leaf size=83 \[ \frac{10 a^3 b^2 x^n}{n}+\frac{5 a^2 b^3 x^{2 n}}{n}+5 a^4 b \log (x)-\frac{a^5 x^{-n}}{n}+\frac{5 a b^4 x^{3 n}}{3 n}+\frac{b^5 x^{4 n}}{4 n} \]
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Rubi [A] time = 0.0369853, antiderivative size = 83, 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} \[ \frac{10 a^3 b^2 x^n}{n}+\frac{5 a^2 b^3 x^{2 n}}{n}+5 a^4 b \log (x)-\frac{a^5 x^{-n}}{n}+\frac{5 a b^4 x^{3 n}}{3 n}+\frac{b^5 x^{4 n}}{4 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (10 a^3 b^2+\frac{a^5}{x^2}+\frac{5 a^4 b}{x}+10 a^2 b^3 x+5 a b^4 x^2+b^5 x^3\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-n}}{n}+\frac{10 a^3 b^2 x^n}{n}+\frac{5 a^2 b^3 x^{2 n}}{n}+\frac{5 a b^4 x^{3 n}}{3 n}+\frac{b^5 x^{4 n}}{4 n}+5 a^4 b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0374077, size = 73, normalized size = 0.88 \[ \frac{10 a^3 b^2 x^n+5 a^2 b^3 x^{2 n}+5 a^4 b n \log (x)-a^5 x^{-n}+\frac{5}{3} a b^4 x^{3 n}+\frac{1}{4} b^5 x^{4 n}}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 98, normalized size = 1.2 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 5\,{a}^{4}b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}-{\frac{{a}^{5}}{n}}+{\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{4\,n}}+{\frac{5\,a{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{3\,n}}+5\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+10\,{\frac{{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \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 = 1.37175, size = 169, normalized size = 2.04 \begin{align*} \frac{60 \, a^{4} b n x^{n} \log \left (x\right ) + 3 \, b^{5} x^{5 \, n} + 20 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} + 120 \, a^{3} b^{2} x^{2 \, n} - 12 \, a^{5}}{12 \, n x^{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 = 1.22391, size = 101, normalized size = 1.22 \begin{align*} \frac{60 \, a^{4} b n x^{n} \log \left (x\right ) + 3 \, b^{5} x^{5 \, n} + 20 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} + 120 \, a^{3} b^{2} x^{2 \, n} - 12 \, a^{5}}{12 \, n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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