Optimal. Leaf size=64 \[ \frac{a^2 x^n}{b^3 n}-\frac{a^3 \log \left (a+b x^n\right )}{b^4 n}-\frac{a x^{2 n}}{2 b^2 n}+\frac{x^{3 n}}{3 b n} \]
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Rubi [A] time = 0.0337004, antiderivative size = 64, 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} \[ \frac{a^2 x^n}{b^3 n}-\frac{a^3 \log \left (a+b x^n\right )}{b^4 n}-\frac{a x^{2 n}}{2 b^2 n}+\frac{x^{3 n}}{3 b n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{-1+4 n}}{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{b^3}-\frac{a x}{b^2}+\frac{x^2}{b}-\frac{a^3}{b^3 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{a^2 x^n}{b^3 n}-\frac{a x^{2 n}}{2 b^2 n}+\frac{x^{3 n}}{3 b n}-\frac{a^3 \log \left (a+b x^n\right )}{b^4 n}\\ \end{align*}
Mathematica [A] time = 0.0309288, size = 52, normalized size = 0.81 \[ \frac{b x^n \left (6 a^2-3 a b x^n+2 b^2 x^{2 n}\right )-6 a^3 \log \left (a+b x^n\right )}{6 b^4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 69, normalized size = 1.1 \begin{align*}{\frac{{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{3}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,bn}}-{\frac{a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,{b}^{2}n}}-{\frac{{a}^{3}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{4}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976348, size = 81, normalized size = 1.27 \begin{align*} -\frac{a^{3} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{4} n} + \frac{2 \, b^{2} x^{3 \, n} - 3 \, a b x^{2 \, n} + 6 \, a^{2} x^{n}}{6 \, b^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0474, size = 113, normalized size = 1.77 \begin{align*} \frac{2 \, b^{3} x^{3 \, n} - 3 \, a b^{2} x^{2 \, n} + 6 \, a^{2} b x^{n} - 6 \, a^{3} \log \left (b x^{n} + a\right )}{6 \, b^{4} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4 \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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