Optimal. Leaf size=66 \[ \frac{a^3}{b^4 n \left (a+b x^n\right )}+\frac{3 a^2 \log \left (a+b x^n\right )}{b^4 n}-\frac{2 a x^n}{b^3 n}+\frac{x^{2 n}}{2 b^2 n} \]
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Rubi [A] time = 0.0406642, antiderivative size = 66, 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} \[ \frac{a^3}{b^4 n \left (a+b x^n\right )}+\frac{3 a^2 \log \left (a+b x^n\right )}{b^4 n}-\frac{2 a x^n}{b^3 n}+\frac{x^{2 n}}{2 b^2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{-1+4 n}}{\left (a+b x^n\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{2 a}{b^3}+\frac{x}{b^2}-\frac{a^3}{b^3 (a+b x)^2}+\frac{3 a^2}{b^3 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{2 a x^n}{b^3 n}+\frac{x^{2 n}}{2 b^2 n}+\frac{a^3}{b^4 n \left (a+b x^n\right )}+\frac{3 a^2 \log \left (a+b x^n\right )}{b^4 n}\\ \end{align*}
Mathematica [A] time = 0.0502157, size = 54, normalized size = 0.82 \[ \frac{\frac{2 a^3}{a+b x^n}+6 a^2 \log \left (a+b x^n\right )-4 a b x^n+b^2 x^{2 n}}{2 b^4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 78, normalized size = 1.2 \begin{align*}{\frac{1}{a+b{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 3\,{\frac{{a}^{3}}{{b}^{4}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,bn}}-{\frac{3\,a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,{b}^{2}n}} \right ) }+3\,{\frac{{a}^{2}\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 = 0.972431, size = 105, normalized size = 1.59 \begin{align*} \frac{b^{3} x^{3 \, n} - 3 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} + 2 \, a^{3}}{2 \,{\left (b^{5} n x^{n} + a b^{4} n\right )}} + \frac{3 \, a^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{4} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.05756, size = 159, normalized size = 2.41 \begin{align*} \frac{b^{3} x^{3 \, n} - 3 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} + 2 \, a^{3} + 6 \,{\left (a^{2} b x^{n} + a^{3}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (b^{5} n x^{n} + a b^{4} n\right )}} \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}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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