Optimal. Leaf size=82 \[ -\frac{a^3 x^n}{b^4 n}+\frac{a^2 x^{2 n}}{2 b^3 n}+\frac{a^4 \log \left (a+b x^n\right )}{b^5 n}-\frac{a x^{3 n}}{3 b^2 n}+\frac{x^{4 n}}{4 b n} \]
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Rubi [A] time = 0.0398073, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {266, 43} \[ -\frac{a^3 x^n}{b^4 n}+\frac{a^2 x^{2 n}}{2 b^3 n}+\frac{a^4 \log \left (a+b x^n\right )}{b^5 n}-\frac{a x^{3 n}}{3 b^2 n}+\frac{x^{4 n}}{4 b n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{4+5 (-1+n)}}{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^3}{b^4}+\frac{a^2 x}{b^3}-\frac{a x^2}{b^2}+\frac{x^3}{b}+\frac{a^4}{b^4 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^3 x^n}{b^4 n}+\frac{a^2 x^{2 n}}{2 b^3 n}-\frac{a x^{3 n}}{3 b^2 n}+\frac{x^{4 n}}{4 b n}+\frac{a^4 \log \left (a+b x^n\right )}{b^5 n}\\ \end{align*}
Mathematica [A] time = 0.0326856, size = 65, normalized size = 0.79 \[ \frac{b x^n \left (6 a^2 b x^n-12 a^3-4 a b^2 x^{2 n}+3 b^3 x^{3 n}\right )+12 a^4 \log \left (a+b x^n\right )}{12 b^5 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 87, normalized size = 1.1 \begin{align*}{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,bn}}-{\frac{a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,{b}^{2}n}}+{\frac{{a}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,{b}^{3}n}}-{\frac{{a}^{3}{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{4}n}}+{\frac{{a}^{4}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{5}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972402, size = 97, normalized size = 1.18 \begin{align*} \frac{a^{4} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{5} n} + \frac{3 \, b^{3} x^{4 \, n} - 4 \, a b^{2} x^{3 \, n} + 6 \, a^{2} b x^{2 \, n} - 12 \, a^{3} x^{n}}{12 \, b^{4} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0318, size = 144, normalized size = 1.76 \begin{align*} \frac{3 \, b^{4} x^{4 \, n} - 4 \, a b^{3} x^{3 \, n} + 6 \, a^{2} b^{2} x^{2 \, n} - 12 \, a^{3} b x^{n} + 12 \, a^{4} \log \left (b x^{n} + a\right )}{12 \, b^{5} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 99.1336, size = 87, normalized size = 1.06 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{5 n}}{5 a n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{a^{4} \log{\left (\frac{a}{b} + x^{n} \right )}}{b^{5} n} - \frac{a^{3} x^{n}}{b^{4} n} + \frac{a^{2} x^{2 n}}{2 b^{3} n} - \frac{a x^{3 n}}{3 b^{2} n} + \frac{x^{4 n}}{4 b n} & \text{otherwise} \end{cases} \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^{5 \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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