Optimal. Leaf size=114 \[ \frac{a^3 x^4 \left (c x^n\right )^{-4/n}}{b^4 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 a^2 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}-\frac{2 a x^4 \left (c x^n\right )^{-3/n}}{b^3}+\frac{x^4 \left (c x^n\right )^{-2/n}}{2 b^2} \]
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Rubi [A] time = 0.0437005, antiderivative size = 114, 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 = {368, 43} \[ \frac{a^3 x^4 \left (c x^n\right )^{-4/n}}{b^4 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 a^2 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}-\frac{2 a x^4 \left (c x^n\right )^{-3/n}}{b^3}+\frac{x^4 \left (c x^n\right )^{-2/n}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 368
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx &=\left (x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x^4 \left (c x^n\right )^{-4/n}\right ) \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,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=-\frac{2 a x^4 \left (c x^n\right )^{-3/n}}{b^3}+\frac{x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac{a^3 x^4 \left (c x^n\right )^{-4/n}}{b^4 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 a^2 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0721593, size = 89, normalized size = 0.78 \[ \frac{x^4 \left (c x^n\right )^{-4/n} \left (\frac{2 a^3}{a+b \left (c x^n\right )^{\frac{1}{n}}}+6 a^2 \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )-4 a b \left (c x^n\right )^{\frac{1}{n}}+b^2 \left (c x^n\right )^{2/n}\right )}{2 b^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.121, size = 661, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x^{4}}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}} - 3 \, \int \frac{x^{3}}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53457, size = 208, normalized size = 1.82 \begin{align*} \frac{b^{3} c^{\frac{3}{n}} x^{3} - 3 \, a b^{2} c^{\frac{2}{n}} x^{2} - 4 \, a^{2} b c^{\left (\frac{1}{n}\right )} x + 2 \, a^{3} + 6 \,{\left (a^{2} b c^{\left (\frac{1}{n}\right )} x + a^{3}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \,{\left (b^{5} c^{\frac{5}{n}} x + a b^{4} c^{\frac{4}{n}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}}\, dx \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^{3}}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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