Optimal. Leaf size=101 \[ \frac{a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac{a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}-\frac{a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac{x^4 \left (c x^n\right )^{-1/n}}{3 b} \]
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Rubi [A] time = 0.0370195, antiderivative size = 101, 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} \[ \frac{a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac{a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}-\frac{a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac{x^4 \left (c x^n\right )^{-1/n}}{3 b} \]
Antiderivative was successfully verified.
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Rule 368
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx &=\left (x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{x^3}{a+b x} \, 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{a^2}{b^3}-\frac{a x}{b^2}+\frac{x^2}{b}-\frac{a^3}{b^3 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac{a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac{x^4 \left (c x^n\right )^{-1/n}}{3 b}-\frac{a^3 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 = 0.0504328, size = 87, normalized size = 0.86 \[ \frac{x^4 \left (c x^n\right )^{-4/n} \left (b \left (c x^n\right )^{\frac{1}{n}} \left (6 a^2-3 a b \left (c x^n\right )^{\frac{1}{n}}+2 b^2 \left (c x^n\right )^{2/n}\right )-6 a^3 \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )\right )}{6 b^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.164, size = 552, normalized size = 5.5 \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*} \int \frac{x^{3}}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32212, size = 149, normalized size = 1.48 \begin{align*} \frac{2 \, b^{3} c^{\frac{3}{n}} x^{3} - 3 \, a b^{2} c^{\frac{2}{n}} x^{2} + 6 \, a^{2} b c^{\left (\frac{1}{n}\right )} x - 6 \, a^{3} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{6 \, b^{4} c^{\frac{4}{n}}} \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}}{a + b \left (c x^{n}\right )^{\frac{1}{n}}}\, 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 (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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