Optimal. Leaf size=77 \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}-\frac{a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac{x^3 \left (c x^n\right )^{-1/n}}{2 b} \]
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Rubi [A] time = 0.0269364, antiderivative size = 77, 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} \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}-\frac{a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac{x^3 \left (c x^n\right )^{-1/n}}{2 b} \]
Antiderivative was successfully verified.
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Rule 368
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx &=\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=-\frac{a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac{x^3 \left (c x^n\right )^{-1/n}}{2 b}+\frac{a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0287248, size = 67, normalized size = 0.87 \[ \frac{x^3 \left (c x^n\right )^{-3/n} \left (2 a^2 \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )+b \left (c x^n\right )^{\frac{1}{n}} \left (b \left (c x^n\right )^{\frac{1}{n}}-2 a\right )\right )}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.105, size = 438, normalized size = 5.7 \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^{2}}{\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 = 1.30868, size = 113, normalized size = 1.47 \begin{align*} \frac{b^{2} c^{\frac{2}{n}} x^{2} - 2 \, a b c^{\left (\frac{1}{n}\right )} x + 2 \, a^{2} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \, b^{3} c^{\frac{3}{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^{2}}{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^{2}}{\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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