Optimal. Leaf size=125 \[ \frac{b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 b^2 \log (x) \left (c x^n\right )^{2/n}}{a^4 x^2}-\frac{3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^4 x^2}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}}}{a^3 x^2}-\frac{1}{2 a^2 x^2} \]
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Rubi [A] time = 0.0501029, antiderivative size = 125, 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, 44} \[ \frac{b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 b^2 \log (x) \left (c x^n\right )^{2/n}}{a^4 x^2}-\frac{3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^4 x^2}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}}}{a^3 x^2}-\frac{1}{2 a^2 x^2} \]
Antiderivative was successfully verified.
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Rule 368
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{x^2}\\ &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^3}-\frac{2 b}{a^3 x^2}+\frac{3 b^2}{a^4 x}-\frac{b^3}{a^3 (a+b x)^2}-\frac{3 b^3}{a^4 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{x^2}\\ &=-\frac{1}{2 a^2 x^2}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}}}{a^3 x^2}+\frac{b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 b^2 \left (c x^n\right )^{2/n} \log (x)}{a^4 x^2}-\frac{3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^4 x^2}\\ \end{align*}
Mathematica [A] time = 0.187761, size = 99, normalized size = 0.79 \[ \frac{\left (c x^n\right )^{2/n} \left (a \left (\frac{2 b^2}{a+b \left (c x^n\right )^{\frac{1}{n}}}-a \left (c x^n\right )^{-2/n}+4 b \left (c x^n\right )^{-1/n}\right )-6 b^2 \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )+6 b^2 \log (x)\right )}{2 a^4 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.107, size = 558, normalized size = 4.5 \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*} \frac{1}{a b c^{\left (\frac{1}{n}\right )} x^{2}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2} x^{2}} + 3 \, \int \frac{1}{a b c^{\left (\frac{1}{n}\right )} x^{3}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57348, size = 262, normalized size = 2.1 \begin{align*} \frac{6 \, b^{3} c^{\frac{3}{n}} x^{3} \log \left (x\right ) + 3 \, a^{2} b c^{\left (\frac{1}{n}\right )} x - a^{3} + 6 \,{\left (a b^{2} x^{2} \log \left (x\right ) + a b^{2} x^{2}\right )} c^{\frac{2}{n}} - 6 \,{\left (b^{3} c^{\frac{3}{n}} x^{3} + a b^{2} c^{\frac{2}{n}} x^{2}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \,{\left (a^{4} b c^{\left (\frac{1}{n}\right )} x^{3} + a^{5} x^{2}\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{1}{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{1}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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