Optimal. Leaf size=94 \[ -\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}-\frac{2 b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^3 x}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x}-\frac{1}{a^2 x} \]
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Rubi [A] time = 0.0376218, antiderivative size = 94, 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} \[ -\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}-\frac{2 b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^3 x}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x}-\frac{1}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 368
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{x}\\ &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^2}-\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a+b x)^2}+\frac{2 b^2}{a^3 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{x}\\ &=-\frac{1}{a^2 x}-\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}-\frac{2 b \left (c x^n\right )^{\frac{1}{n}} \log (x)}{a^3 x}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x}\\ \end{align*}
Mathematica [A] time = 0.117199, size = 71, normalized size = 0.76 \[ -\frac{\left (c x^n\right )^{\frac{1}{n}} \left (a \left (\frac{b}{a+b \left (c x^n\right )^{\frac{1}{n}}}+\left (c x^n\right )^{-1/n}\right )-2 b \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )+2 b \log (x)\right )}{a^3 x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.101, size = 440, normalized size = 4.7 \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*} \frac{1}{a b c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2} x} + 2 \, \int \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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55057, size = 212, normalized size = 2.26 \begin{align*} -\frac{2 \, b^{2} c^{\frac{2}{n}} x^{2} \log \left (x\right ) + a^{2} + 2 \,{\left (a b x \log \left (x\right ) + a b x\right )} c^{\left (\frac{1}{n}\right )} - 2 \,{\left (b^{2} c^{\frac{2}{n}} x^{2} + a b c^{\left (\frac{1}{n}\right )} x\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{a^{3} b c^{\left (\frac{1}{n}\right )} x^{2} + a^{4} x} \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^{2} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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