Optimal. Leaf size=87 \[ \frac{b^2 \log (x) \left (c x^n\right )^{2/n}}{a^3 x^2}-\frac{b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x^2}+\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x^2}-\frac{1}{2 a x^2} \]
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Rubi [A] time = 0.0321322, antiderivative size = 87, 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} \[ \frac{b^2 \log (x) \left (c x^n\right )^{2/n}}{a^3 x^2}-\frac{b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x^2}+\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x^2}-\frac{1}{2 a x^2} \]
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 )} \, dx &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)} \, 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 x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{x^2}\\ &=-\frac{1}{2 a x^2}+\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x^2}+\frac{b^2 \left (c x^n\right )^{2/n} \log (x)}{a^3 x^2}-\frac{b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x^2}\\ \end{align*}
Mathematica [A] time = 0.0423497, size = 75, normalized size = 0.86 \[ -\frac{a^2+2 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )-2 a b \left (c x^n\right )^{\frac{1}{n}}-2 b^2 \log (x) \left (c x^n\right )^{2/n}}{2 a^3 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.107, size = 448, normalized size = 5.2 \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{1}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56592, size = 146, normalized size = 1.68 \begin{align*} -\frac{2 \, b^{2} c^{\frac{2}{n}} x^{2} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) - 2 \, b^{2} c^{\frac{2}{n}} x^{2} \log \left (x\right ) - 2 \, a b c^{\left (\frac{1}{n}\right )} x + a^{2}}{2 \, a^{3} x^{2}} \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 )}\, 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 )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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