Optimal. Leaf size=142 \[ -\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{d \sqrt{d-e x} \sqrt{d+e x}} \]
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Rubi [A] time = 0.399996, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {2342, 2335, 277, 216} \[ -\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{d \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 2342
Rule 2335
Rule 277
Rule 216
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^2 \sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \int \frac{a+b \log \left (c x^n\right )}{x^2 \sqrt{1-\frac{e^2 x^2}{d^2}}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{\sqrt{1-\frac{e^2 x^2}{d^2}}}{x^2} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{1}{\sqrt{1-\frac{e^2 x^2}{d^2}}} \, dx}{d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{d \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.214314, size = 70, normalized size = 0.49 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (a+b \log \left (c x^n\right )+b n\right )+b e n x \tan ^{-1}\left (\frac{e x}{\sqrt{d-e x} \sqrt{d+e x}}\right )}{d^2 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.657, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}}{\frac{1}{\sqrt{-ex+d}}}{\frac{1}{\sqrt{ex+d}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43258, size = 182, normalized size = 1.28 \begin{align*} \frac{2 \, b e n x \arctan \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{e x}\right ) -{\left (b n \log \left (x\right ) + b n + b \log \left (c\right ) + a\right )} \sqrt{e x + d} \sqrt{-e x + d}}{d^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{b \log \left (c x^{n}\right ) + a}{\sqrt{e x + d} \sqrt{-e x + d} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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