Optimal. Leaf size=489 \[ -\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{PolyLog}\left (2,-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}}+1}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}} \]
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Rubi [A] time = 0.724349, antiderivative size = 489, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2342, 266, 51, 63, 208, 2350, 47, 5984, 5918, 2402, 2315} \[ -\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{PolyLog}\left (2,-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}}+1}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 2342
Rule 266
Rule 51
Rule 63
Rule 208
Rule 2350
Rule 47
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^3 \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^3 \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 )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \left (-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}}}{2 x^3}-\frac{e^2 \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 x}\right ) \, 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 )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \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^3} \, dx}{2 \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{\tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{x} \, dx}{2 d^2 \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 )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{e^2 x}{d^2}}}{x^2} \, dx,x,x^2\right )}{4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\sqrt{1-\frac{e^2 x}{d^2}}\right )}{x} \, dx,x,x^2\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \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 )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{e^2 x}{d^2}}} \, dx,x,x^2\right )}{8 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}(x)}{-1+x^2} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \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 )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{d^2 x^2}{e^2}} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{1-x} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \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 )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \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 )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \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 )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{Li}_2\left (1-\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 0.858847, size = 255, normalized size = 0.52 \[ \frac{\frac{b n \left (e^2 x^2-d^2\right ) \left (2 d^3 \, _3F_2\left (\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{5}{2},\frac{5}{2};\frac{d^2}{e^2 x^2}\right )+9 e^2 x^2 (2 \log (x)+1) \left (d \sqrt{1-\frac{d^2}{e^2 x^2}}-e x \sin ^{-1}\left (\frac{d}{e x}\right )\right )\right )}{e^2 x^4 \sqrt{1-\frac{d^2}{e^2 x^2}} \sqrt{d-e x} \sqrt{d+e x}}-18 e^2 \log \left (\sqrt{d-e x} \sqrt{d+e x}+d\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-\frac{18 d \sqrt{d-e x} \sqrt{d+e x} \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{x^2}+18 e^2 \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{36 d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.659, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{3}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{e x + d} \sqrt{-e x + d} b \log \left (c x^{n}\right ) + \sqrt{e x + d} \sqrt{-e x + d} a}{e^{2} x^{5} - d^{2} x^{3}}, x\right ) \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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