Optimal. Leaf size=258 \[ \frac{b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{4 d^{3/2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{3/2}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{4 d^{3/2}}+\frac{b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 d^{3/2}}-\frac{b n \sqrt{d+e x^2}}{4 d x^2} \]
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Rubi [A] time = 0.372716, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {266, 51, 63, 208, 2350, 12, 14, 47, 5984, 5918, 2402, 2315} \[ \frac{b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{4 d^{3/2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{3/2}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{4 d^{3/2}}+\frac{b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 d^{3/2}}-\frac{b n \sqrt{d+e x^2}}{4 d x^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rule 2350
Rule 12
Rule 14
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^2}} \, dx &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}-(b n) \int \frac{-\frac{\sqrt{d+e x^2}}{d}+\frac{e x^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{d^{3/2}}}{2 x^3} \, dx\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}-\frac{1}{2} (b n) \int \frac{-\frac{\sqrt{d+e x^2}}{d}+\frac{e x^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{d^{3/2}}}{x^3} \, dx\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}-\frac{1}{2} (b n) \int \left (-\frac{\sqrt{d+e x^2}}{d x^3}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{d^{3/2} x}\right ) \, dx\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}+\frac{(b n) \int \frac{\sqrt{d+e x^2}}{x^3} \, dx}{2 d}-\frac{(b e n) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{x} \, dx}{2 d^{3/2}}\\ &=-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x^2} \, dx,x,x^2\right )}{4 d}-\frac{(b e n) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx,x,x^2\right )}{4 d^{3/2}}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 d x^2}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}-\frac{(b e n) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{d}}\right )}{-d+x^2} \, dx,x,\sqrt{d+e x^2}\right )}{2 d^{3/2}}+\frac{(b e n) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{8 d}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 d x^2}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{3/2}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{4 d}+\frac{(b e n) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{d}}\right )}{1-\frac{x}{\sqrt{d}}} \, dx,x,\sqrt{d+e x^2}\right )}{2 d^2}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 d x^2}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{4 d^{3/2}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{3/2}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{2 d^{3/2}}-\frac{(b e n) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-\frac{x}{\sqrt{d}}}\right )}{1-\frac{x^2}{d}} \, dx,x,\sqrt{d+e x^2}\right )}{2 d^2}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 d x^2}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{4 d^{3/2}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{3/2}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{2 d^{3/2}}+\frac{(b e n) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{d+e x^2}}{\sqrt{d}}}\right )}{2 d^{3/2}}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 d x^2}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{4 d^{3/2}}-\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{3/2}}-\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2}}+\frac{b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{2 d^{3/2}}+\frac{b e n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x^2}}{\sqrt{d}}}\right )}{4 d^{3/2}}\\ \end{align*}
Mathematica [C] time = 1.06489, size = 229, normalized size = 0.89 \[ \frac{\frac{b n \sqrt{\frac{d}{e x^2}+1} \left (2 d^{3/2} \, _3F_2\left (\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{5}{2},\frac{5}{2};-\frac{d}{e x^2}\right )+9 e x^2 (2 \log (x)+1) \left (\sqrt{e} x \sinh ^{-1}\left (\frac{\sqrt{d}}{\sqrt{e} x}\right )-\sqrt{d} \sqrt{\frac{d}{e x^2}+1}\right )\right )}{x^2 \sqrt{d+e x^2}}-\frac{18 \sqrt{d} \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{x^2}+18 e \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-18 e \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{36 d^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.413, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{3}}{\frac{1}{\sqrt{e{x}^{2}+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^{2} + d} b \log \left (c x^{n}\right ) + \sqrt{e x^{2} + d} a}{e x^{5} + d x^{3}}, x\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{a + b \log{\left (c x^{n} \right )}}{x^{3} \sqrt{d + e x^{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{b \log \left (c x^{n}\right ) + a}{\sqrt{e x^{2} + d} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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