Optimal. Leaf size=337 \[ \frac{5 b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{4 d^{7/2}}-\frac{5 e \left (a+b \log \left (c x^n\right )\right )}{2 d^3 \sqrt{d+e x^2}}-\frac{5 e \left (a+b \log \left (c x^n\right )\right )}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac{5 e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{7/2}}-\frac{a+b \log \left (c x^n\right )}{2 d x^2 \left (d+e x^2\right )^{3/2}}-\frac{b n \sqrt{d+e x^2}}{4 d^3 x^2}+\frac{b e n}{3 d^3 \sqrt{d+e x^2}}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{7/2}}-\frac{31 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{12 d^{7/2}}+\frac{5 b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 d^{7/2}} \]
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Rubi [A] time = 0.487747, antiderivative size = 341, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {266, 51, 63, 208, 2350, 1251, 897, 1259, 453, 5984, 5918, 2402, 2315} \[ \frac{5 b e n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right )}{4 d^{7/2}}-\frac{5 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 x^2}+\frac{5 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^2 \sqrt{d+e x^2}}+\frac{5 e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{7/2}}+\frac{a+b \log \left (c x^n\right )}{3 d x^2 \left (d+e x^2\right )^{3/2}}-\frac{b n \sqrt{d+e x^2}}{4 d^3 x^2}+\frac{b e n}{3 d^3 \sqrt{d+e x^2}}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{7/2}}-\frac{31 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{12 d^{7/2}}+\frac{5 b e n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^2}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 d^{7/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rule 2350
Rule 1251
Rule 897
Rule 1259
Rule 453
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^{5/2}} \, dx &=\frac{a+b \log \left (c x^n\right )}{3 d x^2 \left (d+e x^2\right )^{3/2}}+\frac{5 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{5 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 x^2}+\frac{5 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^{7/2}}-(b n) \int \left (-\frac{3 d^2+20 d e x^2+15 e^2 x^4}{6 d^3 x^3 \left (d+e x^2\right )^{3/2}}+\frac{5 e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 d^{7/2} x}\right ) \, dx\\ &=\frac{a+b \log \left (c x^n\right )}{3 d x^2 \left (d+e x^2\right )^{3/2}}+\frac{5 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{5 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 x^2}+\frac{5 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^{7/2}}+\frac{(b n) \int \frac{3 d^2+20 d e x^2+15 e^2 x^4}{x^3 \left (d+e x^2\right )^{3/2}} \, dx}{6 d^3}-\frac{(5 b e n) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{x} \, dx}{2 d^{7/2}}\\ &=\frac{a+b \log \left (c x^n\right )}{3 d x^2 \left (d+e x^2\right )^{3/2}}+\frac{5 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{5 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 x^2}+\frac{5 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^{7/2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{3 d^2+20 d e x+15 e^2 x^2}{x^2 (d+e x)^{3/2}} \, dx,x,x^2\right )}{12 d^3}-\frac{(5 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^{7/2}}\\ &=\frac{a+b \log \left (c x^n\right )}{3 d x^2 \left (d+e x^2\right )^{3/2}}+\frac{5 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{5 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 x^2}+\frac{5 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^{7/2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{-2 d^2-10 d x^2+15 x^4}{x^2 \left (-\frac{d}{e}+\frac{x^2}{e}\right )^2} \, dx,x,\sqrt{d+e x^2}\right )}{6 d^3 e}-\frac{(5 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^{7/2}}\\ &=-\frac{b n \sqrt{d+e x^2}}{4 d^3 x^2}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{7/2}}+\frac{a+b \log \left (c x^n\right )}{3 d x^2 \left (d+e x^2\right )^{3/2}}+\frac{5 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{5 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 x^2}+\frac{5 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^{7/2}}+\frac{(5 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^4}-\frac{\left (b e^3 n\right ) \operatorname{Subst}\left (\int \frac{-\frac{4 d^3}{e^3}-\frac{27 d^2 x^2}{e^3}}{x^2 \left (-\frac{d}{e}+\frac{x^2}{e}\right )} \, dx,x,\sqrt{d+e x^2}\right )}{12 d^5}\\ &=\frac{b e n}{3 d^3 \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{4 d^3 x^2}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{7/2}}+\frac{a+b \log \left (c x^n\right )}{3 d x^2 \left (d+e x^2\right )^{3/2}}+\frac{5 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{5 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 x^2}+\frac{5 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^{7/2}}+\frac{5 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^{7/2}}+\frac{(31 b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{12 d^3}-\frac{(5 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^4}\\ &=\frac{b e n}{3 d^3 \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{4 d^3 x^2}-\frac{31 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{12 d^{7/2}}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{7/2}}+\frac{a+b \log \left (c x^n\right )}{3 d x^2 \left (d+e x^2\right )^{3/2}}+\frac{5 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{5 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 x^2}+\frac{5 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^{7/2}}+\frac{5 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^{7/2}}+\frac{(5 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^{7/2}}\\ &=\frac{b e n}{3 d^3 \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{4 d^3 x^2}-\frac{31 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{12 d^{7/2}}-\frac{5 b e n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )^2}{4 d^{7/2}}+\frac{a+b \log \left (c x^n\right )}{3 d x^2 \left (d+e x^2\right )^{3/2}}+\frac{5 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^2 \sqrt{d+e x^2}}-\frac{5 \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 x^2}+\frac{5 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^{7/2}}+\frac{5 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^{7/2}}+\frac{5 b e n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x^2}}{\sqrt{d}}}\right )}{4 d^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.308776, size = 227, normalized size = 0.67 \[ \frac{b n \sqrt{\frac{d}{e x^2}+1} \left (5 \, _3F_2\left (\frac{7}{2},\frac{7}{2},\frac{7}{2};\frac{9}{2},\frac{9}{2};-\frac{d}{e x^2}\right )-7 (2 \log (x)+1) \, _2F_1\left (\frac{5}{2},\frac{7}{2};\frac{9}{2};-\frac{d}{e x^2}\right )\right )}{98 e^2 x^6 \sqrt{d+e x^2}}-\frac{\left (3 d^2+20 d e x^2+15 e^2 x^4\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{6 d^3 x^2 \left (d+e x^2\right )^{3/2}}-\frac{5 e \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{2 d^{7/2}}+\frac{5 e \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{2 d^{7/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.411, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{3}} \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, 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^{3} x^{9} + 3 \, d e^{2} x^{7} + 3 \, d^{2} e x^{5} + d^{3} 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}{{\left (e x^{2} + d\right )}^{\frac{5}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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