Optimal. Leaf size=225 \[ -\frac{2 b n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right )}{d^{3/2} r^2}+2 \left (\frac{1}{d r \sqrt{d+e x^r}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{d^{3/2} r^2}+\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r^2}-\frac{4 b n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.342926, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {266, 51, 63, 208, 2348, 5984, 5918, 2402, 2315} \[ -\frac{2 b n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right )}{d^{3/2} r^2}+2 \left (\frac{1}{d r \sqrt{d+e x^r}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{d^{3/2} r^2}+\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r^2}-\frac{4 b n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 51
Rule 63
Rule 208
Rule 2348
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^{3/2}} \, dx &=2 \left (\frac{1}{d r \sqrt{d+e x^r}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac{2}{d r x \sqrt{d+e x^r}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r x}\right ) \, dx\\ &=2 \left (\frac{1}{d r \sqrt{d+e x^r}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{(2 b n) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{x} \, dx}{d^{3/2} r}-\frac{(2 b n) \int \frac{1}{x \sqrt{d+e x^r}} \, dx}{d r}\\ &=2 \left (\frac{1}{d r \sqrt{d+e x^r}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{(2 b n) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx,x,x^r\right )}{d^{3/2} r^2}-\frac{(2 b n) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^r\right )}{d r^2}\\ &=2 \left (\frac{1}{d r \sqrt{d+e x^r}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{(4 b n) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{d}}\right )}{-d+x^2} \, dx,x,\sqrt{d+e x^r}\right )}{d^{3/2} r^2}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^r}\right )}{d e r^2}\\ &=\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r^2}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{d^{3/2} r^2}+2 \left (\frac{1}{d r \sqrt{d+e x^r}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(4 b 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^r}\right )}{d^2 r^2}\\ &=\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r^2}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{d^{3/2} r^2}+2 \left (\frac{1}{d r \sqrt{d+e x^r}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right )}{d^{3/2} r^2}+\frac{(4 b 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^r}\right )}{d^2 r^2}\\ &=\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r^2}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{d^{3/2} r^2}+2 \left (\frac{1}{d r \sqrt{d+e x^r}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right )}{d^{3/2} r^2}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{d+e x^r}}{\sqrt{d}}}\right )}{d^{3/2} r^2}\\ &=\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r^2}+\frac{2 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{d^{3/2} r^2}+2 \left (\frac{1}{d r \sqrt{d+e x^r}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{d^{3/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right )}{d^{3/2} r^2}-\frac{2 b n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x^r}}{\sqrt{d}}}\right )}{d^{3/2} r^2}\\ \end{align*}
Mathematica [F] time = 0.293638, size = 0, normalized size = 0. \[ \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.477, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{x} \left ( d+e{x}^{r} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]