Optimal. Leaf size=240 \[ -\frac{2 b \sqrt{d} n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right )}{r^2}+2 \left (\frac{\sqrt{d+e x^r}}{r}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 b n \sqrt{d+e x^r}}{r^2}+\frac{2 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{r^2}+\frac{4 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r^2}-\frac{4 b \sqrt{d} n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r^2} \]
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Rubi [A] time = 0.317036, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {266, 50, 63, 208, 2348, 5984, 5918, 2402, 2315} \[ -\frac{2 b \sqrt{d} n \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right )}{r^2}+2 \left (\frac{\sqrt{d+e x^r}}{r}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 b n \sqrt{d+e x^r}}{r^2}+\frac{2 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{r^2}+\frac{4 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r^2}-\frac{4 b \sqrt{d} n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}-\sqrt{d+e x^r}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rule 2348
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x^r} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=2 \left (\frac{\sqrt{d+e x^r}}{r}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac{2 \sqrt{d+e x^r}}{r x}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r x}\right ) \, dx\\ &=2 \left (\frac{\sqrt{d+e x^r}}{r}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(2 b n) \int \frac{\sqrt{d+e x^r}}{x} \, dx}{r}+\frac{\left (2 b \sqrt{d} n\right ) \int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{x} \, dx}{r}\\ &=2 \left (\frac{\sqrt{d+e x^r}}{r}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(2 b n) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x} \, dx,x,x^r\right )}{r^2}+\frac{\left (2 b \sqrt{d} n\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{x} \, dx,x,x^r\right )}{r^2}\\ &=-\frac{4 b n \sqrt{d+e x^r}}{r^2}+2 \left (\frac{\sqrt{d+e x^r}}{r}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{\left (4 b \sqrt{d} n\right ) \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 )}{r^2}-\frac{(2 b d n) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^r\right )}{r^2}\\ &=-\frac{4 b n \sqrt{d+e x^r}}{r^2}+\frac{2 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{r^2}+2 \left (\frac{\sqrt{d+e x^r}}{r}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{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 )}{r^2}-\frac{(4 b d n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^r}\right )}{e r^2}\\ &=-\frac{4 b n \sqrt{d+e x^r}}{r^2}+\frac{4 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r^2}+\frac{2 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{r^2}+2 \left (\frac{\sqrt{d+e x^r}}{r}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 b \sqrt{d} 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 )}{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 )}{r^2}\\ &=-\frac{4 b n \sqrt{d+e x^r}}{r^2}+\frac{4 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r^2}+\frac{2 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{r^2}+2 \left (\frac{\sqrt{d+e x^r}}{r}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 b \sqrt{d} 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 )}{r^2}-\frac{\left (4 b \sqrt{d} n\right ) \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 )}{r^2}\\ &=-\frac{4 b n \sqrt{d+e x^r}}{r^2}+\frac{4 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r^2}+\frac{2 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )^2}{r^2}+2 \left (\frac{\sqrt{d+e x^r}}{r}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^r}}{\sqrt{d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 b \sqrt{d} 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 )}{r^2}-\frac{2 b \sqrt{d} n \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{d+e x^r}}{\sqrt{d}}}\right )}{r^2}\\ \end{align*}
Mathematica [F] time = 0.339452, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d+e x^r} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.662, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{x}\sqrt{d+e{x}^{r}}}\, 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(-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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c x^{n} \right )}\right ) \sqrt{d + e x^{r}}}{x}\, 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{\sqrt{e x^{r} + d}{\left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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