Optimal. Leaf size=374 \[ -\frac{b d \text{PolyLog}\left (2,1-\frac{2}{c \sqrt{x}+1}\right )}{e^2}+\frac{b d \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{2 e^2}+\frac{b d \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{2 e^2}+\frac{2 d \log \left (\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{e^2}+\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e}-\frac{b \tanh ^{-1}\left (c \sqrt{x}\right )}{c^2 e}+\frac{b \sqrt{x}}{c e} \]
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Rubi [A] time = 0.484596, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {43, 5980, 5916, 321, 206, 6044, 5920, 2402, 2315, 2447} \[ -\frac{b d \text{PolyLog}\left (2,1-\frac{2}{c \sqrt{x}+1}\right )}{e^2}+\frac{b d \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{2 e^2}+\frac{b d \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{2 e^2}+\frac{2 d \log \left (\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}-\sqrt{e}\right )}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{x}+1\right ) \left (c \sqrt{-d}+\sqrt{e}\right )}\right )}{e^2}+\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e}-\frac{b \tanh ^{-1}\left (c \sqrt{x}\right )}{c^2 e}+\frac{b \sqrt{x}}{c e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 5980
Rule 5916
Rule 321
Rule 206
Rule 6044
Rule 5920
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{d+e x} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{d+e x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \operatorname{Subst}\left (\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt{x}\right )}{e}-\frac{(2 d) \operatorname{Subst}\left (\int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{d+e x^2} \, dx,x,\sqrt{x}\right )}{e}\\ &=\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e}-\frac{(b c) \operatorname{Subst}\left (\int \frac{x^2}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{e}-\frac{(2 d) \operatorname{Subst}\left (\int \left (-\frac{a+b \tanh ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \tanh ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx,x,\sqrt{x}\right )}{e}\\ &=\frac{b \sqrt{x}}{c e}+\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e}+\frac{d \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx,x,\sqrt{x}\right )}{e^{3/2}}-\frac{d \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx,x,\sqrt{x}\right )}{e^{3/2}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{c e}\\ &=\frac{b \sqrt{x}}{c e}-\frac{b \tanh ^{-1}\left (c \sqrt{x}\right )}{c^2 e}+\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e}+\frac{2 d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1+c \sqrt{x}}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{e^2}-2 \frac{(b c d) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{e^2}+\frac{(b c d) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{e^2}+\frac{(b c d) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{e^2}\\ &=\frac{b \sqrt{x}}{c e}-\frac{b \tanh ^{-1}\left (c \sqrt{x}\right )}{c^2 e}+\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e}+\frac{2 d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1+c \sqrt{x}}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{e^2}+\frac{b d \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{2 e^2}+\frac{b d \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{2 e^2}-2 \frac{(b d) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c \sqrt{x}}\right )}{e^2}\\ &=\frac{b \sqrt{x}}{c e}-\frac{b \tanh ^{-1}\left (c \sqrt{x}\right )}{c^2 e}+\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{e}+\frac{2 d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1+c \sqrt{x}}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{e^2}-\frac{d \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{e^2}-\frac{b d \text{Li}_2\left (1-\frac{2}{1+c \sqrt{x}}\right )}{e^2}+\frac{b d \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}-\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{2 e^2}+\frac{b d \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} \sqrt{x}\right )}{\left (c \sqrt{-d}+\sqrt{e}\right ) \left (1+c \sqrt{x}\right )}\right )}{2 e^2}\\ \end{align*}
Mathematica [A] time = 1.44733, size = 337, normalized size = 0.9 \[ \frac{\frac{2 b \left (-c^2 d \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+\tanh ^{-1}\left (c \sqrt{x}\right ) \left (2 c^2 d \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )+c^2 e x-e\right )+c^2 d \tanh ^{-1}\left (c \sqrt{x}\right )^2+c e \sqrt{x}\right )}{c^2}-b d \left (\text{PolyLog}\left (2,-\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d-2 c \sqrt{-d} \sqrt{e}-e}\right )+\text{PolyLog}\left (2,-\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d+2 c \sqrt{-d} \sqrt{e}-e}\right )+2 \tanh ^{-1}\left (c \sqrt{x}\right ) \left (\log \left (\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d-2 c \sqrt{-d} \sqrt{e}-e}+1\right )+\log \left (\frac{\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt{x}\right )}}{c^2 d+2 c \sqrt{-d} \sqrt{e}-e}+1\right )-\tanh ^{-1}\left (c \sqrt{x}\right )\right )\right )-2 a d \log (d+e x)+2 a e x}{2 e^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.057, size = 539, normalized size = 1.4 \begin{align*}{\frac{ax}{e}}-{\frac{ad\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{{e}^{2}}}+{\frac{bx}{e}{\it Artanh} \left ( c\sqrt{x} \right ) }-{\frac{bd\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{{e}^{2}}{\it Artanh} \left ( c\sqrt{x} \right ) }-{\frac{bd\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{2\,{e}^{2}}\ln \left ( c\sqrt{x}-1 \right ) }+{\frac{bd}{2\,{e}^{2}}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({ \left ( c\sqrt{-de}-e \left ( c\sqrt{x}-1 \right ) -e \right ) \left ( c\sqrt{-de}-e \right ) ^{-1}} \right ) }+{\frac{bd}{2\,{e}^{2}}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({ \left ( c\sqrt{-de}+e \left ( c\sqrt{x}-1 \right ) +e \right ) \left ( c\sqrt{-de}+e \right ) ^{-1}} \right ) }+{\frac{bd}{2\,{e}^{2}}{\it dilog} \left ({ \left ( c\sqrt{-de}-e \left ( c\sqrt{x}-1 \right ) -e \right ) \left ( c\sqrt{-de}-e \right ) ^{-1}} \right ) }+{\frac{bd}{2\,{e}^{2}}{\it dilog} \left ({ \left ( c\sqrt{-de}+e \left ( c\sqrt{x}-1 \right ) +e \right ) \left ( c\sqrt{-de}+e \right ) ^{-1}} \right ) }+{\frac{bd\ln \left ({c}^{2}ex+{c}^{2}d \right ) }{2\,{e}^{2}}\ln \left ( 1+c\sqrt{x} \right ) }-{\frac{bd}{2\,{e}^{2}}\ln \left ( 1+c\sqrt{x} \right ) \ln \left ({ \left ( c\sqrt{-de}-e \left ( 1+c\sqrt{x} \right ) +e \right ) \left ( c\sqrt{-de}+e \right ) ^{-1}} \right ) }-{\frac{bd}{2\,{e}^{2}}\ln \left ( 1+c\sqrt{x} \right ) \ln \left ({ \left ( c\sqrt{-de}+e \left ( 1+c\sqrt{x} \right ) -e \right ) \left ( c\sqrt{-de}-e \right ) ^{-1}} \right ) }-{\frac{bd}{2\,{e}^{2}}{\it dilog} \left ({ \left ( c\sqrt{-de}-e \left ( 1+c\sqrt{x} \right ) +e \right ) \left ( c\sqrt{-de}+e \right ) ^{-1}} \right ) }-{\frac{bd}{2\,{e}^{2}}{\it dilog} \left ({ \left ( c\sqrt{-de}+e \left ( 1+c\sqrt{x} \right ) -e \right ) \left ( c\sqrt{-de}-e \right ) ^{-1}} \right ) }+{\frac{b}{ce}\sqrt{x}}+{\frac{b}{2\,{c}^{2}e}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{b}{2\,{c}^{2}e}\ln \left ( 1+c\sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} a{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + b \int \frac{x \log \left (c \sqrt{x} + 1\right )}{2 \,{\left (e x + d\right )}}\,{d x} - b \int \frac{x \log \left (-c \sqrt{x} + 1\right )}{2 \,{\left (e x + d\right )}}\,{d x} \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{b x \operatorname{artanh}\left (c \sqrt{x}\right ) + a x}{e x + d}, 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{{\left (b \operatorname{artanh}\left (c \sqrt{x}\right ) + a\right )} x}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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