Optimal. Leaf size=160 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{1-c \sqrt{x}}\right )}{c^6}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^2}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^4}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^6}+\frac{2 \log \left (\frac{2}{1-c \sqrt{x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^6}-\frac{b x^{3/2}}{6 c^3}-\frac{3 b \sqrt{x}}{2 c^5}+\frac{3 b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 c^6} \]
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Rubi [A] time = 0.428567, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {43, 5980, 5916, 302, 206, 321, 5984, 5918, 2402, 2315} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{1-c \sqrt{x}}\right )}{c^6}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^2}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^4}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^6}+\frac{2 \log \left (\frac{2}{1-c \sqrt{x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^6}-\frac{b x^{3/2}}{6 c^3}-\frac{3 b \sqrt{x}}{2 c^5}+\frac{3 b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 c^6} \]
Antiderivative was successfully verified.
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Rule 43
Rule 5980
Rule 5916
Rule 302
Rule 206
Rule 321
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{1-c^2 x} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \operatorname{Subst}\left (\int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt{x}\right )}{c^2}+\frac{2 \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{c^2}\\ &=-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^2}-\frac{2 \operatorname{Subst}\left (\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt{x}\right )}{c^4}+\frac{2 \operatorname{Subst}\left (\int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{c^4}+\frac{b \operatorname{Subst}\left (\int \frac{x^4}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{2 c}\\ &=-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^4}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^2}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^6}+\frac{2 \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx,x,\sqrt{x}\right )}{c^5}+\frac{b \operatorname{Subst}\left (\int \frac{x^2}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{c^3}+\frac{b \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x^2}{c^2}+\frac{1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx,x,\sqrt{x}\right )}{2 c}\\ &=-\frac{3 b \sqrt{x}}{2 c^5}-\frac{b x^{3/2}}{6 c^3}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^4}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^2}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^6}+\frac{2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1-c \sqrt{x}}\right )}{c^6}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{2 c^5}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{c^5}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{c^5}\\ &=-\frac{3 b \sqrt{x}}{2 c^5}-\frac{b x^{3/2}}{6 c^3}+\frac{3 b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 c^6}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^4}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^2}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^6}+\frac{2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1-c \sqrt{x}}\right )}{c^6}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c \sqrt{x}}\right )}{c^6}\\ &=-\frac{3 b \sqrt{x}}{2 c^5}-\frac{b x^{3/2}}{6 c^3}+\frac{3 b \tanh ^{-1}\left (c \sqrt{x}\right )}{2 c^6}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^4}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^2}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^6}+\frac{2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1-c \sqrt{x}}\right )}{c^6}+\frac{b \text{Li}_2\left (1-\frac{2}{1-c \sqrt{x}}\right )}{c^6}\\ \end{align*}
Mathematica [A] time = 0.379533, size = 130, normalized size = 0.81 \[ -\frac{6 b \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+3 a c^4 x^2+6 a c^2 x+6 a \log \left (1-c^2 x\right )+b c^3 x^{3/2}+3 b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (c^4 x^2+2 c^2 x-4 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )-3\right )+9 b c \sqrt{x}-6 b \tanh ^{-1}\left (c \sqrt{x}\right )^2}{6 c^6} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.05, size = 276, normalized size = 1.7 \begin{align*} -{\frac{a{x}^{2}}{2\,{c}^{2}}}-{\frac{ax}{{c}^{4}}}-{\frac{a}{{c}^{6}}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{a}{{c}^{6}}\ln \left ( 1+c\sqrt{x} \right ) }-{\frac{b{x}^{2}}{2\,{c}^{2}}{\it Artanh} \left ( c\sqrt{x} \right ) }-{\frac{bx}{{c}^{4}}{\it Artanh} \left ( c\sqrt{x} \right ) }-{\frac{b}{{c}^{6}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) }-{\frac{b}{{c}^{6}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{b}{4\,{c}^{6}} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}+{\frac{b}{{c}^{6}}{\it dilog} \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{b}{2\,{c}^{6}}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{b}{2\,{c}^{6}}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }-{\frac{b}{2\,{c}^{6}}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }+{\frac{b}{4\,{c}^{6}} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}}-{\frac{b}{6\,{c}^{3}}{x}^{{\frac{3}{2}}}}-{\frac{3\,b}{2\,{c}^{5}}\sqrt{x}}-{\frac{3\,b}{4\,{c}^{6}}\ln \left ( c\sqrt{x}-1 \right ) }+{\frac{3\,b}{4\,{c}^{6}}\ln \left ( 1+c\sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.75142, size = 281, normalized size = 1.76 \begin{align*} -\frac{1}{2} \, a{\left (\frac{c^{2} x^{2} + 2 \, x}{c^{4}} + \frac{2 \, \log \left (c^{2} x - 1\right )}{c^{6}}\right )} - \frac{{\left (\log \left (c \sqrt{x} + 1\right ) \log \left (-\frac{1}{2} \, c \sqrt{x} + \frac{1}{2}\right ) +{\rm Li}_2\left (\frac{1}{2} \, c \sqrt{x} + \frac{1}{2}\right )\right )} b}{c^{6}} + \frac{3 \, b \log \left (c \sqrt{x} + 1\right )}{4 \, c^{6}} - \frac{3 \, b \log \left (c \sqrt{x} - 1\right )}{4 \, c^{6}} - \frac{2 \, b c^{3} x^{\frac{3}{2}} + 3 \, b \log \left (c \sqrt{x} + 1\right )^{2} - 3 \, b \log \left (-c \sqrt{x} + 1\right )^{2} + 18 \, b c \sqrt{x} + 3 \,{\left (b c^{4} x^{2} + 2 \, b c^{2} x\right )} \log \left (c \sqrt{x} + 1\right ) - 3 \,{\left (b c^{4} x^{2} + 2 \, b c^{2} x + 2 \, b \log \left (c \sqrt{x} + 1\right )\right )} \log \left (-c \sqrt{x} + 1\right )}{12 \, c^{6}} \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^{2} \operatorname{artanh}\left (c \sqrt{x}\right ) + a x^{2}}{c^{2} x - 1}, 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 x^{2}}{c^{2} x - 1}\, dx - \int \frac{b x^{2} \operatorname{atanh}{\left (c \sqrt{x} \right )}}{c^{2} x - 1}\, 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{{\left (b \operatorname{artanh}\left (c \sqrt{x}\right ) + a\right )} x^{2}}{c^{2} x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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