Optimal. Leaf size=195 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{1-c \sqrt{x}}\right )}{c^8}-\frac{x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{3 c^2}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^4}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^6}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^8}+\frac{2 \log \left (\frac{2}{1-c \sqrt{x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^8}-\frac{b x^{5/2}}{15 c^3}-\frac{5 b x^{3/2}}{18 c^5}-\frac{11 b \sqrt{x}}{6 c^7}+\frac{11 b \tanh ^{-1}\left (c \sqrt{x}\right )}{6 c^8} \]
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Rubi [A] time = 0.600731, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 19, 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} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{1-c \sqrt{x}}\right )}{c^8}-\frac{x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{3 c^2}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^4}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^6}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^8}+\frac{2 \log \left (\frac{2}{1-c \sqrt{x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^8}-\frac{b x^{5/2}}{15 c^3}-\frac{5 b x^{3/2}}{18 c^5}-\frac{11 b \sqrt{x}}{6 c^7}+\frac{11 b \tanh ^{-1}\left (c \sqrt{x}\right )}{6 c^8} \]
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^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{1-c^2 x} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^7 \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^5 \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt{x}\right )}{c^2}+\frac{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 )}{c^2}\\ &=-\frac{x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{3 c^2}-\frac{2 \operatorname{Subst}\left (\int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt{x}\right )}{c^4}+\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^4}+\frac{b \operatorname{Subst}\left (\int \frac{x^6}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{3 c}\\ &=-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^4}-\frac{x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{3 c^2}-\frac{2 \operatorname{Subst}\left (\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt{x}\right )}{c^6}+\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^6}+\frac{b \operatorname{Subst}\left (\int \frac{x^4}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{2 c^3}+\frac{b \operatorname{Subst}\left (\int \left (-\frac{1}{c^6}-\frac{x^2}{c^4}-\frac{x^4}{c^2}+\frac{1}{c^6 \left (1-c^2 x^2\right )}\right ) \, dx,x,\sqrt{x}\right )}{3 c}\\ &=-\frac{b \sqrt{x}}{3 c^7}-\frac{b x^{3/2}}{9 c^5}-\frac{b x^{5/2}}{15 c^3}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^6}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^4}-\frac{x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{3 c^2}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^8}+\frac{2 \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx,x,\sqrt{x}\right )}{c^7}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{3 c^7}+\frac{b \operatorname{Subst}\left (\int \frac{x^2}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{c^5}+\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^3}\\ &=-\frac{11 b \sqrt{x}}{6 c^7}-\frac{5 b x^{3/2}}{18 c^5}-\frac{b x^{5/2}}{15 c^3}+\frac{b \tanh ^{-1}\left (c \sqrt{x}\right )}{3 c^8}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^6}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^4}-\frac{x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{3 c^2}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^8}+\frac{2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1-c \sqrt{x}}\right )}{c^8}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{2 c^7}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )}{c^7}-\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^7}\\ &=-\frac{11 b \sqrt{x}}{6 c^7}-\frac{5 b x^{3/2}}{18 c^5}-\frac{b x^{5/2}}{15 c^3}+\frac{11 b \tanh ^{-1}\left (c \sqrt{x}\right )}{6 c^8}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^6}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^4}-\frac{x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{3 c^2}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^8}+\frac{2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1-c \sqrt{x}}\right )}{c^8}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c \sqrt{x}}\right )}{c^8}\\ &=-\frac{11 b \sqrt{x}}{6 c^7}-\frac{5 b x^{3/2}}{18 c^5}-\frac{b x^{5/2}}{15 c^3}+\frac{11 b \tanh ^{-1}\left (c \sqrt{x}\right )}{6 c^8}-\frac{x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{c^6}-\frac{x^2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{2 c^4}-\frac{x^3 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )}{3 c^2}-\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b c^8}+\frac{2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (\frac{2}{1-c \sqrt{x}}\right )}{c^8}+\frac{b \text{Li}_2\left (1-\frac{2}{1-c \sqrt{x}}\right )}{c^8}\\ \end{align*}
Mathematica [A] time = 0.552083, size = 160, normalized size = 0.82 \[ -\frac{90 b \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )+30 a c^6 x^3+45 a c^4 x^2+90 a c^2 x+90 a \log \left (1-c^2 x\right )+6 b c^5 x^{5/2}+25 b c^3 x^{3/2}+15 b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (2 c^6 x^3+3 c^4 x^2+6 c^2 x-12 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}+1\right )-11\right )+165 b c \sqrt{x}-90 b \tanh ^{-1}\left (c \sqrt{x}\right )^2}{90 c^8} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.053, size = 309, normalized size = 1.6 \begin{align*} -{\frac{{x}^{3}a}{3\,{c}^{2}}}-{\frac{a{x}^{2}}{2\,{c}^{4}}}-{\frac{ax}{{c}^{6}}}-{\frac{a}{{c}^{8}}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{a}{{c}^{8}}\ln \left ( 1+c\sqrt{x} \right ) }-{\frac{b{x}^{3}}{3\,{c}^{2}}{\it Artanh} \left ( c\sqrt{x} \right ) }-{\frac{b{x}^{2}}{2\,{c}^{4}}{\it Artanh} \left ( c\sqrt{x} \right ) }-{\frac{bx}{{c}^{6}}{\it Artanh} \left ( c\sqrt{x} \right ) }-{\frac{b}{{c}^{8}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) }-{\frac{b}{{c}^{8}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{b}{4\,{c}^{8}} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}+{\frac{b}{{c}^{8}}{\it dilog} \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{b}{2\,{c}^{8}}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{b}{2\,{c}^{8}}\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}^{8}}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }+{\frac{b}{4\,{c}^{8}} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}}-{\frac{b}{15\,{c}^{3}}{x}^{{\frac{5}{2}}}}-{\frac{5\,b}{18\,{c}^{5}}{x}^{{\frac{3}{2}}}}-{\frac{11\,b}{6\,{c}^{7}}\sqrt{x}}-{\frac{11\,b}{12\,{c}^{8}}\ln \left ( c\sqrt{x}-1 \right ) }+{\frac{11\,b}{12\,{c}^{8}}\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 = 1.79632, size = 332, normalized size = 1.7 \begin{align*} -\frac{1}{6} \, a{\left (\frac{2 \, c^{4} x^{3} + 3 \, c^{2} x^{2} + 6 \, x}{c^{6}} + \frac{6 \, \log \left (c^{2} x - 1\right )}{c^{8}}\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^{8}} + \frac{11 \, b \log \left (c \sqrt{x} + 1\right )}{12 \, c^{8}} - \frac{11 \, b \log \left (c \sqrt{x} - 1\right )}{12 \, c^{8}} - \frac{12 \, b c^{5} x^{\frac{5}{2}} + 50 \, b c^{3} x^{\frac{3}{2}} + 45 \, b \log \left (c \sqrt{x} + 1\right )^{2} - 45 \, b \log \left (-c \sqrt{x} + 1\right )^{2} + 330 \, b c \sqrt{x} + 15 \,{\left (2 \, b c^{6} x^{3} + 3 \, b c^{4} x^{2} + 6 \, b c^{2} x\right )} \log \left (c \sqrt{x} + 1\right ) - 15 \,{\left (2 \, b c^{6} x^{3} + 3 \, b c^{4} x^{2} + 6 \, b c^{2} x + 6 \, b \log \left (c \sqrt{x} + 1\right )\right )} \log \left (-c \sqrt{x} + 1\right )}{180 \, c^{8}} \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^{3} \operatorname{artanh}\left (c \sqrt{x}\right ) + a x^{3}}{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^{3}}{c^{2} x - 1}\, dx - \int \frac{b x^{3} \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^{3}}{c^{2} x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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