Optimal. Leaf size=188 \[ -\frac{a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac{1}{8} b c^{4/3} \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt{x}+1\right )+\frac{1}{8} b c^{4/3} \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt{x}+1\right )+\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} \sqrt{x}}{\sqrt{3}}\right )-\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c} \sqrt{x}+1}{\sqrt{3}}\right )+\frac{1}{2} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )-\frac{3 b c}{2 \sqrt{x}} \]
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Rubi [A] time = 0.286581, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {6097, 325, 329, 296, 634, 618, 204, 628, 206} \[ -\frac{a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac{1}{8} b c^{4/3} \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt{x}+1\right )+\frac{1}{8} b c^{4/3} \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt{x}+1\right )+\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} \sqrt{x}}{\sqrt{3}}\right )-\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c} \sqrt{x}+1}{\sqrt{3}}\right )+\frac{1}{2} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )-\frac{3 b c}{2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 325
Rule 329
Rule 296
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x^3} \, dx &=-\frac{a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}+\frac{1}{4} (3 b c) \int \frac{1}{x^{3/2} \left (1-c^2 x^3\right )} \, dx\\ &=-\frac{3 b c}{2 \sqrt{x}}-\frac{a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}+\frac{1}{4} \left (3 b c^3\right ) \int \frac{x^{3/2}}{1-c^2 x^3} \, dx\\ &=-\frac{3 b c}{2 \sqrt{x}}-\frac{a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}+\frac{1}{2} \left (3 b c^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{1-c^2 x^6} \, dx,x,\sqrt{x}\right )\\ &=-\frac{3 b c}{2 \sqrt{x}}-\frac{a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}+\frac{1}{2} \left (b c^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^{2/3} x^2} \, dx,x,\sqrt{x}\right )+\frac{1}{2} \left (b c^{5/3}\right ) \operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )+\frac{1}{2} \left (b c^{5/3}\right ) \operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{3 b c}{2 \sqrt{x}}+\frac{1}{2} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )-\frac{a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac{1}{8} \left (b c^{4/3}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )+\frac{1}{8} \left (b c^{4/3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )-\frac{1}{8} \left (3 b c^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )-\frac{1}{8} \left (3 b c^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{3 b c}{2 \sqrt{x}}+\frac{1}{2} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )-\frac{a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac{1}{8} b c^{4/3} \log \left (1-\sqrt [3]{c} \sqrt{x}+c^{2/3} x\right )+\frac{1}{8} b c^{4/3} \log \left (1+\sqrt [3]{c} \sqrt{x}+c^{2/3} x\right )-\frac{1}{4} \left (3 b c^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} \sqrt{x}\right )+\frac{1}{4} \left (3 b c^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} \sqrt{x}\right )\\ &=-\frac{3 b c}{2 \sqrt{x}}+\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} \sqrt{x}}{\sqrt{3}}\right )-\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left (\frac{1+2 \sqrt [3]{c} \sqrt{x}}{\sqrt{3}}\right )+\frac{1}{2} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )-\frac{a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac{1}{8} b c^{4/3} \log \left (1-\sqrt [3]{c} \sqrt{x}+c^{2/3} x\right )+\frac{1}{8} b c^{4/3} \log \left (1+\sqrt [3]{c} \sqrt{x}+c^{2/3} x\right )\\ \end{align*}
Mathematica [A] time = 0.058352, size = 220, normalized size = 1.17 \[ -\frac{a}{2 x^2}-\frac{1}{4} b c^{4/3} \log \left (1-\sqrt [3]{c} \sqrt{x}\right )+\frac{1}{4} b c^{4/3} \log \left (\sqrt [3]{c} \sqrt{x}+1\right )-\frac{1}{8} b c^{4/3} \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt{x}+1\right )+\frac{1}{8} b c^{4/3} \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt{x}+1\right )-\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c} \sqrt{x}-1}{\sqrt{3}}\right )-\frac{1}{4} \sqrt{3} b c^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c} \sqrt{x}+1}{\sqrt{3}}\right )-\frac{b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac{3 b c}{2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 180, normalized size = 1. \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{b}{2\,{x}^{2}}{\it Artanh} \left ( c{x}^{{\frac{3}{2}}} \right ) }-{\frac{3\,bc}{2}{\frac{1}{\sqrt{x}}}}-{\frac{bc}{4}\ln \left ( \sqrt{x}-\sqrt [3]{{c}^{-1}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}+{\frac{bc}{8}\ln \left ( x+\sqrt [3]{{c}^{-1}}\sqrt{x}+ \left ({c}^{-1} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}-{\frac{bc\sqrt{3}}{4}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt{x}}{\sqrt [3]{{c}^{-1}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}+{\frac{bc}{4}\ln \left ( \sqrt{x}+\sqrt [3]{{c}^{-1}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}-{\frac{bc}{8}\ln \left ( x-\sqrt [3]{{c}^{-1}}\sqrt{x}+ \left ({c}^{-1} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}-{\frac{bc\sqrt{3}}{4}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt{x}}{\sqrt [3]{{c}^{-1}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}} \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 [A] time = 1.99585, size = 618, normalized size = 3.29 \begin{align*} -\frac{2 \, \sqrt{3} b \left (-c\right )^{\frac{1}{3}} c x^{2} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (-c\right )^{\frac{1}{3}} \sqrt{x} - \frac{1}{3} \, \sqrt{3}\right ) + 2 \, \sqrt{3} b c^{\frac{4}{3}} x^{2} \arctan \left (\frac{2}{3} \, \sqrt{3} c^{\frac{1}{3}} \sqrt{x} - \frac{1}{3} \, \sqrt{3}\right ) + b \left (-c\right )^{\frac{1}{3}} c x^{2} \log \left (c x + \left (-c\right )^{\frac{2}{3}} \sqrt{x} - \left (-c\right )^{\frac{1}{3}}\right ) + b c^{\frac{4}{3}} x^{2} \log \left (c x - c^{\frac{2}{3}} \sqrt{x} + c^{\frac{1}{3}}\right ) - 2 \, b \left (-c\right )^{\frac{1}{3}} c x^{2} \log \left (c \sqrt{x} - \left (-c\right )^{\frac{2}{3}}\right ) - 2 \, b c^{\frac{4}{3}} x^{2} \log \left (c \sqrt{x} + c^{\frac{2}{3}}\right ) + 12 \, b c x^{\frac{3}{2}} + 2 \, b \log \left (-\frac{c^{2} x^{3} + 2 \, c x^{\frac{3}{2}} + 1}{c^{2} x^{3} - 1}\right ) + 4 \, a}{8 \, x^{2}} \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 [A] time = 1.51374, size = 270, normalized size = 1.44 \begin{align*} -\frac{1}{8} \, b c^{3}{\left (\frac{2 \, \sqrt{3}{\left | c \right |}^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{x} + \frac{1}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{c^{2}} + \frac{2 \, \sqrt{3}{\left | c \right |}^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{x} - \frac{1}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{c^{2}} - \frac{{\left | c \right |}^{\frac{1}{3}} \log \left (x + \frac{\sqrt{x}}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{2}} + \frac{{\left | c \right |}^{\frac{1}{3}} \log \left (x - \frac{\sqrt{x}}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{2}} - \frac{2 \,{\left | c \right |}^{\frac{1}{3}} \log \left (\sqrt{x} + \frac{1}{{\left | c \right |}^{\frac{1}{3}}}\right )}{c^{2}} + \frac{2 \,{\left | c \right |}^{\frac{1}{3}} \log \left ({\left | \sqrt{x} - \frac{1}{{\left | c \right |}^{\frac{1}{3}}} \right |}\right )}{c^{2}}\right )} - \frac{b \log \left (-\frac{c x^{\frac{3}{2}} + 1}{c x^{\frac{3}{2}} - 1}\right )}{4 \, x^{2}} - \frac{3 \, b c x^{\frac{3}{2}} + a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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