Optimal. Leaf size=301 \[ -\frac{2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}-\frac{b c^{3/4} \log \left (\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{3 \sqrt{2} d^{5/2}}+\frac{b c^{3/4} \log \left (\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{3 \sqrt{2} d^{5/2}}+\frac{2 b c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{\sqrt{2} b c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{\sqrt{2} b c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}+1\right )}{3 d^{5/2}}+\frac{2 b c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}} \]
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Rubi [A] time = 0.249511, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {6097, 16, 329, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ -\frac{2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}-\frac{b c^{3/4} \log \left (\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{3 \sqrt{2} d^{5/2}}+\frac{b c^{3/4} \log \left (\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{3 \sqrt{2} d^{5/2}}+\frac{2 b c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{\sqrt{2} b c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{\sqrt{2} b c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}+1\right )}{3 d^{5/2}}+\frac{2 b c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 16
Rule 329
Rule 214
Rule 212
Rule 208
Rule 205
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^2\right )}{(d x)^{5/2}} \, dx &=-\frac{2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac{(4 b c) \int \frac{x}{(d x)^{3/2} \left (1-c^2 x^4\right )} \, dx}{3 d}\\ &=-\frac{2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac{(4 b c) \int \frac{1}{\sqrt{d x} \left (1-c^2 x^4\right )} \, dx}{3 d^2}\\ &=-\frac{2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac{(8 b c) \operatorname{Subst}\left (\int \frac{1}{1-\frac{c^2 x^8}{d^4}} \, dx,x,\sqrt{d x}\right )}{3 d^3}\\ &=-\frac{2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac{(4 b c) \operatorname{Subst}\left (\int \frac{1}{d^2-c x^4} \, dx,x,\sqrt{d x}\right )}{3 d}+\frac{(4 b c) \operatorname{Subst}\left (\int \frac{1}{d^2+c x^4} \, dx,x,\sqrt{d x}\right )}{3 d}\\ &=-\frac{2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac{(2 b c) \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{c} x^2} \, dx,x,\sqrt{d x}\right )}{3 d^2}+\frac{(2 b c) \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{c} x^2} \, dx,x,\sqrt{d x}\right )}{3 d^2}+\frac{(2 b c) \operatorname{Subst}\left (\int \frac{d-\sqrt{c} x^2}{d^2+c x^4} \, dx,x,\sqrt{d x}\right )}{3 d^2}+\frac{(2 b c) \operatorname{Subst}\left (\int \frac{d+\sqrt{c} x^2}{d^2+c x^4} \, dx,x,\sqrt{d x}\right )}{3 d^2}\\ &=\frac{2 b c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac{2 b c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{\left (b c^{3/4}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{d}}{\sqrt [4]{c}}+2 x}{-\frac{d}{\sqrt{c}}-\frac{\sqrt{2} \sqrt{d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{d x}\right )}{3 \sqrt{2} d^{5/2}}-\frac{\left (b c^{3/4}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{d}}{\sqrt [4]{c}}-2 x}{-\frac{d}{\sqrt{c}}+\frac{\sqrt{2} \sqrt{d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{d x}\right )}{3 \sqrt{2} d^{5/2}}+\frac{\left (b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d}{\sqrt{c}}-\frac{\sqrt{2} \sqrt{d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d x}\right )}{3 d^2}+\frac{\left (b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d}{\sqrt{c}}+\frac{\sqrt{2} \sqrt{d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{d x}\right )}{3 d^2}\\ &=\frac{2 b c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac{2 b c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{b c^{3/4} \log \left (\sqrt{d}+\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt{d x}\right )}{3 \sqrt{2} d^{5/2}}+\frac{b c^{3/4} \log \left (\sqrt{d}+\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt{d x}\right )}{3 \sqrt{2} d^{5/2}}+\frac{\left (\sqrt{2} b c^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{\left (\sqrt{2} b c^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}\\ &=\frac{2 b c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{\sqrt{2} b c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}+\frac{\sqrt{2} b c^{3/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac{2 b c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{b c^{3/4} \log \left (\sqrt{d}+\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt{d x}\right )}{3 \sqrt{2} d^{5/2}}+\frac{b c^{3/4} \log \left (\sqrt{d}+\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt{d x}\right )}{3 \sqrt{2} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.105879, size = 268, normalized size = 0.89 \[ -\frac{x \left (4 a+2 b c^{3/4} x^{3/2} \log \left (1-\sqrt [4]{c} \sqrt{x}\right )-2 b c^{3/4} x^{3/2} \log \left (\sqrt [4]{c} \sqrt{x}+1\right )+\sqrt{2} b c^{3/4} x^{3/2} \log \left (\sqrt{c} x-\sqrt{2} \sqrt [4]{c} \sqrt{x}+1\right )-\sqrt{2} b c^{3/4} x^{3/2} \log \left (\sqrt{c} x+\sqrt{2} \sqrt [4]{c} \sqrt{x}+1\right )+2 \sqrt{2} b c^{3/4} x^{3/2} \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{c} \sqrt{x}\right )-2 \sqrt{2} b c^{3/4} x^{3/2} \tan ^{-1}\left (\sqrt{2} \sqrt [4]{c} \sqrt{x}+1\right )-4 b c^{3/4} x^{3/2} \tan ^{-1}\left (\sqrt [4]{c} \sqrt{x}\right )+4 b \tanh ^{-1}\left (c x^2\right )\right )}{6 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 280, normalized size = 0.9 \begin{align*} -{\frac{2\,a}{3\,d} \left ( dx \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,b{\it Artanh} \left ( c{x}^{2} \right ) }{3\,d} \left ( dx \right ) ^{-{\frac{3}{2}}}}+{\frac{bc\sqrt{2}}{6\,{d}^{3}}\sqrt [4]{{\frac{{d}^{2}}{c}}}\ln \left ({ \left ( dx+\sqrt [4]{{\frac{{d}^{2}}{c}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{{d}^{2}}{c}}} \right ) \left ( dx-\sqrt [4]{{\frac{{d}^{2}}{c}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{{d}^{2}}{c}}} \right ) ^{-1}} \right ) }+{\frac{bc\sqrt{2}}{3\,{d}^{3}}\sqrt [4]{{\frac{{d}^{2}}{c}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}}+1 \right ) }+{\frac{bc\sqrt{2}}{3\,{d}^{3}}\sqrt [4]{{\frac{{d}^{2}}{c}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}}-1 \right ) }+{\frac{bc}{3\,{d}^{3}}\sqrt [4]{{\frac{{d}^{2}}{c}}}\ln \left ({ \left ( \sqrt{dx}+\sqrt [4]{{\frac{{d}^{2}}{c}}} \right ) \left ( \sqrt{dx}-\sqrt [4]{{\frac{{d}^{2}}{c}}} \right ) ^{-1}} \right ) }+{\frac{2\,bc}{3\,{d}^{3}}\sqrt [4]{{\frac{{d}^{2}}{c}}}\arctan \left ({\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}} \right ) } \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 = 2.26762, size = 89, normalized size = 0.3 \begin{align*} -\frac{\sqrt{d x}{\left (b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a\right )}}{3 \, d^{3} 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 [B] time = 3.18579, size = 698, normalized size = 2.32 \begin{align*} \frac{1}{6} \, b c d^{2}{\left (\frac{2 \, \sqrt{2} \left (c^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{d^{2}}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{d^{2}}{c}\right )^{\frac{1}{4}}}\right )}{c d^{5}} + \frac{2 \, \sqrt{2} \left (c^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{d^{2}}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{d^{2}}{c}\right )^{\frac{1}{4}}}\right )}{c d^{5}} + \frac{2 \, \sqrt{2} \left (-c^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-\frac{d^{2}}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (-\frac{d^{2}}{c}\right )^{\frac{1}{4}}}\right )}{c d^{5}} + \frac{2 \, \sqrt{2} \left (-c^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-\frac{d^{2}}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (-\frac{d^{2}}{c}\right )^{\frac{1}{4}}}\right )}{c d^{5}} + \frac{\sqrt{2} \left (c^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x + \sqrt{2} \sqrt{d x} \left (\frac{d^{2}}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{d^{2}}{c}}\right )}{c d^{5}} - \frac{\sqrt{2} \left (c^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x - \sqrt{2} \sqrt{d x} \left (\frac{d^{2}}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{d^{2}}{c}}\right )}{c d^{5}} + \frac{\sqrt{2} \left (-c^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x + \sqrt{2} \sqrt{d x} \left (-\frac{d^{2}}{c}\right )^{\frac{1}{4}} + \sqrt{-\frac{d^{2}}{c}}\right )}{c d^{5}} - \frac{\sqrt{2} \left (-c^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x - \sqrt{2} \sqrt{d x} \left (-\frac{d^{2}}{c}\right )^{\frac{1}{4}} + \sqrt{-\frac{d^{2}}{c}}\right )}{c d^{5}}\right )} - \frac{\frac{b \log \left (-\frac{c d^{2} x^{2} + d^{2}}{c d^{2} x^{2} - d^{2}}\right )}{\sqrt{d x} d x} + \frac{2 \, a}{\sqrt{d x} d x}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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