Optimal. Leaf size=301 \[ \frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}+\frac{b \sqrt{d} \log \left (\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{3 \sqrt{2} c^{3/4}}-\frac{b \sqrt{d} \log \left (\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{3 \sqrt{2} c^{3/4}}+\frac{2 b \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}-\frac{\sqrt{2} b \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}+\frac{\sqrt{2} b \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}+1\right )}{3 c^{3/4}}-\frac{2 b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}} \]
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Rubi [A] time = 0.250824, 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, 301, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ \frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}+\frac{b \sqrt{d} \log \left (\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{3 \sqrt{2} c^{3/4}}-\frac{b \sqrt{d} \log \left (\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{3 \sqrt{2} c^{3/4}}+\frac{2 b \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}-\frac{\sqrt{2} b \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}+\frac{\sqrt{2} b \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}+1\right )}{3 c^{3/4}}-\frac{2 b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 16
Rule 329
Rule 301
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \sqrt{d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac{(4 b c) \int \frac{x (d x)^{3/2}}{1-c^2 x^4} \, dx}{3 d}\\ &=\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac{(4 b c) \int \frac{(d x)^{5/2}}{1-c^2 x^4} \, dx}{3 d^2}\\ &=\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac{(8 b c) \operatorname{Subst}\left (\int \frac{x^6}{1-\frac{c^2 x^8}{d^4}} \, dx,x,\sqrt{d x}\right )}{3 d^3}\\ &=\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac{1}{3} (4 b d) \operatorname{Subst}\left (\int \frac{x^2}{d^2-c x^4} \, dx,x,\sqrt{d x}\right )+\frac{1}{3} (4 b d) \operatorname{Subst}\left (\int \frac{x^2}{d^2+c x^4} \, dx,x,\sqrt{d x}\right )\\ &=\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac{(2 b d) \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{c} x^2} \, dx,x,\sqrt{d x}\right )}{3 \sqrt{c}}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{c} x^2} \, dx,x,\sqrt{d x}\right )}{3 \sqrt{c}}-\frac{(2 b d) \operatorname{Subst}\left (\int \frac{d-\sqrt{c} x^2}{d^2+c x^4} \, dx,x,\sqrt{d x}\right )}{3 \sqrt{c}}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{d+\sqrt{c} x^2}{d^2+c x^4} \, dx,x,\sqrt{d x}\right )}{3 \sqrt{c}}\\ &=\frac{2 b \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}+\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac{2 b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}+\frac{\left (b \sqrt{d}\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} c^{3/4}}+\frac{\left (b \sqrt{d}\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} c^{3/4}}+\frac{(b d) \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 c}+\frac{(b d) \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 c}\\ &=\frac{2 b \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}+\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac{2 b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}+\frac{b \sqrt{d} \log \left (\sqrt{d}+\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt{d x}\right )}{3 \sqrt{2} c^{3/4}}-\frac{b \sqrt{d} \log \left (\sqrt{d}+\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt{d x}\right )}{3 \sqrt{2} c^{3/4}}+\frac{\left (\sqrt{2} b \sqrt{d}\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 c^{3/4}}-\frac{\left (\sqrt{2} b \sqrt{d}\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 c^{3/4}}\\ &=\frac{2 b \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}-\frac{\sqrt{2} b \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}+\frac{\sqrt{2} b \sqrt{d} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}+\frac{2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac{2 b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{3 c^{3/4}}+\frac{b \sqrt{d} \log \left (\sqrt{d}+\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt{d x}\right )}{3 \sqrt{2} c^{3/4}}-\frac{b \sqrt{d} \log \left (\sqrt{d}+\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt{d x}\right )}{3 \sqrt{2} c^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.071632, size = 227, normalized size = 0.75 \[ \frac{\sqrt{d x} \left (4 a c^{3/4} x^{3/2}+4 b c^{3/4} x^{3/2} \tanh ^{-1}\left (c x^2\right )+2 b \log \left (1-\sqrt [4]{c} \sqrt{x}\right )-2 b \log \left (\sqrt [4]{c} \sqrt{x}+1\right )+\sqrt{2} b \log \left (\sqrt{c} x-\sqrt{2} \sqrt [4]{c} \sqrt{x}+1\right )-\sqrt{2} b \log \left (\sqrt{c} x+\sqrt{2} \sqrt [4]{c} \sqrt{x}+1\right )-2 \sqrt{2} b \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{c} \sqrt{x}\right )+2 \sqrt{2} b \tan ^{-1}\left (\sqrt{2} \sqrt [4]{c} \sqrt{x}+1\right )+4 b \tan ^{-1}\left (\sqrt [4]{c} \sqrt{x}\right )\right )}{6 c^{3/4} \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, 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{db\sqrt{2}}{6\,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{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}}+{\frac{db\sqrt{2}}{3\,c}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}}+{\frac{db\sqrt{2}}{3\,c}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}}+{\frac{2\,db}{3\,c}\arctan \left ({\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}}-{\frac{db}{3\,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{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}} \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 = 2.15992, size = 80, normalized size = 0.27 \begin{align*} \frac{1}{3} \,{\left (b x \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a x\right )} \sqrt{d x} \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 \sqrt{d x}{\left (b \operatorname{artanh}\left (c x^{2}\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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