Optimal. Leaf size=287 \[ \frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\sqrt{x} (A+B x)}{2 a \left (a+c x^2\right )} \]
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Rubi [A] time = 0.246486, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {823, 827, 1168, 1162, 617, 204, 1165, 628} \[ \frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\sqrt{x} (A+B x)}{2 a \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 823
Rule 827
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x}{\sqrt{x} \left (a+c x^2\right )^2} \, dx &=\frac{\sqrt{x} (A+B x)}{2 a \left (a+c x^2\right )}-\frac{\int \frac{-\frac{3}{2} a A c-\frac{1}{2} a B c x}{\sqrt{x} \left (a+c x^2\right )} \, dx}{2 a^2 c}\\ &=\frac{\sqrt{x} (A+B x)}{2 a \left (a+c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{2} a A c-\frac{1}{2} a B c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{a^2 c}\\ &=\frac{\sqrt{x} (A+B x)}{2 a \left (a+c x^2\right )}-\frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{4 a^{3/2} c}+\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{4 a^{3/2} c}\\ &=\frac{\sqrt{x} (A+B x)}{2 a \left (a+c x^2\right )}+\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{8 a^{3/2} c}+\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{8 a^{3/2} c}+\frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}\\ &=\frac{\sqrt{x} (A+B x)}{2 a \left (a+c x^2\right )}+\frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} c^{3/4}}\\ &=\frac{\sqrt{x} (A+B x)}{2 a \left (a+c x^2\right )}-\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (\sqrt{a} B+3 A \sqrt{c}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} B-3 A \sqrt{c}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.2365, size = 304, normalized size = 1.06 \[ \frac{\frac{8 a A \sqrt{x}}{a+c x^2}-\frac{3 \sqrt{2} \sqrt [4]{a} A \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{3 \sqrt{2} \sqrt [4]{a} A \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}-\frac{6 \sqrt{2} \sqrt [4]{a} A \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac{6 \sqrt{2} \sqrt [4]{a} A \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}-\frac{4 (-a)^{3/4} B \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac{4 (-a)^{3/4} B \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac{8 a B x^{3/2}}{a+c x^2}}{16 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 313, normalized size = 1.1 \begin{align*}{\frac{A}{2\,a \left ( c{x}^{2}+a \right ) }\sqrt{x}}+{\frac{3\,A\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,A\sqrt{2}}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,A\sqrt{2}}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{B}{2\,a \left ( c{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{B\sqrt{2}}{16\,ac}\ln \left ({ \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{B\sqrt{2}}{8\,ac}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{B\sqrt{2}}{8\,ac}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{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 [B] time = 1.53396, size = 1862, normalized size = 6.49 \begin{align*} \text{result too large to display} \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 = 1.3659, size = 369, normalized size = 1.29 \begin{align*} \frac{B x^{\frac{3}{2}} + A \sqrt{x}}{2 \,{\left (c x^{2} + a\right )} a} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} c^{3}} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} c^{3}} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a^{2} c^{3}} - \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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