Optimal. Leaf size=292 \[ -\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{5/4} c^{5/4}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{5/4} c^{5/4}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} c^{5/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} c^{5/4}}-\frac{\sqrt{x} (a B-A c x)}{2 a c \left (a+c x^2\right )} \]
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Rubi [A] time = 0.226462, antiderivative size = 292, 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 = {821, 827, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{5/4} c^{5/4}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{5/4} c^{5/4}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} c^{5/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} c^{5/4}}-\frac{\sqrt{x} (a B-A c x)}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 821
Rule 827
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt{x} (A+B x)}{\left (a+c x^2\right )^2} \, dx &=-\frac{\sqrt{x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac{\int \frac{\frac{a B}{2}+\frac{A c x}{2}}{\sqrt{x} \left (a+c x^2\right )} \, dx}{2 a c}\\ &=-\frac{\sqrt{x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{a B}{2}+\frac{1}{2} A c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{a c}\\ &=-\frac{\sqrt{x} (a B-A c x)}{2 a c \left (a+c x^2\right )}-\frac{\left (A-\frac{\sqrt{a} B}{\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 c}+\frac{\left (A+\frac{\sqrt{a} B}{\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 c}\\ &=-\frac{\sqrt{x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac{\left (A+\frac{\sqrt{a} B}{\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 c}+\frac{\left (A+\frac{\sqrt{a} B}{\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 c}+\frac{\left (A-\frac{\sqrt{a} B}{\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^{5/4} c^{3/4}}+\frac{\left (A-\frac{\sqrt{a} B}{\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^{5/4} c^{3/4}}\\ &=-\frac{\sqrt{x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac{\left (A-\frac{\sqrt{a} B}{\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^{5/4} c^{3/4}}-\frac{\left (A-\frac{\sqrt{a} B}{\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^{5/4} c^{3/4}}+\frac{\left (\sqrt{a} B+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^{5/4} c^{5/4}}-\frac{\left (\sqrt{a} B+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^{5/4} c^{5/4}}\\ &=-\frac{\sqrt{x} (a B-A c x)}{2 a c \left (a+c x^2\right )}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} c^{5/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} c^{5/4}}+\frac{\left (A-\frac{\sqrt{a} B}{\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^{5/4} c^{3/4}}-\frac{\left (A-\frac{\sqrt{a} B}{\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^{5/4} c^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.244507, size = 315, normalized size = 1.08 \[ \frac{-\frac{\sqrt{2} a^{5/4} B \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{c^{5/4}}+\frac{\sqrt{2} a^{5/4} B \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{c^{5/4}}-\frac{2 \sqrt{2} a^{5/4} B \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{c^{5/4}}+\frac{2 \sqrt{2} a^{5/4} B \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{c^{5/4}}-\frac{4 (-a)^{3/4} A \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac{4 (-a)^{3/4} A \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac{8 a A x^{3/2}}{a+c x^2}+\frac{8 a B x^{5/2}}{a+c x^2}-\frac{8 a B \sqrt{x}}{c}}{16 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 316, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{c{x}^{2}+a} \left ( 1/4\,{\frac{A{x}^{3/2}}{a}}-1/4\,{\frac{B\sqrt{x}}{c}} \right ) }+{\frac{B\sqrt{2}}{8\,ac}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{B\sqrt{2}}{8\,ac}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{B\sqrt{2}}{16\,ac}\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{A\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{A\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{A\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 = 1.52791, size = 1831, normalized size = 6.27 \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 [A] time = 114.659, size = 151, normalized size = 0.52 \begin{align*} \frac{2 A x^{\frac{3}{2}}}{4 a^{2} + 4 a c x^{2}} + 2 A \operatorname{RootSum}{\left (65536 t^{4} a^{5} c^{3} + 1, \left ( t \mapsto t \log{\left (4096 t^{3} a^{4} c^{2} + \sqrt{x} \right )} \right )\right )} - \frac{2 B a \sqrt{x}}{4 a^{2} c + 4 a c^{2} x^{2}} - \frac{2 B a \operatorname{RootSum}{\left (65536 t^{4} a^{7} c + 81, \left ( t \mapsto t \log{\left (\frac{16 t a^{2}}{3} + \sqrt{x} \right )} \right )\right )}}{c} + \frac{2 B \operatorname{RootSum}{\left (256 t^{4} a^{3} c + 1, \left ( t \mapsto t \log{\left (4 t a + \sqrt{x} \right )} \right )\right )}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34301, size = 379, normalized size = 1.3 \begin{align*} \frac{A c x^{\frac{3}{2}} - B a \sqrt{x}}{2 \,{\left (c x^{2} + a\right )} a c} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c + \left (a c^{3}\right )^{\frac{3}{4}} A\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 (\left (a c^{3}\right )^{\frac{1}{4}} B a c - \left (a c^{3}\right )^{\frac{3}{4}} A\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 (\left (a c^{3}\right )^{\frac{1}{4}} B a c^{3} + \left (a c^{3}\right )^{\frac{3}{4}} A c^{2}\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^{5}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c^{3} - \left (a c^{3}\right )^{\frac{3}{4}} A c^{2}\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^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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