Optimal. Leaf size=304 \[ -\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{9/4} \sqrt [4]{c}}+\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{9/4} \sqrt [4]{c}}-\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} \sqrt [4]{c}}+\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4} \sqrt [4]{c}}-\frac{5 A}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )} \]
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Rubi [A] time = 0.299308, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {823, 829, 827, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{9/4} \sqrt [4]{c}}+\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{9/4} \sqrt [4]{c}}-\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} \sqrt [4]{c}}+\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4} \sqrt [4]{c}}-\frac{5 A}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 823
Rule 829
Rule 827
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{3/2} \left (a+c x^2\right )^2} \, dx &=\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )}-\frac{\int \frac{-\frac{5}{2} a A c-\frac{3}{2} a B c x}{x^{3/2} \left (a+c x^2\right )} \, dx}{2 a^2 c}\\ &=-\frac{5 A}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )}-\frac{\int \frac{-\frac{3}{2} a^2 B c+\frac{5}{2} a A c^2 x}{\sqrt{x} \left (a+c x^2\right )} \, dx}{2 a^3 c}\\ &=-\frac{5 A}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{2} a^2 B c+\frac{5}{2} a A c^2 x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{a^3 c}\\ &=-\frac{5 A}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )}-\frac{\left (5 A-\frac{3 \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^2}+\frac{\left (5 A+\frac{3 \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^2}\\ &=-\frac{5 A}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )}-\frac{\left (5 A-\frac{3 \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^2}-\frac{\left (5 A-\frac{3 \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^2}-\frac{\left (3 \sqrt{a} B+5 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^{9/4} \sqrt [4]{c}}-\frac{\left (3 \sqrt{a} B+5 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^{9/4} \sqrt [4]{c}}\\ &=-\frac{5 A}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )}-\frac{\left (3 \sqrt{a} B+5 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^{9/4} \sqrt [4]{c}}+\frac{\left (3 \sqrt{a} B+5 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^{9/4} \sqrt [4]{c}}-\frac{\left (\left (5 A-\frac{3 \sqrt{a} B}{\sqrt{c}}\right ) \sqrt [4]{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^{9/4}}+\frac{\left (\left (5 A-\frac{3 \sqrt{a} B}{\sqrt{c}}\right ) \sqrt [4]{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^{9/4}}\\ &=-\frac{5 A}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )}+\frac{\left (5 A-\frac{3 \sqrt{a} B}{\sqrt{c}}\right ) \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4}}-\frac{\left (5 A-\frac{3 \sqrt{a} B}{\sqrt{c}}\right ) \sqrt [4]{c} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4}}-\frac{\left (3 \sqrt{a} B+5 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^{9/4} \sqrt [4]{c}}+\frac{\left (3 \sqrt{a} B+5 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^{9/4} \sqrt [4]{c}}\\ \end{align*}
Mathematica [C] time = 0.474794, size = 256, normalized size = 0.84 \[ \frac{\sqrt [4]{a} \left (\frac{8 a^{3/4} A}{\sqrt{x} \left (a+c x^2\right )}+\frac{8 a^{3/4} B \sqrt{x}}{a+c x^2}-\frac{3 \sqrt{2} B \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{3 \sqrt{2} B \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}-\frac{6 \sqrt{2} B \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac{6 \sqrt{2} B \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}\right )-\frac{40 A \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\frac{c x^2}{a}\right )}{\sqrt{x}}}{16 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 314, normalized size = 1. \begin{align*} -2\,{\frac{A}{{a}^{2}\sqrt{x}}}-{\frac{Ac}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{B}{2\,a \left ( c{x}^{2}+a \right ) }\sqrt{x}}+{\frac{3\,B\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\,B\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\,B\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{5\,A\sqrt{2}}{16\,{a}^{2}}\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{5\,A\sqrt{2}}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{5\,A\sqrt{2}}{8\,{a}^{2}}\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 = 1.41684, size = 1914, normalized size = 6.3 \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 = 1.4373, size = 379, normalized size = 1.25 \begin{align*} -\frac{5 \, A c x^{2} - B a x + 4 \, A a}{2 \,{\left (c x^{\frac{5}{2}} + a \sqrt{x}\right )} a^{2}} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 5 \, \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^{3} c^{2}} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 5 \, \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^{3} c^{2}} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 5 \, \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^{3} c^{2}} - \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 5 \, \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^{3} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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