Optimal. Leaf size=306 \[ \frac{c^{5/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}-\frac{c^{5/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}-\frac{c^{5/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{11/4}}+\frac{c^{5/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{11/4}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{2 B c}{a^2 \sqrt{x}}-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}} \]
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Rubi [A] time = 0.368087, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {829, 827, 1168, 1162, 617, 204, 1165, 628} \[ \frac{c^{5/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}-\frac{c^{5/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}-\frac{c^{5/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{11/4}}+\frac{c^{5/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{11/4}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{2 B c}{a^2 \sqrt{x}}-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
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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^{9/2} \left (a+c x^2\right )} \, dx &=-\frac{2 A}{7 a x^{7/2}}+\frac{\int \frac{a B-A c x}{x^{7/2} \left (a+c x^2\right )} \, dx}{a}\\ &=-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}}+\frac{\int \frac{-a A c-a B c x}{x^{5/2} \left (a+c x^2\right )} \, dx}{a^2}\\ &=-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{\int \frac{-a^2 B c+a A c^2 x}{x^{3/2} \left (a+c x^2\right )} \, dx}{a^3}\\ &=-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{2 B c}{a^2 \sqrt{x}}+\frac{\int \frac{a^2 A c^2+a^2 B c^2 x}{\sqrt{x} \left (a+c x^2\right )} \, dx}{a^4}\\ &=-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{2 B c}{a^2 \sqrt{x}}+\frac{2 \operatorname{Subst}\left (\int \frac{a^2 A c^2+a^2 B c^2 x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{a^4}\\ &=-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{2 B c}{a^2 \sqrt{x}}-\frac{\left (\left (\sqrt{a} B-A \sqrt{c}\right ) c\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{a^{5/2}}+\frac{\left (\left (\sqrt{a} B+A \sqrt{c}\right ) c\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{a^{5/2}}\\ &=-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{2 B c}{a^2 \sqrt{x}}+\frac{\left (\left (\sqrt{a} B+A \sqrt{c}\right ) 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 )}{2 a^{5/2}}+\frac{\left (\left (\sqrt{a} B+A \sqrt{c}\right ) 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 )}{2 a^{5/2}}+\frac{\left (\left (\sqrt{a} B-A \sqrt{c}\right ) c^{5/4}\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 )}{2 \sqrt{2} a^{11/4}}+\frac{\left (\left (\sqrt{a} B-A \sqrt{c}\right ) c^{5/4}\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 )}{2 \sqrt{2} a^{11/4}}\\ &=-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{2 B c}{a^2 \sqrt{x}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) c^{5/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}-\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) c^{5/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}+\frac{\left (\left (\sqrt{a} B+A \sqrt{c}\right ) c^{5/4}\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 )}{\sqrt{2} a^{11/4}}-\frac{\left (\left (\sqrt{a} B+A \sqrt{c}\right ) c^{5/4}\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 )}{\sqrt{2} a^{11/4}}\\ &=-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{2 B c}{a^2 \sqrt{x}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{11/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) c^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{11/4}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) c^{5/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}-\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) c^{5/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}\\ \end{align*}
Mathematica [C] time = 0.0141569, size = 54, normalized size = 0.18 \[ -\frac{2 \left (5 A \, _2F_1\left (-\frac{7}{4},1;-\frac{3}{4};-\frac{c x^2}{a}\right )+7 B x \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};-\frac{c x^2}{a}\right )\right )}{35 a x^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 318, normalized size = 1. \begin{align*} -{\frac{2\,A}{7\,a}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{5\,a}{x}^{-{\frac{5}{2}}}}+{\frac{2\,Ac}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+2\,{\frac{Bc}{{a}^{2}\sqrt{x}}}+{\frac{A{c}^{2}\sqrt{2}}{4\,{a}^{3}}\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{c}^{2}\sqrt{2}}{2\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{A{c}^{2}\sqrt{2}}{2\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{Bc\sqrt{2}}{4\,{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{Bc\sqrt{2}}{2\,{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{Bc\sqrt{2}}{2\,{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 = 1.63339, size = 1782, normalized size = 5.82 \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 = 1.3126, size = 373, normalized size = 1.22 \begin{align*} \frac{\sqrt{2}{\left (\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 )}{2 \, a^{3} c} + \frac{\sqrt{2}{\left (\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 )}{2 \, a^{3} c} + \frac{\sqrt{2}{\left (\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 )}{4 \, a^{3} c} - \frac{\sqrt{2}{\left (\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 )}{4 \, a^{3} c} + \frac{2 \,{\left (105 \, B c x^{3} + 35 \, A c x^{2} - 21 \, B a x - 15 \, A a\right )}}{105 \, a^{2} x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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