Optimal. Leaf size=333 \[ \frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}-\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.354311, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ \frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}-\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2} \]
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 )^3} \, dx &=\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}-\frac{\int \frac{-\frac{9}{2} a A c-\frac{7}{2} a B c x}{x^{3/2} \left (a+c x^2\right )^2} \, dx}{4 a^2 c}\\ &=\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}+\frac{\int \frac{\frac{45}{4} a^2 A c^2+\frac{21}{4} a^2 B c^2 x}{x^{3/2} \left (a+c x^2\right )} \, dx}{8 a^4 c^2}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}+\frac{\int \frac{\frac{21}{4} a^3 B c^2-\frac{45}{4} a^2 A c^3 x}{\sqrt{x} \left (a+c x^2\right )} \, dx}{8 a^5 c^2}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{21}{4} a^3 B c^2-\frac{45}{4} a^2 A c^3 x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{4 a^5 c^2}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}-\frac{\left (3 \left (15 A-\frac{7 \sqrt{a} B}{\sqrt{c}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{32 a^3}+\frac{\left (3 \left (15 A+\frac{7 \sqrt{a} B}{\sqrt{c}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{32 a^3}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}-\frac{\left (3 \left (15 A-\frac{7 \sqrt{a} B}{\sqrt{c}}\right )\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 )}{64 a^3}-\frac{\left (3 \left (15 A-\frac{7 \sqrt{a} B}{\sqrt{c}}\right )\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 )}{64 a^3}-\frac{\left (3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right )\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 )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{\left (3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right )\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 )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}-\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{\left (3 \left (15 A-\frac{7 \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 )}{32 \sqrt{2} a^{13/4}}+\frac{\left (3 \left (15 A-\frac{7 \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 )}{32 \sqrt{2} a^{13/4}}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}+\frac{3 \left (15 A-\frac{7 \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 )}{32 \sqrt{2} a^{13/4}}-\frac{3 \left (15 A-\frac{7 \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 )}{32 \sqrt{2} a^{13/4}}-\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}\\ \end{align*}
Mathematica [C] time = 0.302213, size = 300, normalized size = 0.9 \[ \frac{\sqrt [4]{a} \left (\frac{32 a^{7/4} A}{\sqrt{x} \left (a+c x^2\right )^2}+\frac{72 a^{3/4} A}{\sqrt{x} \left (a+c x^2\right )}+\frac{32 a^{7/4} B \sqrt{x}}{\left (a+c x^2\right )^2}+\frac{56 a^{3/4} B \sqrt{x}}{a+c x^2}-\frac{21 \sqrt{2} B \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{21 \sqrt{2} B \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}-\frac{42 \sqrt{2} B \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac{42 \sqrt{2} B \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}\right )-\frac{360 A \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\frac{c x^2}{a}\right )}{\sqrt{x}}}{128 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 354, normalized size = 1.1 \begin{align*} -2\,{\frac{A}{{a}^{3}\sqrt{x}}}-{\frac{13\,A{c}^{2}}{16\,{a}^{3} \left ( c{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{7\,Bc}{16\,{a}^{2} \left ( c{x}^{2}+a \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{17\,Ac}{16\,{a}^{2} \left ( c{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{11\,B}{16\,a \left ( c{x}^{2}+a \right ) ^{2}}\sqrt{x}}+{\frac{21\,B\sqrt{2}}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{21\,B\sqrt{2}}{128\,{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{21\,B\sqrt{2}}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{45\,A\sqrt{2}}{128\,{a}^{3}}\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{45\,A\sqrt{2}}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{45\,A\sqrt{2}}{64\,{a}^{3}}\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 = 1.91955, size = 2283, normalized size = 6.86 \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 = 1.4102, size = 410, normalized size = 1.23 \begin{align*} -\frac{2 \, A}{a^{3} \sqrt{x}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 15 \, \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 )}{64 \, a^{4} c^{2}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 15 \, \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 )}{64 \, a^{4} c^{2}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 15 \, \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 )}{128 \, a^{4} c^{2}} - \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 15 \, \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 )}{128 \, a^{4} c^{2}} - \frac{13 \, A c^{2} x^{\frac{7}{2}} - 7 \, B a c x^{\frac{5}{2}} + 17 \, A a c x^{\frac{3}{2}} - 11 \, B a^{2} \sqrt{x}}{16 \,{\left (c x^{2} + a\right )}^{2} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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