Optimal. Leaf size=373 \[ \frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}+\frac{7 A+5 B x}{6 a^2 e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x}}+\frac{7 A \sqrt{c} x \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{7 A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{11/4} e \sqrt{e x} \sqrt{a+c x^2}}+\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.42177, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {823, 835, 842, 840, 1198, 220, 1196} \[ \frac{7 A+5 B x}{6 a^2 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x}}+\frac{7 A \sqrt{c} x \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{7 A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{11/4} e \sqrt{e x} \sqrt{a+c x^2}}+\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 823
Rule 835
Rule 842
Rule 840
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx &=\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}}-\frac{\int \frac{-\frac{7}{2} a A c e^2-\frac{5}{2} a B c e^2 x}{(e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx}{3 a^2 c e^2}\\ &=\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}}+\frac{7 A+5 B x}{6 a^2 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{\int \frac{\frac{21}{4} a^2 A c^2 e^4+\frac{5}{4} a^2 B c^2 e^4 x}{(e x)^{3/2} \sqrt{a+c x^2}} \, dx}{3 a^4 c^2 e^4}\\ &=\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}}+\frac{7 A+5 B x}{6 a^2 e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x}}-\frac{2 \int \frac{-\frac{5}{8} a^3 B c^2 e^5-\frac{21}{8} a^2 A c^3 e^5 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{3 a^5 c^2 e^6}\\ &=\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}}+\frac{7 A+5 B x}{6 a^2 e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x}}-\frac{\left (2 \sqrt{x}\right ) \int \frac{-\frac{5}{8} a^3 B c^2 e^5-\frac{21}{8} a^2 A c^3 e^5 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{3 a^5 c^2 e^6 \sqrt{e x}}\\ &=\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}}+\frac{7 A+5 B x}{6 a^2 e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x}}-\frac{\left (4 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{5}{8} a^3 B c^2 e^5-\frac{21}{8} a^2 A c^3 e^5 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{3 a^5 c^2 e^6 \sqrt{e x}}\\ &=\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}}+\frac{7 A+5 B x}{6 a^2 e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x}}+\frac{\left (\left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{6 a^{5/2} e \sqrt{e x}}-\frac{\left (7 A \sqrt{c} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{2 a^{5/2} e \sqrt{e x}}\\ &=\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}}+\frac{7 A+5 B x}{6 a^2 e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x}}+\frac{7 A \sqrt{c} x \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{7 A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{11/4} e \sqrt{e x} \sqrt{a+c x^2}}+\frac{\left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0970794, size = 137, normalized size = 0.37 \[ \frac{x \left (-21 A \left (a+c x^2\right ) \sqrt{\frac{c x^2}{a}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c x^2}{a}\right )+9 a A+5 B x \left (a+c x^2\right ) \sqrt{\frac{c x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{a}\right )+7 a B x+7 A c x^2+5 B c x^3\right )}{6 a^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 605, normalized size = 1.6 \begin{align*} -{\frac{1}{12\,e{a}^{3}c} \left ( 21\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}a{c}^{2}-42\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}a{c}^{2}-5\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{x}^{2}ac+21\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c-42\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c-5\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{a}^{2}+42\,A{c}^{3}{x}^{4}-10\,aB{c}^{2}{x}^{3}+70\,aA{c}^{2}{x}^{2}-14\,{a}^{2}Bcx+24\,A{a}^{2}c \right ){\frac{1}{\sqrt{ex}}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{e x}}{c^{3} e^{2} x^{8} + 3 \, a c^{2} e^{2} x^{6} + 3 \, a^{2} c e^{2} x^{4} + a^{3} e^{2} x^{2}}, x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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