Optimal. Leaf size=339 \[ \frac{4 c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{35 \sqrt [4]{a} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 c \sqrt{a+c x^2} (5 A+21 B x)}{35 e^3 (e x)^{3/2}}-\frac{2 \left (a+c x^2\right )^{3/2} (5 A+7 B x)}{35 e (e x)^{7/2}}+\frac{24 B c^{3/2} x \sqrt{a+c x^2}}{5 e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{24 \sqrt [4]{a} B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^4 \sqrt{e x} \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.338008, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {811, 842, 840, 1198, 220, 1196} \[ \frac{4 c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (21 \sqrt{a} B+5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{35 \sqrt [4]{a} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 c \sqrt{a+c x^2} (5 A+21 B x)}{35 e^3 (e x)^{3/2}}-\frac{2 \left (a+c x^2\right )^{3/2} (5 A+7 B x)}{35 e (e x)^{7/2}}+\frac{24 B c^{3/2} x \sqrt{a+c x^2}}{5 e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{24 \sqrt [4]{a} B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^4 \sqrt{e x} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 811
Rule 842
Rule 840
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{(e x)^{9/2}} \, dx &=-\frac{2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}-\frac{6 \int \frac{\left (-5 a A c e^2-7 a B c e^2 x\right ) \sqrt{a+c x^2}}{(e x)^{5/2}} \, dx}{35 a e^4}\\ &=-\frac{4 c (5 A+21 B x) \sqrt{a+c x^2}}{35 e^3 (e x)^{3/2}}-\frac{2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}+\frac{4 \int \frac{5 a^2 A c^2 e^4+21 a^2 B c^2 e^4 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{35 a^2 e^8}\\ &=-\frac{4 c (5 A+21 B x) \sqrt{a+c x^2}}{35 e^3 (e x)^{3/2}}-\frac{2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}+\frac{\left (4 \sqrt{x}\right ) \int \frac{5 a^2 A c^2 e^4+21 a^2 B c^2 e^4 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{35 a^2 e^8 \sqrt{e x}}\\ &=-\frac{4 c (5 A+21 B x) \sqrt{a+c x^2}}{35 e^3 (e x)^{3/2}}-\frac{2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}+\frac{\left (8 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{5 a^2 A c^2 e^4+21 a^2 B c^2 e^4 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{35 a^2 e^8 \sqrt{e x}}\\ &=-\frac{4 c (5 A+21 B x) \sqrt{a+c x^2}}{35 e^3 (e x)^{3/2}}-\frac{2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}-\frac{\left (24 \sqrt{a} B c^{3/2} \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 )}{5 e^4 \sqrt{e x}}+\frac{\left (8 \left (21 \sqrt{a} B+5 A \sqrt{c}\right ) c^{3/2} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{35 e^4 \sqrt{e x}}\\ &=-\frac{4 c (5 A+21 B x) \sqrt{a+c x^2}}{35 e^3 (e x)^{3/2}}+\frac{24 B c^{3/2} x \sqrt{a+c x^2}}{5 e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}-\frac{24 \sqrt [4]{a} B c^{5/4} \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 )}{5 e^4 \sqrt{e x} \sqrt{a+c x^2}}+\frac{4 \left (21 \sqrt{a} B+5 A \sqrt{c}\right ) c^{5/4} \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 )}{35 \sqrt [4]{a} e^4 \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0353726, size = 89, normalized size = 0.26 \[ -\frac{2 a \sqrt{e x} \sqrt{a+c x^2} \left (5 A \, _2F_1\left (-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{c x^2}{a}\right )+7 B x \, _2F_1\left (-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{c x^2}{a}\right )\right )}{35 e^5 x^4 \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 337, normalized size = 1. \begin{align*}{\frac{2}{35\,{x}^{3}{e}^{4}} \left ( 10\,A\sqrt{-ac}\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}^{3}c+84\,B\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}^{3}ac-42\,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 ){x}^{3}ac-49\,B{c}^{2}{x}^{5}-15\,A{c}^{2}{x}^{4}-56\,aBc{x}^{3}-20\,aAc{x}^{2}-7\,{a}^{2}Bx-5\,A{a}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \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{{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{\left (e x\right )^{\frac{9}{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{{\left (B c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt{c x^{2} + a} \sqrt{e x}}{e^{5} x^{5}}, 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{{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{\left (e x\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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