Optimal. Leaf size=339 \[ \frac{4 \sqrt [4]{a} c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+9 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{15 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 c \sqrt{a+c x^2} (9 A-5 B x)}{15 e^3 \sqrt{e x}}-\frac{2 \left (a+c x^2\right )^{3/2} (3 A+5 B x)}{15 e (e x)^{5/2}}+\frac{24 A c^{3/2} x \sqrt{a+c x^2}}{5 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{24 \sqrt [4]{a} A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^3 \sqrt{e x} \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.344087, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {811, 813, 842, 840, 1198, 220, 1196} \[ \frac{4 \sqrt [4]{a} c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+9 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 c \sqrt{a+c x^2} (9 A-5 B x)}{15 e^3 \sqrt{e x}}-\frac{2 \left (a+c x^2\right )^{3/2} (3 A+5 B x)}{15 e (e x)^{5/2}}+\frac{24 A c^{3/2} x \sqrt{a+c x^2}}{5 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{24 \sqrt [4]{a} A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^3 \sqrt{e x} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 811
Rule 813
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)^{7/2}} \, dx &=-\frac{2 (3 A+5 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{5/2}}-\frac{2 \int \frac{\left (-3 a A c e^2-5 a B c e^2 x\right ) \sqrt{a+c x^2}}{(e x)^{3/2}} \, dx}{5 a e^4}\\ &=-\frac{4 c (9 A-5 B x) \sqrt{a+c x^2}}{15 e^3 \sqrt{e x}}-\frac{2 (3 A+5 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{5/2}}+\frac{4 \int \frac{5 a^2 B c e^3+9 a A c^2 e^3 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{15 a e^6}\\ &=-\frac{4 c (9 A-5 B x) \sqrt{a+c x^2}}{15 e^3 \sqrt{e x}}-\frac{2 (3 A+5 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{5/2}}+\frac{\left (4 \sqrt{x}\right ) \int \frac{5 a^2 B c e^3+9 a A c^2 e^3 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{15 a e^6 \sqrt{e x}}\\ &=-\frac{4 c (9 A-5 B x) \sqrt{a+c x^2}}{15 e^3 \sqrt{e x}}-\frac{2 (3 A+5 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{5/2}}+\frac{\left (8 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{5 a^2 B c e^3+9 a A c^2 e^3 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{15 a e^6 \sqrt{e x}}\\ &=-\frac{4 c (9 A-5 B x) \sqrt{a+c x^2}}{15 e^3 \sqrt{e x}}-\frac{2 (3 A+5 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{5/2}}+\frac{\left (8 \sqrt{a} \left (5 \sqrt{a} B+9 A \sqrt{c}\right ) c \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{15 e^3 \sqrt{e x}}-\frac{\left (24 \sqrt{a} A 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^3 \sqrt{e x}}\\ &=-\frac{4 c (9 A-5 B x) \sqrt{a+c x^2}}{15 e^3 \sqrt{e x}}+\frac{24 A c^{3/2} x \sqrt{a+c x^2}}{5 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 (3 A+5 B x) \left (a+c x^2\right )^{3/2}}{15 e (e x)^{5/2}}-\frac{24 \sqrt [4]{a} A 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^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{4 \sqrt [4]{a} \left (5 \sqrt{a} B+9 A \sqrt{c}\right ) c^{3/4} \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 )}{15 e^3 \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0344829, size = 84, normalized size = 0.25 \[ -\frac{2 a x \sqrt{a+c x^2} \left (3 A \, _2F_1\left (-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{c x^2}{a}\right )+5 B x \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{c x^2}{a}\right )\right )}{15 (e x)^{7/2} \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 329, normalized size = 1. \begin{align*} -{\frac{2}{15\,{x}^{2}{e}^{3}} \left ( 18\,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}ac-36\,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}ac-10\,B\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}^{2}a-5\,B{c}^{2}{x}^{5}+21\,A{c}^{2}{x}^{4}+24\,aAc{x}^{2}+5\,{a}^{2}Bx+3\,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{7}{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^{4} x^{4}}, 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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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