Optimal. Leaf size=437 \[ -\frac{8 a^{15/4} e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (231 \sqrt{a} B+221 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{51051 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a^3 e \sqrt{e x} \sqrt{a+c x^2} (221 A+231 B x)}{51051 c}-\frac{4 a^2 e \sqrt{e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{51051 c}-\frac{16 a^4 B e^2 x \sqrt{a+c x^2}}{221 c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{16 a^{17/4} B e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{221 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 a e \sqrt{e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{36465 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c} \]
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Rubi [A] time = 0.568721, antiderivative size = 437, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {833, 815, 842, 840, 1198, 220, 1196} \[ -\frac{8 a^{15/4} e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (231 \sqrt{a} B+221 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{51051 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a^3 e \sqrt{e x} \sqrt{a+c x^2} (221 A+231 B x)}{51051 c}-\frac{4 a^2 e \sqrt{e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{51051 c}-\frac{16 a^4 B e^2 x \sqrt{a+c x^2}}{221 c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{16 a^{17/4} B e^2 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{221 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 a e \sqrt{e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{36465 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c} \]
Antiderivative was successfully verified.
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Rule 833
Rule 815
Rule 842
Rule 840
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{5/2} \, dx &=\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac{2 \int \sqrt{e x} \left (-\frac{3}{2} a B e+\frac{17}{2} A c e x\right ) \left (a+c x^2\right )^{5/2} \, dx}{17 c}\\ &=\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac{4 \int \frac{\left (-\frac{17}{4} a A c e^2-\frac{45}{4} a B c e^2 x\right ) \left (a+c x^2\right )^{5/2}}{\sqrt{e x}} \, dx}{255 c^2}\\ &=-\frac{2 a e \sqrt{e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac{16 \int \frac{\left (-\frac{221}{8} a^2 A c^2 e^4-\frac{495}{8} a^2 B c^2 e^4 x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt{e x}} \, dx}{7293 c^3 e^2}\\ &=-\frac{4 a^2 e \sqrt{e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac{2 a e \sqrt{e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac{64 \int \frac{\left (-\frac{1989}{16} a^3 A c^3 e^6-\frac{3465}{16} a^3 B c^3 e^6 x\right ) \sqrt{a+c x^2}}{\sqrt{e x}} \, dx}{153153 c^4 e^4}\\ &=-\frac{8 a^3 e \sqrt{e x} (221 A+231 B x) \sqrt{a+c x^2}}{51051 c}-\frac{4 a^2 e \sqrt{e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac{2 a e \sqrt{e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac{256 \int \frac{-\frac{9945}{32} a^4 A c^4 e^8-\frac{10395}{32} a^4 B c^4 e^8 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{2297295 c^5 e^6}\\ &=-\frac{8 a^3 e \sqrt{e x} (221 A+231 B x) \sqrt{a+c x^2}}{51051 c}-\frac{4 a^2 e \sqrt{e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac{2 a e \sqrt{e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac{\left (256 \sqrt{x}\right ) \int \frac{-\frac{9945}{32} a^4 A c^4 e^8-\frac{10395}{32} a^4 B c^4 e^8 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{2297295 c^5 e^6 \sqrt{e x}}\\ &=-\frac{8 a^3 e \sqrt{e x} (221 A+231 B x) \sqrt{a+c x^2}}{51051 c}-\frac{4 a^2 e \sqrt{e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac{2 a e \sqrt{e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac{\left (512 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{9945}{32} a^4 A c^4 e^8-\frac{10395}{32} a^4 B c^4 e^8 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{2297295 c^5 e^6 \sqrt{e x}}\\ &=-\frac{8 a^3 e \sqrt{e x} (221 A+231 B x) \sqrt{a+c x^2}}{51051 c}-\frac{4 a^2 e \sqrt{e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac{2 a e \sqrt{e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac{\left (16 a^{9/2} B e^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 )}{221 c^{3/2} \sqrt{e x}}-\frac{\left (16 a^4 \left (231 \sqrt{a} B+221 A \sqrt{c}\right ) e^2 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{51051 c^{3/2} \sqrt{e x}}\\ &=-\frac{8 a^3 e \sqrt{e x} (221 A+231 B x) \sqrt{a+c x^2}}{51051 c}-\frac{16 a^4 B e^2 x \sqrt{a+c x^2}}{221 c^{3/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{4 a^2 e \sqrt{e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac{2 a e \sqrt{e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac{2 A e \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac{16 a^{17/4} B e^2 \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 )}{221 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a^{15/4} \left (231 \sqrt{a} B+221 A \sqrt{c}\right ) e^2 \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 )}{51051 c^{7/4} \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.138882, size = 124, normalized size = 0.28 \[ \frac{2 e \sqrt{e x} \sqrt{a+c x^2} \left (-17 a^3 A \, _2F_1\left (-\frac{5}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{a}\right )-15 a^3 B x \, _2F_1\left (-\frac{5}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{a}\right )+\left (a+c x^2\right )^3 \sqrt{\frac{c x^2}{a}+1} (17 A+15 B x)\right )}{255 c \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 390, normalized size = 0.9 \begin{align*} -{\frac{2\,e}{255255\,x{c}^{2}}\sqrt{ex} \left ( -15015\,B{x}^{10}{c}^{5}-17017\,A{x}^{9}{c}^{5}-56595\,B{x}^{8}a{c}^{4}-66521\,A{x}^{7}a{c}^{4}-75845\,B{x}^{6}{a}^{2}{c}^{3}+4420\,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 ) \sqrt{-ac}{a}^{4}-95251\,A{x}^{5}{a}^{2}{c}^{3}+9240\,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 ){a}^{5}-4620\,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 ){a}^{5}-37345\,B{x}^{4}{a}^{3}{c}^{2}-54587\,A{x}^{3}{a}^{3}{c}^{2}-3080\,B{x}^{2}{a}^{4}c-8840\,Ax{a}^{4}c \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \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{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )} \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 ({\left (B c^{2} e x^{6} + A c^{2} e x^{5} + 2 \, B a c e x^{4} + 2 \, A a c e x^{3} + B a^{2} e x^{2} + A a^{2} e x\right )} \sqrt{c x^{2} + a} \sqrt{e x}, 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{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )} \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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