Optimal. Leaf size=427 \[ \frac{2 a^{11/4} e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (539 \sqrt{a} B+325 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{15015 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 a^2 e^3 \sqrt{e x} \sqrt{a+c x^2} (325 A+539 B x)}{15015 c^2}+\frac{28 a^3 B e^4 x \sqrt{a+c x^2}}{195 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{28 a^{13/4} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{10 a A e^3 \sqrt{e x} \left (a+c x^2\right )^{3/2}}{77 c^2}+\frac{2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac{14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c} \]
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Rubi [A] time = 0.589852, antiderivative size = 427, 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} \[ \frac{2 a^2 e^3 \sqrt{e x} \sqrt{a+c x^2} (325 A+539 B x)}{15015 c^2}+\frac{2 a^{11/4} e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (539 \sqrt{a} B+325 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15015 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{28 a^3 B e^4 x \sqrt{a+c x^2}}{195 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{28 a^{13/4} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{10 a A e^3 \sqrt{e x} \left (a+c x^2\right )^{3/2}}{77 c^2}+\frac{2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac{14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c} \]
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)^{7/2} (A+B x) \sqrt{a+c x^2} \, dx &=\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac{2 \int (e x)^{5/2} \left (-\frac{7}{2} a B e+\frac{13}{2} A c e x\right ) \sqrt{a+c x^2} \, dx}{13 c}\\ &=\frac{2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac{4 \int (e x)^{3/2} \left (-\frac{65}{4} a A c e^2-\frac{77}{4} a B c e^2 x\right ) \sqrt{a+c x^2} \, dx}{143 c^2}\\ &=-\frac{14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac{2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac{8 \int \sqrt{e x} \left (\frac{231}{8} a^2 B c e^3-\frac{585}{8} a A c^2 e^3 x\right ) \sqrt{a+c x^2} \, dx}{1287 c^3}\\ &=-\frac{10 a A e^3 \sqrt{e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac{14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac{2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac{16 \int \frac{\left (\frac{585}{16} a^2 A c^2 e^4+\frac{1617}{16} a^2 B c^2 e^4 x\right ) \sqrt{a+c x^2}}{\sqrt{e x}} \, dx}{9009 c^4}\\ &=\frac{2 a^2 e^3 \sqrt{e x} (325 A+539 B x) \sqrt{a+c x^2}}{15015 c^2}-\frac{10 a A e^3 \sqrt{e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac{14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac{2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac{64 \int \frac{\frac{2925}{32} a^3 A c^3 e^6+\frac{4851}{32} a^3 B c^3 e^6 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{135135 c^5 e^2}\\ &=\frac{2 a^2 e^3 \sqrt{e x} (325 A+539 B x) \sqrt{a+c x^2}}{15015 c^2}-\frac{10 a A e^3 \sqrt{e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac{14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac{2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac{\left (64 \sqrt{x}\right ) \int \frac{\frac{2925}{32} a^3 A c^3 e^6+\frac{4851}{32} a^3 B c^3 e^6 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{135135 c^5 e^2 \sqrt{e x}}\\ &=\frac{2 a^2 e^3 \sqrt{e x} (325 A+539 B x) \sqrt{a+c x^2}}{15015 c^2}-\frac{10 a A e^3 \sqrt{e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac{14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac{2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac{\left (128 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{\frac{2925}{32} a^3 A c^3 e^6+\frac{4851}{32} a^3 B c^3 e^6 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{135135 c^5 e^2 \sqrt{e x}}\\ &=\frac{2 a^2 e^3 \sqrt{e x} (325 A+539 B x) \sqrt{a+c x^2}}{15015 c^2}-\frac{10 a A e^3 \sqrt{e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac{14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac{2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac{\left (28 a^{7/2} B e^4 \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 )}{195 c^{5/2} \sqrt{e x}}+\frac{\left (4 a^3 \left (539 \sqrt{a} B+325 A \sqrt{c}\right ) e^4 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{15015 c^{5/2} \sqrt{e x}}\\ &=\frac{2 a^2 e^3 \sqrt{e x} (325 A+539 B x) \sqrt{a+c x^2}}{15015 c^2}+\frac{28 a^3 B e^4 x \sqrt{a+c x^2}}{195 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{10 a A e^3 \sqrt{e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac{14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac{2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac{2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac{28 a^{13/4} B e^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 )}{195 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 a^{11/4} \left (539 \sqrt{a} B+325 A \sqrt{c}\right ) e^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 )}{15015 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.138412, size = 142, normalized size = 0.33 \[ \frac{2 e^3 \sqrt{e x} \sqrt{a+c x^2} \left (585 a^2 A \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{a}\right )+539 a^2 B x \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{a}\right )-\left (a+c x^2\right ) \sqrt{\frac{c x^2}{a}+1} \left (a (585 A+539 B x)-63 c x^2 (13 A+11 B x)\right )\right )}{9009 c^2 \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 368, normalized size = 0.9 \begin{align*}{\frac{2\,{e}^{3}}{45045\,x{c}^{3}}\sqrt{ex} \left ( 3465\,B{c}^{4}{x}^{8}+4095\,A{c}^{4}{x}^{7}+4235\,aB{c}^{3}{x}^{6}+975\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\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{2}\sqrt{-ac}{a}^{3}+5265\,aA{c}^{3}{x}^{5}+3234\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\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 ) \sqrt{2}{a}^{4}-1617\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\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{2}{a}^{4}-308\,{a}^{2}B{c}^{2}{x}^{4}-780\,{a}^{2}A{c}^{2}{x}^{3}-1078\,{a}^{3}Bc{x}^{2}-1950\,{a}^{3}Acx \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 \sqrt{c x^{2} + a}{\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 ({\left (B e^{3} x^{4} + A e^{3} x^{3}\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 \sqrt{c x^{2} + a}{\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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