Optimal. Leaf size=363 \[ -\frac{c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+9 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{15 a^{7/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{6 A c^{3/2} x \sqrt{a+c x^2}}{5 a^2 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{6 A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{6 A c \sqrt{a+c x^2}}{5 a^2 e^3 \sqrt{e x}}-\frac{2 A \sqrt{a+c x^2}}{5 a e (e x)^{5/2}}-\frac{2 B \sqrt{a+c x^2}}{3 a e^2 (e x)^{3/2}} \]
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Rubi [A] time = 0.391311, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {835, 842, 840, 1198, 220, 1196} \[ -\frac{c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+9 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{7/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{6 A c^{3/2} x \sqrt{a+c x^2}}{5 a^2 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{6 A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{6 A c \sqrt{a+c x^2}}{5 a^2 e^3 \sqrt{e x}}-\frac{2 A \sqrt{a+c x^2}}{5 a e (e x)^{5/2}}-\frac{2 B \sqrt{a+c x^2}}{3 a e^2 (e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 835
Rule 842
Rule 840
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{A+B x}{(e x)^{7/2} \sqrt{a+c x^2}} \, dx &=-\frac{2 A \sqrt{a+c x^2}}{5 a e (e x)^{5/2}}-\frac{2 \int \frac{-\frac{5}{2} a B e+\frac{3}{2} A c e x}{(e x)^{5/2} \sqrt{a+c x^2}} \, dx}{5 a e^2}\\ &=-\frac{2 A \sqrt{a+c x^2}}{5 a e (e x)^{5/2}}-\frac{2 B \sqrt{a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac{4 \int \frac{-\frac{9}{4} a A c e^2-\frac{5}{4} a B c e^2 x}{(e x)^{3/2} \sqrt{a+c x^2}} \, dx}{15 a^2 e^4}\\ &=-\frac{2 A \sqrt{a+c x^2}}{5 a e (e x)^{5/2}}-\frac{2 B \sqrt{a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac{6 A c \sqrt{a+c x^2}}{5 a^2 e^3 \sqrt{e x}}-\frac{8 \int \frac{\frac{5}{8} a^2 B c e^3+\frac{9}{8} a A c^2 e^3 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{15 a^3 e^6}\\ &=-\frac{2 A \sqrt{a+c x^2}}{5 a e (e x)^{5/2}}-\frac{2 B \sqrt{a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac{6 A c \sqrt{a+c x^2}}{5 a^2 e^3 \sqrt{e x}}-\frac{\left (8 \sqrt{x}\right ) \int \frac{\frac{5}{8} a^2 B c e^3+\frac{9}{8} a A c^2 e^3 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{15 a^3 e^6 \sqrt{e x}}\\ &=-\frac{2 A \sqrt{a+c x^2}}{5 a e (e x)^{5/2}}-\frac{2 B \sqrt{a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac{6 A c \sqrt{a+c x^2}}{5 a^2 e^3 \sqrt{e x}}-\frac{\left (16 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{\frac{5}{8} a^2 B c e^3+\frac{9}{8} a A c^2 e^3 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{15 a^3 e^6 \sqrt{e x}}\\ &=-\frac{2 A \sqrt{a+c x^2}}{5 a e (e x)^{5/2}}-\frac{2 B \sqrt{a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac{6 A c \sqrt{a+c x^2}}{5 a^2 e^3 \sqrt{e x}}-\frac{\left (2 \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 a^{3/2} e^3 \sqrt{e x}}+\frac{\left (6 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 a^{3/2} e^3 \sqrt{e x}}\\ &=-\frac{2 A \sqrt{a+c x^2}}{5 a e (e x)^{5/2}}-\frac{2 B \sqrt{a+c x^2}}{3 a e^2 (e x)^{3/2}}+\frac{6 A c \sqrt{a+c x^2}}{5 a^2 e^3 \sqrt{e x}}-\frac{6 A c^{3/2} x \sqrt{a+c x^2}}{5 a^2 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{6 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 a^{7/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{\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 a^{7/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0288517, size = 83, normalized size = 0.23 \[ -\frac{2 x \sqrt{\frac{c x^2}{a}+1} \left (3 A \, _2F_1\left (-\frac{5}{4},\frac{1}{2};-\frac{1}{4};-\frac{c x^2}{a}\right )+5 B x \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{c x^2}{a}\right )\right )}{15 (e x)^{7/2} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 331, normalized size = 0.9 \begin{align*}{\frac{1}{15\,{x}^{2}{e}^{3}{a}^{2}} \left ( 9\,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-18\,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-5\,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+18\,A{c}^{2}{x}^{4}-10\,aBc{x}^{3}+12\,aAc{x}^{2}-10\,{a}^{2}Bx-6\,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{B x + A}{\sqrt{c x^{2} + a} \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{\sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{e x}}{c e^{4} x^{6} + a 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{B x + A}{\sqrt{c x^{2} + a} \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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