Optimal. Leaf size=432 \[ -\frac{c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B+77 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{20 a^{15/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}-\frac{77 A c^{3/2} x \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{77 A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{15/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{77 A c \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.578926, antiderivative size = 432, 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 = {823, 835, 842, 840, 1198, 220, 1196} \[ -\frac{c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B+77 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{20 a^{15/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}-\frac{77 A c^{3/2} x \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{77 A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{15/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{77 A c \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 823
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} \left (a+c x^2\right )^{5/2}} \, dx &=\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}-\frac{\int \frac{-\frac{11}{2} a A c e^2-\frac{9}{2} a B c e^2 x}{(e x)^{7/2} \left (a+c x^2\right )^{3/2}} \, dx}{3 a^2 c e^2}\\ &=\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}+\frac{\int \frac{\frac{77}{4} a^2 A c^2 e^4+\frac{45}{4} a^2 B c^2 e^4 x}{(e x)^{7/2} \sqrt{a+c x^2}} \, dx}{3 a^4 c^2 e^4}\\ &=\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{2 \int \frac{-\frac{225}{8} a^3 B c^2 e^5+\frac{231}{8} a^2 A c^3 e^5 x}{(e x)^{5/2} \sqrt{a+c x^2}} \, dx}{15 a^5 c^2 e^6}\\ &=\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac{4 \int \frac{-\frac{693}{16} a^3 A c^3 e^6-\frac{225}{16} a^3 B c^3 e^6 x}{(e x)^{3/2} \sqrt{a+c x^2}} \, dx}{45 a^6 c^2 e^8}\\ &=\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac{77 A c \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x}}-\frac{8 \int \frac{\frac{225}{32} a^4 B c^3 e^7+\frac{693}{32} a^3 A c^4 e^7 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{45 a^7 c^2 e^{10}}\\ &=\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac{77 A c \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x}}-\frac{\left (8 \sqrt{x}\right ) \int \frac{\frac{225}{32} a^4 B c^3 e^7+\frac{693}{32} a^3 A c^4 e^7 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{45 a^7 c^2 e^{10} \sqrt{e x}}\\ &=\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac{77 A c \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x}}-\frac{\left (16 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{\frac{225}{32} a^4 B c^3 e^7+\frac{693}{32} a^3 A c^4 e^7 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{45 a^7 c^2 e^{10} \sqrt{e x}}\\ &=\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac{77 A c \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x}}-\frac{\left (\left (25 \sqrt{a} B+77 A \sqrt{c}\right ) c \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{10 a^{7/2} e^3 \sqrt{e x}}+\frac{\left (77 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 )}{10 a^{7/2} e^3 \sqrt{e x}}\\ &=\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac{77 A c \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x}}-\frac{77 A c^{3/2} x \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{77 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 )}{10 a^{15/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{\left (25 \sqrt{a} B+77 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 )}{20 a^{15/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0997915, size = 137, normalized size = 0.32 \[ \frac{x \left (-77 A \left (a+c x^2\right ) \sqrt{\frac{c x^2}{a}+1} \, _2F_1\left (-\frac{5}{4},\frac{1}{2};-\frac{1}{4};-\frac{c x^2}{a}\right )+65 a A-75 B x \left (a+c x^2\right ) \sqrt{\frac{c x^2}{a}+1} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{c x^2}{a}\right )+55 a B x+55 A c x^2+45 B c x^3\right )}{30 a^2 (e x)^{7/2} \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 632, normalized size = 1.5 \begin{align*}{\frac{1}{60\,{x}^{2}{a}^{4}{e}^{3}} \left ( 231\,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}^{4}a{c}^{2}-462\,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}^{4}a{c}^{2}-75\,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}^{4}ac+231\,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}{a}^{2}c-462\,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}{a}^{2}c-75\,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 ) \sqrt{-ac}{x}^{2}{a}^{2}+462\,A{c}^{3}{x}^{6}-150\,aB{c}^{2}{x}^{5}+770\,aA{c}^{2}{x}^{4}-210\,{a}^{2}Bc{x}^{3}+264\,{a}^{2}Ac{x}^{2}-40\,{a}^{3}Bx-24\,A{a}^{3} \right ){\frac{1}{\sqrt{ex}}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \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}{{\left (c x^{2} + a\right )}^{\frac{5}{2}} \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^{3} e^{4} x^{10} + 3 \, a c^{2} e^{4} x^{8} + 3 \, a^{2} c e^{4} x^{6} + a^{3} 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}{{\left (c x^{2} + a\right )}^{\frac{5}{2}} \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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