Optimal. Leaf size=369 \[ -\frac{7 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-13 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{195 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{14 b^2 x^{3/2} \left (b+c x^2\right ) (11 b B-13 A c)}{195 c^{7/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{14 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-13 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{2 x^{5/2} \sqrt{b x^2+c x^4} (11 b B-13 A c)}{117 c^2}+\frac{14 b \sqrt{x} \sqrt{b x^2+c x^4} (11 b B-13 A c)}{585 c^3}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c} \]
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Rubi [A] time = 0.444791, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2039, 2024, 2032, 329, 305, 220, 1196} \[ -\frac{14 b^2 x^{3/2} \left (b+c x^2\right ) (11 b B-13 A c)}{195 c^{7/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{7 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-13 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{15/4} \sqrt{b x^2+c x^4}}+\frac{14 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-13 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{2 x^{5/2} \sqrt{b x^2+c x^4} (11 b B-13 A c)}{117 c^2}+\frac{14 b \sqrt{x} \sqrt{b x^2+c x^4} (11 b B-13 A c)}{585 c^3}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c} \]
Antiderivative was successfully verified.
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Rule 2039
Rule 2024
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^{11/2} \left (A+B x^2\right )}{\sqrt{b x^2+c x^4}} \, dx &=\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c}-\frac{\left (2 \left (\frac{11 b B}{2}-\frac{13 A c}{2}\right )\right ) \int \frac{x^{11/2}}{\sqrt{b x^2+c x^4}} \, dx}{13 c}\\ &=-\frac{2 (11 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c^2}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c}+\frac{(7 b (11 b B-13 A c)) \int \frac{x^{7/2}}{\sqrt{b x^2+c x^4}} \, dx}{117 c^2}\\ &=\frac{14 b (11 b B-13 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^3}-\frac{2 (11 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c^2}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c}-\frac{\left (7 b^2 (11 b B-13 A c)\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{195 c^3}\\ &=\frac{14 b (11 b B-13 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^3}-\frac{2 (11 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c^2}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c}-\frac{\left (7 b^2 (11 b B-13 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{195 c^3 \sqrt{b x^2+c x^4}}\\ &=\frac{14 b (11 b B-13 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^3}-\frac{2 (11 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c^2}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c}-\frac{\left (14 b^2 (11 b B-13 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{195 c^3 \sqrt{b x^2+c x^4}}\\ &=\frac{14 b (11 b B-13 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^3}-\frac{2 (11 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c^2}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c}-\frac{\left (14 b^{5/2} (11 b B-13 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{195 c^{7/2} \sqrt{b x^2+c x^4}}+\frac{\left (14 b^{5/2} (11 b B-13 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{b}}}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{195 c^{7/2} \sqrt{b x^2+c x^4}}\\ &=-\frac{14 b^2 (11 b B-13 A c) x^{3/2} \left (b+c x^2\right )}{195 c^{7/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{14 b (11 b B-13 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^3}-\frac{2 (11 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c^2}+\frac{2 B x^{9/2} \sqrt{b x^2+c x^4}}{13 c}+\frac{14 b^{9/4} (11 b B-13 A c) x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{7 b^{9/4} (11 b B-13 A c) x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{15/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.138221, size = 122, normalized size = 0.33 \[ \frac{2 x^{5/2} \left (\left (b+c x^2\right ) \left (-b c \left (91 A+55 B x^2\right )+5 c^2 x^2 \left (13 A+9 B x^2\right )+77 b^2 B\right )+7 b^2 \sqrt{\frac{c x^2}{b}+1} (13 A c-11 b B) \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right )\right )}{585 c^3 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 437, normalized size = 1.2 \begin{align*}{\frac{1}{585\,{c}^{4}}\sqrt{x} \left ( 90\,B{x}^{8}{c}^{4}+130\,A{x}^{6}{c}^{4}-20\,B{x}^{6}b{c}^{3}+546\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{3}c-273\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{3}c-462\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{4}+231\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{4}-52\,A{x}^{4}b{c}^{3}+44\,B{x}^{4}{b}^{2}{c}^{2}-182\,A{x}^{2}{b}^{2}{c}^{2}+154\,B{x}^{2}{b}^{3}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{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{{\left (B x^{2} + A\right )} x^{\frac{11}{2}}}{\sqrt{c x^{4} + b x^{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 x^{5} + A x^{3}\right )} \sqrt{c x^{4} + b x^{2}} \sqrt{x}}{c x^{2} + b}, 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 (B x^{2} + A\right )} x^{\frac{11}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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