Optimal. Leaf size=369 \[ \frac{2 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{15 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 c^{3/2} x^{3/2} \left (b+c x^2\right ) (3 b B-A c)}{15 b^2 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{4 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4} (3 b B-A c)}{15 b^2 x^{3/2}}-\frac{2 \sqrt{b x^2+c x^4} (3 b B-A c)}{15 b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}} \]
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Rubi [A] time = 0.436171, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2038, 2020, 2025, 2032, 329, 305, 220, 1196} \[ \frac{4 c^{3/2} x^{3/2} \left (b+c x^2\right ) (3 b B-A c)}{15 b^2 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{2 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4} (3 b B-A c)}{15 b^2 x^{3/2}}-\frac{2 \sqrt{b x^2+c x^4} (3 b B-A c)}{15 b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2020
Rule 2025
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{x^{13/2}} \, dx &=-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}}-\frac{\left (2 \left (-\frac{9 b B}{2}+\frac{3 A c}{2}\right )\right ) \int \frac{\sqrt{b x^2+c x^4}}{x^{9/2}} \, dx}{9 b}\\ &=-\frac{2 (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}}+\frac{(2 c (3 b B-A c)) \int \frac{1}{\sqrt{x} \sqrt{b x^2+c x^4}} \, dx}{15 b}\\ &=-\frac{2 (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b x^{7/2}}-\frac{4 c (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}}+\frac{\left (2 c^2 (3 b B-A c)\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{15 b^2}\\ &=-\frac{2 (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b x^{7/2}}-\frac{4 c (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}}+\frac{\left (2 c^2 (3 b B-A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{15 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b x^{7/2}}-\frac{4 c (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}}+\frac{\left (4 c^2 (3 b B-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 )}{15 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b x^{7/2}}-\frac{4 c (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}}+\frac{\left (4 c^{3/2} (3 b B-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 )}{15 b^{3/2} \sqrt{b x^2+c x^4}}-\frac{\left (4 c^{3/2} (3 b B-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 )}{15 b^{3/2} \sqrt{b x^2+c x^4}}\\ &=\frac{4 c^{3/2} (3 b B-A c) x^{3/2} \left (b+c x^2\right )}{15 b^2 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b x^{7/2}}-\frac{4 c (3 b B-A c) \sqrt{b x^2+c x^4}}{15 b^2 x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}}-\frac{4 c^{5/4} (3 b B-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 )}{15 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{2 c^{5/4} (3 b B-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 )}{15 b^{7/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0438959, size = 99, normalized size = 0.27 \[ -\frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (3 x^2 (3 b B-A c) \, _2F_1\left (-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{c x^2}{b}\right )+5 A \left (b+c x^2\right ) \sqrt{\frac{c x^2}{b}+1}\right )}{45 b x^{11/2} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 452, normalized size = 1.2 \begin{align*} -{\frac{2}{ \left ( 45\,c{x}^{2}+45\,b \right ){b}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 6\,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 ){x}^{4}b{c}^{2}-3\,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 ){x}^{4}b{c}^{2}-18\,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 ){x}^{4}{b}^{2}c+9\,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 ){x}^{4}{b}^{2}c-6\,A{c}^{3}{x}^{6}+18\,B{x}^{6}b{c}^{2}-4\,Ab{c}^{2}{x}^{4}+27\,B{x}^{4}{b}^{2}c+7\,A{b}^{2}c{x}^{2}+9\,B{x}^{2}{b}^{3}+5\,A{b}^{3} \right ){x}^{-{\frac{11}{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{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{13}{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^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{13}{2}}}, 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{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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