Optimal. Leaf size=405 \[ -\frac{7 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-11 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{30 b^{15/4} \sqrt{b x^2+c x^4}}-\frac{7 c^{3/2} x^{3/2} \left (b+c x^2\right ) (9 b B-11 A c)}{15 b^4 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{7 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-11 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{15/4} \sqrt{b x^2+c x^4}}+\frac{7 c \sqrt{b x^2+c x^4} (9 b B-11 A c)}{15 b^4 x^{3/2}}+\frac{9 b B-11 A c}{9 b^2 x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 \sqrt{b x^2+c x^4} (9 b B-11 A c)}{45 b^3 x^{7/2}}-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.522016, antiderivative size = 405, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2038, 2023, 2025, 2032, 329, 305, 220, 1196} \[ -\frac{7 c^{3/2} x^{3/2} \left (b+c x^2\right ) (9 b B-11 A c)}{15 b^4 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{7 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-11 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{30 b^{15/4} \sqrt{b x^2+c x^4}}+\frac{7 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-11 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{15/4} \sqrt{b x^2+c x^4}}+\frac{7 c \sqrt{b x^2+c x^4} (9 b B-11 A c)}{15 b^4 x^{3/2}}+\frac{9 b B-11 A c}{9 b^2 x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 \sqrt{b x^2+c x^4} (9 b B-11 A c)}{45 b^3 x^{7/2}}-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2023
Rule 2025
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}}-\frac{\left (2 \left (-\frac{9 b B}{2}+\frac{11 A c}{2}\right )\right ) \int \frac{1}{\sqrt{x} \left (b x^2+c x^4\right )^{3/2}} \, dx}{9 b}\\ &=-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}}+\frac{9 b B-11 A c}{9 b^2 x^{3/2} \sqrt{b x^2+c x^4}}+\frac{(7 (9 b B-11 A c)) \int \frac{1}{x^{5/2} \sqrt{b x^2+c x^4}} \, dx}{18 b^2}\\ &=-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}}+\frac{9 b B-11 A c}{9 b^2 x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 (9 b B-11 A c) \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}-\frac{(7 c (9 b B-11 A c)) \int \frac{1}{\sqrt{x} \sqrt{b x^2+c x^4}} \, dx}{30 b^3}\\ &=-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}}+\frac{9 b B-11 A c}{9 b^2 x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 (9 b B-11 A c) \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac{7 c (9 b B-11 A c) \sqrt{b x^2+c x^4}}{15 b^4 x^{3/2}}-\frac{\left (7 c^2 (9 b B-11 A c)\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{30 b^4}\\ &=-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}}+\frac{9 b B-11 A c}{9 b^2 x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 (9 b B-11 A c) \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac{7 c (9 b B-11 A c) \sqrt{b x^2+c x^4}}{15 b^4 x^{3/2}}-\frac{\left (7 c^2 (9 b B-11 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{30 b^4 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}}+\frac{9 b B-11 A c}{9 b^2 x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 (9 b B-11 A c) \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac{7 c (9 b B-11 A c) \sqrt{b x^2+c x^4}}{15 b^4 x^{3/2}}-\frac{\left (7 c^2 (9 b B-11 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^4 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}}+\frac{9 b B-11 A c}{9 b^2 x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 (9 b B-11 A c) \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac{7 c (9 b B-11 A c) \sqrt{b x^2+c x^4}}{15 b^4 x^{3/2}}-\frac{\left (7 c^{3/2} (9 b B-11 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^{7/2} \sqrt{b x^2+c x^4}}+\frac{\left (7 c^{3/2} (9 b B-11 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^{7/2} \sqrt{b x^2+c x^4}}\\ &=-\frac{2 A}{9 b x^{7/2} \sqrt{b x^2+c x^4}}+\frac{9 b B-11 A c}{9 b^2 x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 c^{3/2} (9 b B-11 A c) x^{3/2} \left (b+c x^2\right )}{15 b^4 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{7 (9 b B-11 A c) \sqrt{b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac{7 c (9 b B-11 A c) \sqrt{b x^2+c x^4}}{15 b^4 x^{3/2}}+\frac{7 c^{5/4} (9 b B-11 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^{15/4} \sqrt{b x^2+c x^4}}-\frac{7 c^{5/4} (9 b B-11 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 )}{30 b^{15/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0447571, size = 79, normalized size = 0.2 \[ \frac{2 x^2 \sqrt{\frac{c x^2}{b}+1} (11 A c-9 b B) \, _2F_1\left (-\frac{5}{4},\frac{3}{2};-\frac{1}{4};-\frac{c x^2}{b}\right )-10 A b}{45 b^2 x^{7/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 450, normalized size = 1.1 \begin{align*}{\frac{c{x}^{2}+b}{90\,{b}^{4}} \left ( 462\,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}-231\,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}-378\,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+189\,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-462\,A{c}^{3}{x}^{6}+378\,B{x}^{6}b{c}^{2}-308\,Ab{c}^{2}{x}^{4}+252\,B{x}^{4}{b}^{2}c+44\,A{b}^{2}c{x}^{2}-36\,B{x}^{2}{b}^{3}-20\,A{b}^{3} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{x}^{-{\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^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{5}{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 )} \sqrt{x}}{c^{2} x^{11} + 2 \, b c x^{9} + b^{2} x^{7}}, 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^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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