Optimal. Leaf size=369 \[ \frac{2 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-13 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{195 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{4 b^2 x^{3/2} \left (b+c x^2\right ) (7 b B-13 A c)}{195 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{4 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-13 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{x} \sqrt{b x^2+c x^4} (7 b B-13 A c)}{585 c^2}-\frac{2 x^{5/2} \sqrt{b x^2+c x^4} (7 b B-13 A c)}{117 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{13 c} \]
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Rubi [A] time = 0.435648, 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 = {2039, 2021, 2024, 2032, 329, 305, 220, 1196} \[ \frac{4 b^2 x^{3/2} \left (b+c x^2\right ) (7 b B-13 A c)}{195 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{2 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-13 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{4 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-13 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{x} \sqrt{b x^2+c x^4} (7 b B-13 A c)}{585 c^2}-\frac{2 x^{5/2} \sqrt{b x^2+c x^4} (7 b B-13 A c)}{117 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{13 c} \]
Antiderivative was successfully verified.
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Rule 2039
Rule 2021
Rule 2024
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int x^{3/2} \left (A+B x^2\right ) \sqrt{b x^2+c x^4} \, dx &=\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{13 c}-\frac{\left (2 \left (\frac{7 b B}{2}-\frac{13 A c}{2}\right )\right ) \int x^{3/2} \sqrt{b x^2+c x^4} \, dx}{13 c}\\ &=-\frac{2 (7 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{13 c}-\frac{(2 b (7 b B-13 A c)) \int \frac{x^{7/2}}{\sqrt{b x^2+c x^4}} \, dx}{117 c}\\ &=-\frac{4 b (7 b B-13 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^2}-\frac{2 (7 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{13 c}+\frac{\left (2 b^2 (7 b B-13 A c)\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{195 c^2}\\ &=-\frac{4 b (7 b B-13 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^2}-\frac{2 (7 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{13 c}+\frac{\left (2 b^2 (7 b B-13 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{195 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{4 b (7 b B-13 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^2}-\frac{2 (7 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{13 c}+\frac{\left (4 b^2 (7 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^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{4 b (7 b B-13 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^2}-\frac{2 (7 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{13 c}+\frac{\left (4 b^{5/2} (7 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^{5/2} \sqrt{b x^2+c x^4}}-\frac{\left (4 b^{5/2} (7 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^{5/2} \sqrt{b x^2+c x^4}}\\ &=\frac{4 b^2 (7 b B-13 A c) x^{3/2} \left (b+c x^2\right )}{195 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{4 b (7 b B-13 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{585 c^2}-\frac{2 (7 b B-13 A c) x^{5/2} \sqrt{b x^2+c x^4}}{117 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{13 c}-\frac{4 b^{9/4} (7 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^{11/4} \sqrt{b x^2+c x^4}}+\frac{2 b^{9/4} (7 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^{11/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.128747, size = 111, normalized size = 0.3 \[ \frac{2 \sqrt{x} \sqrt{x^2 \left (b+c x^2\right )} \left (b (7 b B-13 A c) \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right )-\left (b+c x^2\right ) \sqrt{\frac{c x^2}{b}+1} \left (-13 A c+7 b B-9 B c x^2\right )\right )}{117 c^2 \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 446, normalized size = 1.2 \begin{align*} -{\frac{2}{ \left ( 585\,c{x}^{2}+585\,b \right ){c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( -45\,B{x}^{8}{c}^{4}-65\,A{x}^{6}{c}^{4}-55\,B{x}^{6}b{c}^{3}+78\,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-39\,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-42\,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}+21\,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}-91\,A{x}^{4}b{c}^{3}+4\,B{x}^{4}{b}^{2}{c}^{2}-26\,A{x}^{2}{b}^{2}{c}^{2}+14\,B{x}^{2}{b}^{3}c \right ){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 \sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} x^{\frac{3}{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 (\sqrt{c x^{4} + b x^{2}}{\left (B x^{3} + A x\right )} \sqrt{x}, 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 \sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} x^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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