Optimal. Leaf size=408 \[ \frac{4 b^{13/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-17 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{8 b^3 x^{3/2} \left (b+c x^2\right ) (7 b B-17 A c)}{1105 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{8 b^2 \sqrt{x} \sqrt{b x^2+c x^4} (7 b B-17 A c)}{3315 c^2}-\frac{8 b^{13/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-17 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{5/2} \sqrt{b x^2+c x^4} (7 b B-17 A c)}{663 c}-\frac{2 \sqrt{x} \left (b x^2+c x^4\right )^{3/2} (7 b B-17 A c)}{221 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}} \]
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Rubi [A] time = 0.515229, antiderivative size = 408, 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 = {2039, 2021, 2024, 2032, 329, 305, 220, 1196} \[ \frac{8 b^3 x^{3/2} \left (b+c x^2\right ) (7 b B-17 A c)}{1105 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{8 b^2 \sqrt{x} \sqrt{b x^2+c x^4} (7 b B-17 A c)}{3315 c^2}+\frac{4 b^{13/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-17 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{8 b^{13/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-17 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{5/2} \sqrt{b x^2+c x^4} (7 b B-17 A c)}{663 c}-\frac{2 \sqrt{x} \left (b x^2+c x^4\right )^{3/2} (7 b B-17 A c)}{221 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}} \]
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 \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{\sqrt{x}} \, dx &=\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac{\left (2 \left (\frac{7 b B}{2}-\frac{17 A c}{2}\right )\right ) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{\sqrt{x}} \, dx}{17 c}\\ &=-\frac{2 (7 b B-17 A c) \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{221 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac{(6 b (7 b B-17 A c)) \int x^{3/2} \sqrt{b x^2+c x^4} \, dx}{221 c}\\ &=-\frac{4 b (7 b B-17 A c) x^{5/2} \sqrt{b x^2+c x^4}}{663 c}-\frac{2 (7 b B-17 A c) \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{221 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac{\left (4 b^2 (7 b B-17 A c)\right ) \int \frac{x^{7/2}}{\sqrt{b x^2+c x^4}} \, dx}{663 c}\\ &=-\frac{8 b^2 (7 b B-17 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}-\frac{4 b (7 b B-17 A c) x^{5/2} \sqrt{b x^2+c x^4}}{663 c}-\frac{2 (7 b B-17 A c) \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{221 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}+\frac{\left (4 b^3 (7 b B-17 A c)\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{1105 c^2}\\ &=-\frac{8 b^2 (7 b B-17 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}-\frac{4 b (7 b B-17 A c) x^{5/2} \sqrt{b x^2+c x^4}}{663 c}-\frac{2 (7 b B-17 A c) \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{221 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}+\frac{\left (4 b^3 (7 b B-17 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{1105 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{8 b^2 (7 b B-17 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}-\frac{4 b (7 b B-17 A c) x^{5/2} \sqrt{b x^2+c x^4}}{663 c}-\frac{2 (7 b B-17 A c) \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{221 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}+\frac{\left (8 b^3 (7 b B-17 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 )}{1105 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{8 b^2 (7 b B-17 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}-\frac{4 b (7 b B-17 A c) x^{5/2} \sqrt{b x^2+c x^4}}{663 c}-\frac{2 (7 b B-17 A c) \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{221 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}+\frac{\left (8 b^{7/2} (7 b B-17 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 )}{1105 c^{5/2} \sqrt{b x^2+c x^4}}-\frac{\left (8 b^{7/2} (7 b B-17 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 )}{1105 c^{5/2} \sqrt{b x^2+c x^4}}\\ &=\frac{8 b^3 (7 b B-17 A c) x^{3/2} \left (b+c x^2\right )}{1105 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{8 b^2 (7 b B-17 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{3315 c^2}-\frac{4 b (7 b B-17 A c) x^{5/2} \sqrt{b x^2+c x^4}}{663 c}-\frac{2 (7 b B-17 A c) \sqrt{x} \left (b x^2+c x^4\right )^{3/2}}{221 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{17 c x^{3/2}}-\frac{8 b^{13/4} (7 b B-17 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 )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{13/4} (7 b B-17 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 )}{1105 c^{11/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.143927, size = 115, normalized size = 0.28 \[ \frac{2 \sqrt{x} \sqrt{x^2 \left (b+c x^2\right )} \left (b^2 (7 b B-17 A c) \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right )-\left (b+c x^2\right )^2 \sqrt{\frac{c x^2}{b}+1} \left (-17 A c+7 b B-13 B c x^2\right )\right )}{221 c^2 \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 470, normalized size = 1.2 \begin{align*} -{\frac{2}{3315\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -195\,B{x}^{10}{c}^{5}-255\,A{x}^{8}{c}^{5}-480\,B{x}^{8}b{c}^{4}-680\,A{x}^{6}b{c}^{4}-305\,B{x}^{6}{b}^{2}{c}^{3}+204\,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}^{4}c-102\,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}^{4}c-84\,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}^{5}+42\,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}^{5}-493\,A{x}^{4}{b}^{2}{c}^{3}+8\,B{x}^{4}{b}^{3}{c}^{2}-68\,A{x}^{2}{b}^{3}{c}^{2}+28\,B{x}^{2}{b}^{4}c \right ){x}^{-{\frac{7}{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 (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{\sqrt{x}}\,{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 ({\left (B c x^{5} +{\left (B b + A c\right )} x^{3} + A b x\right )} \sqrt{c x^{4} + b x^{2}} \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 \frac{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{\sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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