Optimal. Leaf size=447 \[ -\frac{4 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-21 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{3315 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4} (11 b B-21 A c)}{13923 c^2}-\frac{8 b^4 x^{3/2} \left (b+c x^2\right ) (11 b B-21 A c)}{3315 c^{7/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{8 b^3 \sqrt{x} \sqrt{b x^2+c x^4} (11 b B-21 A c)}{9945 c^3}+\frac{8 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-21 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3315 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{9/2} \sqrt{b x^2+c x^4} (11 b B-21 A c)}{1547 c}-\frac{2 x^{5/2} \left (b x^2+c x^4\right )^{3/2} (11 b B-21 A c)}{357 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c} \]
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Rubi [A] time = 0.589558, antiderivative size = 447, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{8 b^2 x^{5/2} \sqrt{b x^2+c x^4} (11 b B-21 A c)}{13923 c^2}-\frac{8 b^4 x^{3/2} \left (b+c x^2\right ) (11 b B-21 A c)}{3315 c^{7/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{8 b^3 \sqrt{x} \sqrt{b x^2+c x^4} (11 b B-21 A c)}{9945 c^3}-\frac{4 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-21 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3315 c^{15/4} \sqrt{b x^2+c x^4}}+\frac{8 b^{17/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-21 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3315 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{9/2} \sqrt{b x^2+c x^4} (11 b B-21 A c)}{1547 c}-\frac{2 x^{5/2} \left (b x^2+c x^4\right )^{3/2} (11 b B-21 A c)}{357 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c} \]
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 ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac{\left (2 \left (\frac{11 b B}{2}-\frac{21 A c}{2}\right )\right ) \int x^{3/2} \left (b x^2+c x^4\right )^{3/2} \, dx}{21 c}\\ &=-\frac{2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac{(2 b (11 b B-21 A c)) \int x^{7/2} \sqrt{b x^2+c x^4} \, dx}{119 c}\\ &=-\frac{4 b (11 b B-21 A c) x^{9/2} \sqrt{b x^2+c x^4}}{1547 c}-\frac{2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac{\left (4 b^2 (11 b B-21 A c)\right ) \int \frac{x^{11/2}}{\sqrt{b x^2+c x^4}} \, dx}{1547 c}\\ &=-\frac{8 b^2 (11 b B-21 A c) x^{5/2} \sqrt{b x^2+c x^4}}{13923 c^2}-\frac{4 b (11 b B-21 A c) x^{9/2} \sqrt{b x^2+c x^4}}{1547 c}-\frac{2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c}+\frac{\left (4 b^3 (11 b B-21 A c)\right ) \int \frac{x^{7/2}}{\sqrt{b x^2+c x^4}} \, dx}{1989 c^2}\\ &=\frac{8 b^3 (11 b B-21 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{9945 c^3}-\frac{8 b^2 (11 b B-21 A c) x^{5/2} \sqrt{b x^2+c x^4}}{13923 c^2}-\frac{4 b (11 b B-21 A c) x^{9/2} \sqrt{b x^2+c x^4}}{1547 c}-\frac{2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac{\left (4 b^4 (11 b B-21 A c)\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{3315 c^3}\\ &=\frac{8 b^3 (11 b B-21 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{9945 c^3}-\frac{8 b^2 (11 b B-21 A c) x^{5/2} \sqrt{b x^2+c x^4}}{13923 c^2}-\frac{4 b (11 b B-21 A c) x^{9/2} \sqrt{b x^2+c x^4}}{1547 c}-\frac{2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac{\left (4 b^4 (11 b B-21 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{3315 c^3 \sqrt{b x^2+c x^4}}\\ &=\frac{8 b^3 (11 b B-21 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{9945 c^3}-\frac{8 b^2 (11 b B-21 A c) x^{5/2} \sqrt{b x^2+c x^4}}{13923 c^2}-\frac{4 b (11 b B-21 A c) x^{9/2} \sqrt{b x^2+c x^4}}{1547 c}-\frac{2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac{\left (8 b^4 (11 b B-21 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 )}{3315 c^3 \sqrt{b x^2+c x^4}}\\ &=\frac{8 b^3 (11 b B-21 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{9945 c^3}-\frac{8 b^2 (11 b B-21 A c) x^{5/2} \sqrt{b x^2+c x^4}}{13923 c^2}-\frac{4 b (11 b B-21 A c) x^{9/2} \sqrt{b x^2+c x^4}}{1547 c}-\frac{2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c}-\frac{\left (8 b^{9/2} (11 b B-21 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 )}{3315 c^{7/2} \sqrt{b x^2+c x^4}}+\frac{\left (8 b^{9/2} (11 b B-21 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 )}{3315 c^{7/2} \sqrt{b x^2+c x^4}}\\ &=-\frac{8 b^4 (11 b B-21 A c) x^{3/2} \left (b+c x^2\right )}{3315 c^{7/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{8 b^3 (11 b B-21 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{9945 c^3}-\frac{8 b^2 (11 b B-21 A c) x^{5/2} \sqrt{b x^2+c x^4}}{13923 c^2}-\frac{4 b (11 b B-21 A c) x^{9/2} \sqrt{b x^2+c x^4}}{1547 c}-\frac{2 (11 b B-21 A c) x^{5/2} \left (b x^2+c x^4\right )^{3/2}}{357 c}+\frac{2 B \sqrt{x} \left (b x^2+c x^4\right )^{5/2}}{21 c}+\frac{8 b^{17/4} (11 b B-21 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 )}{3315 c^{15/4} \sqrt{b x^2+c x^4}}-\frac{4 b^{17/4} (11 b B-21 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 )}{3315 c^{15/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.175096, size = 138, normalized size = 0.31 \[ \frac{2 \sqrt{x} \sqrt{x^2 \left (b+c x^2\right )} \left (\left (b+c x^2\right )^2 \sqrt{\frac{c x^2}{b}+1} \left (-b c \left (147 A+143 B x^2\right )+13 c^2 x^2 \left (21 A+17 B x^2\right )+77 b^2 B\right )+7 b^3 (21 A c-11 b B) \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right )\right )}{4641 c^3 \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 494, normalized size = 1.1 \begin{align*}{\frac{2}{69615\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{4}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3315\,B{x}^{12}{c}^{6}+4095\,A{x}^{10}{c}^{6}+7800\,B{x}^{10}b{c}^{5}+10080\,A{x}^{8}b{c}^{5}+4665\,B{x}^{8}{b}^{2}{c}^{4}+6405\,A{x}^{6}{b}^{2}{c}^{4}-40\,B{x}^{6}{b}^{3}{c}^{3}+1764\,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}^{5}c-882\,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}^{5}c-924\,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}^{6}+462\,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}^{6}-168\,A{x}^{4}{b}^{3}{c}^{3}+88\,B{x}^{4}{b}^{4}{c}^{2}-588\,A{x}^{2}{b}^{4}{c}^{2}+308\,B{x}^{2}{b}^{5}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{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{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 ({\left (B c x^{7} +{\left (B b + A c\right )} x^{5} + A b x^{3}\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{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{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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