Optimal. Leaf size=282 \[ -\frac{4 b^{15/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-19 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{4389 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{8 b^3 \sqrt{b x^2+c x^4} (9 b B-19 A c)}{4389 c^3 \sqrt{x}}-\frac{8 b^2 x^{3/2} \sqrt{b x^2+c x^4} (9 b B-19 A c)}{7315 c^2}-\frac{4 b x^{7/2} \sqrt{b x^2+c x^4} (9 b B-19 A c)}{1045 c}-\frac{2 x^{3/2} \left (b x^2+c x^4\right )^{3/2} (9 b B-19 A c)}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}} \]
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Rubi [A] time = 0.424831, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2039, 2021, 2024, 2032, 329, 220} \[ \frac{8 b^3 \sqrt{b x^2+c x^4} (9 b B-19 A c)}{4389 c^3 \sqrt{x}}-\frac{8 b^2 x^{3/2} \sqrt{b x^2+c x^4} (9 b B-19 A c)}{7315 c^2}-\frac{4 b^{15/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-19 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4389 c^{13/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{7/2} \sqrt{b x^2+c x^4} (9 b B-19 A c)}{1045 c}-\frac{2 x^{3/2} \left (b x^2+c x^4\right )^{3/2} (9 b B-19 A c)}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 2039
Rule 2021
Rule 2024
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \sqrt{x} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}}-\frac{\left (2 \left (\frac{9 b B}{2}-\frac{19 A c}{2}\right )\right ) \int \sqrt{x} \left (b x^2+c x^4\right )^{3/2} \, dx}{19 c}\\ &=-\frac{2 (9 b B-19 A c) x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}}-\frac{(2 b (9 b B-19 A c)) \int x^{5/2} \sqrt{b x^2+c x^4} \, dx}{95 c}\\ &=-\frac{4 b (9 b B-19 A c) x^{7/2} \sqrt{b x^2+c x^4}}{1045 c}-\frac{2 (9 b B-19 A c) x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}}-\frac{\left (4 b^2 (9 b B-19 A c)\right ) \int \frac{x^{9/2}}{\sqrt{b x^2+c x^4}} \, dx}{1045 c}\\ &=-\frac{8 b^2 (9 b B-19 A c) x^{3/2} \sqrt{b x^2+c x^4}}{7315 c^2}-\frac{4 b (9 b B-19 A c) x^{7/2} \sqrt{b x^2+c x^4}}{1045 c}-\frac{2 (9 b B-19 A c) x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}}+\frac{\left (4 b^3 (9 b B-19 A c)\right ) \int \frac{x^{5/2}}{\sqrt{b x^2+c x^4}} \, dx}{1463 c^2}\\ &=\frac{8 b^3 (9 b B-19 A c) \sqrt{b x^2+c x^4}}{4389 c^3 \sqrt{x}}-\frac{8 b^2 (9 b B-19 A c) x^{3/2} \sqrt{b x^2+c x^4}}{7315 c^2}-\frac{4 b (9 b B-19 A c) x^{7/2} \sqrt{b x^2+c x^4}}{1045 c}-\frac{2 (9 b B-19 A c) x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}}-\frac{\left (4 b^4 (9 b B-19 A c)\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{4389 c^3}\\ &=\frac{8 b^3 (9 b B-19 A c) \sqrt{b x^2+c x^4}}{4389 c^3 \sqrt{x}}-\frac{8 b^2 (9 b B-19 A c) x^{3/2} \sqrt{b x^2+c x^4}}{7315 c^2}-\frac{4 b (9 b B-19 A c) x^{7/2} \sqrt{b x^2+c x^4}}{1045 c}-\frac{2 (9 b B-19 A c) x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}}-\frac{\left (4 b^4 (9 b B-19 A c) x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{4389 c^3 \sqrt{b x^2+c x^4}}\\ &=\frac{8 b^3 (9 b B-19 A c) \sqrt{b x^2+c x^4}}{4389 c^3 \sqrt{x}}-\frac{8 b^2 (9 b B-19 A c) x^{3/2} \sqrt{b x^2+c x^4}}{7315 c^2}-\frac{4 b (9 b B-19 A c) x^{7/2} \sqrt{b x^2+c x^4}}{1045 c}-\frac{2 (9 b B-19 A c) x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}}-\frac{\left (8 b^4 (9 b B-19 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 )}{4389 c^3 \sqrt{b x^2+c x^4}}\\ &=\frac{8 b^3 (9 b B-19 A c) \sqrt{b x^2+c x^4}}{4389 c^3 \sqrt{x}}-\frac{8 b^2 (9 b B-19 A c) x^{3/2} \sqrt{b x^2+c x^4}}{7315 c^2}-\frac{4 b (9 b B-19 A c) x^{7/2} \sqrt{b x^2+c x^4}}{1045 c}-\frac{2 (9 b B-19 A c) x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{285 c}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{19 c \sqrt{x}}-\frac{4 b^{15/4} (9 b B-19 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 )}{4389 c^{13/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.168598, size = 138, normalized size = 0.49 \[ \frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (\left (b+c x^2\right )^2 \sqrt{\frac{c x^2}{b}+1} \left (-b c \left (95 A+99 B x^2\right )+11 c^2 x^2 \left (19 A+15 B x^2\right )+45 b^2 B\right )+5 b^3 (19 A c-9 b B) \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right )\right )}{3135 c^3 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 331, normalized size = 1.2 \begin{align*}{\frac{2}{21945\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{4}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 1155\,B{x}^{11}{c}^{6}+1463\,A{x}^{9}{c}^{6}+2772\,B{x}^{9}b{c}^{5}+3724\,A{x}^{7}b{c}^{5}+1701\,B{x}^{7}{b}^{2}{c}^{4}+190\,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 ) \sqrt{-bc}{b}^{4}c+2489\,A{x}^{5}{b}^{2}{c}^{4}-90\,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 ) \sqrt{-bc}{b}^{5}-24\,B{x}^{5}{b}^{3}{c}^{3}-152\,A{x}^{3}{b}^{3}{c}^{3}+72\,B{x}^{3}{b}^{4}{c}^{2}-380\,Ax{b}^{4}{c}^{2}+180\,Bx{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 )} \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^{6} +{\left (B b + A c\right )} x^{4} + A b x^{2}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )\, dx \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 )} \sqrt{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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