Optimal. Leaf size=321 \[ \frac{12 b^{19/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (13 b B-23 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{33649 c^{17/4} \sqrt{b x^2+c x^4}}-\frac{8 b^2 x^{7/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{24035 c^2}+\frac{72 b^3 x^{3/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{168245 c^3}-\frac{24 b^4 \sqrt{b x^2+c x^4} (13 b B-23 A c)}{33649 c^4 \sqrt{x}}-\frac{4 b x^{11/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{2185 c}-\frac{2 x^{7/2} \left (b x^2+c x^4\right )^{3/2} (13 b B-23 A c)}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c} \]
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Rubi [A] time = 0.489965, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{8 b^2 x^{7/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{24035 c^2}+\frac{72 b^3 x^{3/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{168245 c^3}-\frac{24 b^4 \sqrt{b x^2+c x^4} (13 b B-23 A c)}{33649 c^4 \sqrt{x}}+\frac{12 b^{19/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (13 b B-23 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{33649 c^{17/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{11/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{2185 c}-\frac{2 x^{7/2} \left (b x^2+c x^4\right )^{3/2} (13 b B-23 A c)}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c} \]
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 x^{5/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}-\frac{\left (2 \left (\frac{13 b B}{2}-\frac{23 A c}{2}\right )\right ) \int x^{5/2} \left (b x^2+c x^4\right )^{3/2} \, dx}{23 c}\\ &=-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}-\frac{(6 b (13 b B-23 A c)) \int x^{9/2} \sqrt{b x^2+c x^4} \, dx}{437 c}\\ &=-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}-\frac{\left (4 b^2 (13 b B-23 A c)\right ) \int \frac{x^{13/2}}{\sqrt{b x^2+c x^4}} \, dx}{2185 c}\\ &=-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}+\frac{\left (36 b^3 (13 b B-23 A c)\right ) \int \frac{x^{9/2}}{\sqrt{b x^2+c x^4}} \, dx}{24035 c^2}\\ &=\frac{72 b^3 (13 b B-23 A c) x^{3/2} \sqrt{b x^2+c x^4}}{168245 c^3}-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}-\frac{\left (36 b^4 (13 b B-23 A c)\right ) \int \frac{x^{5/2}}{\sqrt{b x^2+c x^4}} \, dx}{33649 c^3}\\ &=-\frac{24 b^4 (13 b B-23 A c) \sqrt{b x^2+c x^4}}{33649 c^4 \sqrt{x}}+\frac{72 b^3 (13 b B-23 A c) x^{3/2} \sqrt{b x^2+c x^4}}{168245 c^3}-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}+\frac{\left (12 b^5 (13 b B-23 A c)\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{33649 c^4}\\ &=-\frac{24 b^4 (13 b B-23 A c) \sqrt{b x^2+c x^4}}{33649 c^4 \sqrt{x}}+\frac{72 b^3 (13 b B-23 A c) x^{3/2} \sqrt{b x^2+c x^4}}{168245 c^3}-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}+\frac{\left (12 b^5 (13 b B-23 A c) x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{33649 c^4 \sqrt{b x^2+c x^4}}\\ &=-\frac{24 b^4 (13 b B-23 A c) \sqrt{b x^2+c x^4}}{33649 c^4 \sqrt{x}}+\frac{72 b^3 (13 b B-23 A c) x^{3/2} \sqrt{b x^2+c x^4}}{168245 c^3}-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}+\frac{\left (24 b^5 (13 b B-23 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 )}{33649 c^4 \sqrt{b x^2+c x^4}}\\ &=-\frac{24 b^4 (13 b B-23 A c) \sqrt{b x^2+c x^4}}{33649 c^4 \sqrt{x}}+\frac{72 b^3 (13 b B-23 A c) x^{3/2} \sqrt{b x^2+c x^4}}{168245 c^3}-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}+\frac{12 b^{19/4} (13 b B-23 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 )}{33649 c^{17/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.207018, size = 160, normalized size = 0.5 \[ \frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (15 b^4 (13 b B-23 A c) \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right )-\left (b+c x^2\right )^2 \sqrt{\frac{c x^2}{b}+1} \left (-3 b^2 c \left (115 A+143 B x^2\right )+11 b c^2 x^2 \left (69 A+65 B x^2\right )-55 c^3 x^4 \left (23 A+19 B x^2\right )+195 b^3 B\right )\right )}{24035 c^4 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 355, normalized size = 1.1 \begin{align*} -{\frac{2}{168245\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -7315\,B{x}^{13}{c}^{7}-8855\,A{x}^{11}{c}^{7}-16940\,B{x}^{11}b{c}^{6}-21252\,A{x}^{9}b{c}^{6}-9933\,B{x}^{9}{b}^{2}{c}^{5}-13041\,A{x}^{7}{b}^{2}{c}^{5}+56\,B{x}^{7}{b}^{3}{c}^{4}+690\,A\sqrt{-bc}\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+184\,A{x}^{5}{b}^{3}{c}^{4}-390\,B\sqrt{-bc}\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}-104\,B{x}^{5}{b}^{4}{c}^{3}-552\,A{x}^{3}{b}^{4}{c}^{3}+312\,B{x}^{3}{b}^{5}{c}^{2}-1380\,Ax{b}^{5}{c}^{2}+780\,Bx{b}^{6}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{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 ({\left (B c x^{8} +{\left (B b + A c\right )} x^{6} + A b x^{4}\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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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