Optimal. Leaf size=239 \[ \frac{4 b^{11/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{8 b^2 \sqrt{b x^2+c x^4} (b B-3 A c)}{231 c^2 \sqrt{x}}-\frac{4 b x^{3/2} \sqrt{b x^2+c x^4} (b B-3 A c)}{77 c}-\frac{2 \left (b x^2+c x^4\right )^{3/2} (b B-3 A c)}{33 c \sqrt{x}}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{15 c x^{5/2}} \]
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Rubi [A] time = 0.369354, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{8 b^2 \sqrt{b x^2+c x^4} (b B-3 A c)}{231 c^2 \sqrt{x}}+\frac{4 b^{11/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{3/2} \sqrt{b x^2+c x^4} (b B-3 A c)}{77 c}-\frac{2 \left (b x^2+c x^4\right )^{3/2} (b B-3 A c)}{33 c \sqrt{x}}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{15 c x^{5/2}} \]
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 \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{3/2}} \, dx &=\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{15 c x^{5/2}}-\frac{\left (2 \left (\frac{5 b B}{2}-\frac{15 A c}{2}\right )\right ) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{3/2}} \, dx}{15 c}\\ &=-\frac{2 (b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{33 c \sqrt{x}}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{15 c x^{5/2}}-\frac{(2 b (b B-3 A c)) \int \sqrt{x} \sqrt{b x^2+c x^4} \, dx}{11 c}\\ &=-\frac{4 b (b B-3 A c) x^{3/2} \sqrt{b x^2+c x^4}}{77 c}-\frac{2 (b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{33 c \sqrt{x}}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{15 c x^{5/2}}-\frac{\left (4 b^2 (b B-3 A c)\right ) \int \frac{x^{5/2}}{\sqrt{b x^2+c x^4}} \, dx}{77 c}\\ &=-\frac{8 b^2 (b B-3 A c) \sqrt{b x^2+c x^4}}{231 c^2 \sqrt{x}}-\frac{4 b (b B-3 A c) x^{3/2} \sqrt{b x^2+c x^4}}{77 c}-\frac{2 (b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{33 c \sqrt{x}}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{15 c x^{5/2}}+\frac{\left (4 b^3 (b B-3 A c)\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{231 c^2}\\ &=-\frac{8 b^2 (b B-3 A c) \sqrt{b x^2+c x^4}}{231 c^2 \sqrt{x}}-\frac{4 b (b B-3 A c) x^{3/2} \sqrt{b x^2+c x^4}}{77 c}-\frac{2 (b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{33 c \sqrt{x}}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{15 c x^{5/2}}+\frac{\left (4 b^3 (b B-3 A c) x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{231 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{8 b^2 (b B-3 A c) \sqrt{b x^2+c x^4}}{231 c^2 \sqrt{x}}-\frac{4 b (b B-3 A c) x^{3/2} \sqrt{b x^2+c x^4}}{77 c}-\frac{2 (b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{33 c \sqrt{x}}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{15 c x^{5/2}}+\frac{\left (8 b^3 (b B-3 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 )}{231 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{8 b^2 (b B-3 A c) \sqrt{b x^2+c x^4}}{231 c^2 \sqrt{x}}-\frac{4 b (b B-3 A c) x^{3/2} \sqrt{b x^2+c x^4}}{77 c}-\frac{2 (b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{33 c \sqrt{x}}+\frac{2 B \left (b x^2+c x^4\right )^{5/2}}{15 c x^{5/2}}+\frac{4 b^{11/4} (b B-3 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 )}{231 c^{9/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.142065, size = 115, normalized size = 0.48 \[ \frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (5 b^2 (b B-3 A c) \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right )-\left (b+c x^2\right )^2 \sqrt{\frac{c x^2}{b}+1} \left (-15 A c+5 b B-11 B c x^2\right )\right )}{165 c^2 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 307, normalized size = 1.3 \begin{align*} -{\frac{2}{1155\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -77\,B{x}^{9}{c}^{5}-105\,A{x}^{7}{c}^{5}-196\,B{x}^{7}b{c}^{4}+30\,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}^{3}c-300\,A{x}^{5}b{c}^{4}-10\,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}^{4}-131\,B{x}^{5}{b}^{2}{c}^{3}-255\,A{x}^{3}{b}^{2}{c}^{3}+8\,B{x}^{3}{b}^{3}{c}^{2}-60\,Ax{b}^{3}{c}^{2}+20\,Bx{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 )}}{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^{4} +{\left (B b + A c\right )} x^{2} + A b\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 )}}{x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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