Optimal. Leaf size=354 \[ \frac{12 \sqrt [4]{b} \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+b B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{5 \sqrt{b x^2+c x^4}}-\frac{2 \left (b x^2+c x^4\right )^{3/2} (A c+b B)}{b x^{7/2}}+\frac{12 c \sqrt{x} \sqrt{b x^2+c x^4} (A c+b B)}{5 b}+\frac{24 \sqrt{c} x^{3/2} \left (b+c x^2\right ) (A c+b B)}{5 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{24 \sqrt [4]{b} \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 \sqrt{b x^2+c x^4}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{15/2}} \]
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Rubi [A] time = 0.435397, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2038, 2020, 2021, 2032, 329, 305, 220, 1196} \[ -\frac{2 \left (b x^2+c x^4\right )^{3/2} (A c+b B)}{b x^{7/2}}+\frac{12 c \sqrt{x} \sqrt{b x^2+c x^4} (A c+b B)}{5 b}+\frac{24 \sqrt{c} x^{3/2} \left (b+c x^2\right ) (A c+b B)}{5 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{12 \sqrt [4]{b} \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+b B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 \sqrt{b x^2+c x^4}}-\frac{24 \sqrt [4]{b} \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 \sqrt{b x^2+c x^4}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{15/2}} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2020
Rule 2021
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}}{x^{13/2}} \, dx &=-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{15/2}}+-\frac{\left (2 \left (-\frac{5 b B}{2}-\frac{5 A c}{2}\right )\right ) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{9/2}} \, dx}{5 b}\\ &=-\frac{2 (b B+A c) \left (b x^2+c x^4\right )^{3/2}}{b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{15/2}}+\frac{(6 c (b B+A c)) \int \frac{\sqrt{b x^2+c x^4}}{\sqrt{x}} \, dx}{b}\\ &=\frac{12 c (b B+A c) \sqrt{x} \sqrt{b x^2+c x^4}}{5 b}-\frac{2 (b B+A c) \left (b x^2+c x^4\right )^{3/2}}{b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{15/2}}+\frac{1}{5} (12 c (b B+A c)) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{12 c (b B+A c) \sqrt{x} \sqrt{b x^2+c x^4}}{5 b}-\frac{2 (b B+A c) \left (b x^2+c x^4\right )^{3/2}}{b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{15/2}}+\frac{\left (12 c (b B+A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{5 \sqrt{b x^2+c x^4}}\\ &=\frac{12 c (b B+A c) \sqrt{x} \sqrt{b x^2+c x^4}}{5 b}-\frac{2 (b B+A c) \left (b x^2+c x^4\right )^{3/2}}{b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{15/2}}+\frac{\left (24 c (b B+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 )}{5 \sqrt{b x^2+c x^4}}\\ &=\frac{12 c (b B+A c) \sqrt{x} \sqrt{b x^2+c x^4}}{5 b}-\frac{2 (b B+A c) \left (b x^2+c x^4\right )^{3/2}}{b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{15/2}}+\frac{\left (24 \sqrt{b} \sqrt{c} (b B+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 )}{5 \sqrt{b x^2+c x^4}}-\frac{\left (24 \sqrt{b} \sqrt{c} (b B+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 )}{5 \sqrt{b x^2+c x^4}}\\ &=\frac{24 \sqrt{c} (b B+A c) x^{3/2} \left (b+c x^2\right )}{5 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{12 c (b B+A c) \sqrt{x} \sqrt{b x^2+c x^4}}{5 b}-\frac{2 (b B+A c) \left (b x^2+c x^4\right )^{3/2}}{b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{5 b x^{15/2}}-\frac{24 \sqrt [4]{b} \sqrt [4]{c} (b B+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 )}{5 \sqrt{b x^2+c x^4}}+\frac{12 \sqrt [4]{b} \sqrt [4]{c} (b B+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 )}{5 \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.044802, size = 99, normalized size = 0.28 \[ -\frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (5 b x^2 (A c+b B) \, _2F_1\left (-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{c x^2}{b}\right )+A \left (b+c x^2\right )^2 \sqrt{\frac{c x^2}{b}+1}\right )}{5 b x^{7/2} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 427, normalized size = 1.2 \begin{align*}{\frac{2}{5\, \left ( c{x}^{2}+b \right ) ^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 12\,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 ){x}^{2}bc-6\,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 ){x}^{2}bc+12\,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 ){x}^{2}{b}^{2}-6\,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 ){x}^{2}{b}^{2}+B{c}^{2}{x}^{6}-7\,A{c}^{2}{x}^{4}-4\,B{x}^{4}bc-8\,Abc{x}^{2}-5\,B{x}^{2}{b}^{2}-A{b}^{2} \right ){x}^{-{\frac{11}{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{13}{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 (\frac{{\left (B c x^{4} +{\left (B b + A c\right )} x^{2} + A b\right )} \sqrt{c x^{4} + b x^{2}}}{x^{\frac{9}{2}}}, 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{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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