Optimal. Leaf size=340 \[ -\frac{3 \sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-5 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{10 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{3 x^{3/2} \left (b+c x^2\right ) (7 b B-5 A c)}{5 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{\sqrt{x} \sqrt{b x^2+c x^4} (7 b B-5 A c)}{5 b c^2}+\frac{3 \sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-5 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{x^{9/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.396641, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2037, 2024, 2032, 329, 305, 220, 1196} \[ -\frac{3 x^{3/2} \left (b+c x^2\right ) (7 b B-5 A c)}{5 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{\sqrt{x} \sqrt{b x^2+c x^4} (7 b B-5 A c)}{5 b c^2}-\frac{3 \sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 c^{11/4} \sqrt{b x^2+c x^4}}+\frac{3 \sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-5 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{x^{9/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2037
Rule 2024
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^{11/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{(b B-A c) x^{9/2}}{b c \sqrt{b x^2+c x^4}}+\frac{\left (\frac{7 b B}{2}-\frac{5 A c}{2}\right ) \int \frac{x^{7/2}}{\sqrt{b x^2+c x^4}} \, dx}{b c}\\ &=-\frac{(b B-A c) x^{9/2}}{b c \sqrt{b x^2+c x^4}}+\frac{(7 b B-5 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{5 b c^2}-\frac{(3 (7 b B-5 A c)) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{10 c^2}\\ &=-\frac{(b B-A c) x^{9/2}}{b c \sqrt{b x^2+c x^4}}+\frac{(7 b B-5 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{5 b c^2}-\frac{\left (3 (7 b B-5 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{10 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{(b B-A c) x^{9/2}}{b c \sqrt{b x^2+c x^4}}+\frac{(7 b B-5 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{5 b c^2}-\frac{\left (3 (7 b B-5 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 c^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{(b B-A c) x^{9/2}}{b c \sqrt{b x^2+c x^4}}+\frac{(7 b B-5 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{5 b c^2}-\frac{\left (3 \sqrt{b} (7 b B-5 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 c^{5/2} \sqrt{b x^2+c x^4}}+\frac{\left (3 \sqrt{b} (7 b B-5 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 c^{5/2} \sqrt{b x^2+c x^4}}\\ &=-\frac{(b B-A c) x^{9/2}}{b c \sqrt{b x^2+c x^4}}-\frac{3 (7 b B-5 A c) x^{3/2} \left (b+c x^2\right )}{5 c^{5/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{(7 b B-5 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{5 b c^2}+\frac{3 \sqrt [4]{b} (7 b B-5 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 c^{11/4} \sqrt{b x^2+c x^4}}-\frac{3 \sqrt [4]{b} (7 b B-5 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 )}{10 c^{11/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.102897, size = 85, normalized size = 0.25 \[ \frac{2 x^{5/2} \left (\sqrt{\frac{c x^2}{b}+1} (7 b B-5 A c) \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^2}{b}\right )+5 A c-7 b B+B c x^2\right )}{5 c^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 394, normalized size = 1.2 \begin{align*}{\frac{c{x}^{2}+b}{10\,{c}^{3}}{x}^{{\frac{5}{2}}} \left ( 30\,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 ) bc-15\,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 ) bc-42\,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}^{2}+21\,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}^{2}+4\,B{c}^{2}{x}^{4}-10\,A{x}^{2}{c}^{2}+14\,B{x}^{2}bc \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{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 (B x^{2} + A\right )} x^{\frac{11}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\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 (\frac{\sqrt{c x^{4} + b x^{2}}{\left (B x^{3} + A x\right )} \sqrt{x}}{c^{2} x^{4} + 2 \, b c x^{2} + b^{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 (B x^{2} + A\right )} x^{\frac{11}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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