Optimal. Leaf size=299 \[ \frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{2 b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}-\frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}+\frac{x^{3/2} \left (b+c x^2\right ) (3 b B-A c)}{b c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{x^{5/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.327944, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2037, 2032, 329, 305, 220, 1196} \[ \frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}-\frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}+\frac{x^{3/2} \left (b+c x^2\right ) (3 b B-A c)}{b c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{x^{5/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2037
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^{7/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{(b B-A c) x^{5/2}}{b c \sqrt{b x^2+c x^4}}+\frac{\left (\frac{3 b B}{2}-\frac{A c}{2}\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{b c}\\ &=-\frac{(b B-A c) x^{5/2}}{b c \sqrt{b x^2+c x^4}}+\frac{\left (\left (\frac{3 b B}{2}-\frac{A c}{2}\right ) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{b c \sqrt{b x^2+c x^4}}\\ &=-\frac{(b B-A c) x^{5/2}}{b c \sqrt{b x^2+c x^4}}+\frac{\left (2 \left (\frac{3 b B}{2}-\frac{A c}{2}\right ) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{b c \sqrt{b x^2+c x^4}}\\ &=-\frac{(b B-A c) x^{5/2}}{b c \sqrt{b x^2+c x^4}}+\frac{\left (2 \left (\frac{3 b B}{2}-\frac{A c}{2}\right ) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{\sqrt{b} c^{3/2} \sqrt{b x^2+c x^4}}-\frac{\left (2 \left (\frac{3 b B}{2}-\frac{A c}{2}\right ) 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 )}{\sqrt{b} c^{3/2} \sqrt{b x^2+c x^4}}\\ &=-\frac{(b B-A c) x^{5/2}}{b c \sqrt{b x^2+c x^4}}+\frac{(3 b B-A c) x^{3/2} \left (b+c x^2\right )}{b c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{(3 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 )}{b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}+\frac{(3 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 )}{2 b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.086856, size = 78, normalized size = 0.26 \[ -\frac{2 x^{5/2} \left (\sqrt{\frac{c x^2}{b}+1} (3 b B-A c) \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^2}{b}\right )-3 b B\right )}{3 b c \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 388, normalized size = 1.3 \begin{align*} -{\frac{c{x}^{2}+b}{2\,b{c}^{2}}{x}^{{\frac{5}{2}}} \left ( 2\,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-A\sqrt{{ \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{2}\sqrt{{ \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}}{\it EllipticF} \left ( \sqrt{{ \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ) bc-6\,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}+3\,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}-2\,A{x}^{2}{c}^{2}+2\,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{7}{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^{2} + A\right )} \sqrt{x}}{c^{2} x^{5} + 2 \, b c x^{3} + b^{2} 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 (B x^{2} + A\right )} x^{\frac{7}{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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