Optimal. Leaf size=163 \[ \frac{2 x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+b B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{3 \sqrt [4]{b} \sqrt [4]{c} \sqrt{b x^2+c x^4}}+\frac{2 \sqrt{b x^2+c x^4} (A c+b B)}{3 b \sqrt{x}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}} \]
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Rubi [A] time = 0.242626, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2038, 2021, 2032, 329, 220} \[ \frac{2 \sqrt{b x^2+c x^4} (A c+b B)}{3 b \sqrt{x}}+\frac{2 x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+b B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{b} \sqrt [4]{c} \sqrt{b x^2+c x^4}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2021
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{x^{7/2}} \, dx &=-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+-\frac{\left (2 \left (-\frac{3 b B}{2}-\frac{3 A c}{2}\right )\right ) \int \frac{\sqrt{b x^2+c x^4}}{x^{3/2}} \, dx}{3 b}\\ &=\frac{2 (b B+A c) \sqrt{b x^2+c x^4}}{3 b \sqrt{x}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+\frac{1}{3} (2 (b B+A c)) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{2 (b B+A c) \sqrt{b x^2+c x^4}}{3 b \sqrt{x}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+\frac{\left (2 (b B+A c) x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{3 \sqrt{b x^2+c x^4}}\\ &=\frac{2 (b B+A c) \sqrt{b x^2+c x^4}}{3 b \sqrt{x}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+\frac{\left (4 (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 )}{3 \sqrt{b x^2+c x^4}}\\ &=\frac{2 (b B+A c) \sqrt{b x^2+c x^4}}{3 b \sqrt{x}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+\frac{2 (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 )}{3 \sqrt [4]{b} \sqrt [4]{c} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0435645, size = 97, normalized size = 0.6 \[ \frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (3 x^2 (A c+b B) \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right )-A \left (b+c x^2\right ) \sqrt{\frac{c x^2}{b}+1}\right )}{3 b x^{5/2} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 239, normalized size = 1.5 \begin{align*}{\frac{2}{ \left ( 3\,c{x}^{2}+3\,b \right ) c}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 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 ) \sqrt{-bc}xc+B\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 ) \sqrt{-bc}xb+B{c}^{2}{x}^{4}-A{x}^{2}{c}^{2}+B{x}^{2}bc-Abc \right ){x}^{-{\frac{5}{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{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{7}{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 )}}{x^{\frac{7}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{\frac{7}{2}}}\, dx \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{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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