Optimal. Leaf size=152 \[ \frac{4 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{7 \sqrt [4]{a} c^{9/2} \sqrt{a+b x^2}}-\frac{4 b \sqrt{a+b x^2}}{7 c^3 (c x)^{3/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{7 c (c x)^{7/2}} \]
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Rubi [A] time = 0.0917747, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {277, 329, 220} \[ \frac{4 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{a} c^{9/2} \sqrt{a+b x^2}}-\frac{4 b \sqrt{a+b x^2}}{7 c^3 (c x)^{3/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{7 c (c x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 277
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2}}{(c x)^{9/2}} \, dx &=-\frac{2 \left (a+b x^2\right )^{3/2}}{7 c (c x)^{7/2}}+\frac{(6 b) \int \frac{\sqrt{a+b x^2}}{(c x)^{5/2}} \, dx}{7 c^2}\\ &=-\frac{4 b \sqrt{a+b x^2}}{7 c^3 (c x)^{3/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{7 c (c x)^{7/2}}+\frac{\left (4 b^2\right ) \int \frac{1}{\sqrt{c x} \sqrt{a+b x^2}} \, dx}{7 c^4}\\ &=-\frac{4 b \sqrt{a+b x^2}}{7 c^3 (c x)^{3/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{7 c (c x)^{7/2}}+\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{7 c^5}\\ &=-\frac{4 b \sqrt{a+b x^2}}{7 c^3 (c x)^{3/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{7 c (c x)^{7/2}}+\frac{4 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{a} c^{9/2} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0146829, size = 57, normalized size = 0.38 \[ -\frac{2 a x \sqrt{a+b x^2} \, _2F_1\left (-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{b x^2}{a}\right )}{7 (c x)^{9/2} \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 135, normalized size = 0.9 \begin{align*}{\frac{2}{7\,{x}^{3}{c}^{4}} \left ( 2\,\sqrt{-ab}\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{3}b-3\,{b}^{2}{x}^{4}-4\,ab{x}^{2}-{a}^{2} \right ){\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \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 )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{9}{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 x^{2} + a\right )}^{\frac{3}{2}} \sqrt{c x}}{c^{5} x^{5}}, 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 )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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