Optimal. Leaf size=296 \[ \frac{12 a^{5/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{5 c^{3/2} \sqrt{a+b x^2}}-\frac{24 a^{5/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{3/2} \sqrt{a+b x^2}}+\frac{12 b (c x)^{3/2} \sqrt{a+b x^2}}{5 c^3}+\frac{24 a \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{5 c^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \left (a+b x^2\right )^{3/2}}{c \sqrt{c x}} \]
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Rubi [A] time = 0.229691, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {277, 279, 329, 305, 220, 1196} \[ \frac{12 a^{5/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{3/2} \sqrt{a+b x^2}}-\frac{24 a^{5/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{3/2} \sqrt{a+b x^2}}+\frac{12 b (c x)^{3/2} \sqrt{a+b x^2}}{5 c^3}+\frac{24 a \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{5 c^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \left (a+b x^2\right )^{3/2}}{c \sqrt{c x}} \]
Antiderivative was successfully verified.
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Rule 277
Rule 279
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2}}{(c x)^{3/2}} \, dx &=-\frac{2 \left (a+b x^2\right )^{3/2}}{c \sqrt{c x}}+\frac{(6 b) \int \sqrt{c x} \sqrt{a+b x^2} \, dx}{c^2}\\ &=\frac{12 b (c x)^{3/2} \sqrt{a+b x^2}}{5 c^3}-\frac{2 \left (a+b x^2\right )^{3/2}}{c \sqrt{c x}}+\frac{(12 a b) \int \frac{\sqrt{c x}}{\sqrt{a+b x^2}} \, dx}{5 c^2}\\ &=\frac{12 b (c x)^{3/2} \sqrt{a+b x^2}}{5 c^3}-\frac{2 \left (a+b x^2\right )^{3/2}}{c \sqrt{c x}}+\frac{(24 a b) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{5 c^3}\\ &=\frac{12 b (c x)^{3/2} \sqrt{a+b x^2}}{5 c^3}-\frac{2 \left (a+b x^2\right )^{3/2}}{c \sqrt{c x}}+\frac{\left (24 a^{3/2} \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{5 c^2}-\frac{\left (24 a^{3/2} \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a} c}}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{5 c^2}\\ &=\frac{12 b (c x)^{3/2} \sqrt{a+b x^2}}{5 c^3}+\frac{24 a \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{5 c^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \left (a+b x^2\right )^{3/2}}{c \sqrt{c x}}-\frac{24 a^{5/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{3/2} \sqrt{a+b x^2}}+\frac{12 a^{5/4} \sqrt [4]{b} \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 )}{5 c^{3/2} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0127858, size = 55, normalized size = 0.19 \[ -\frac{2 a x \sqrt{a+b x^2} \, _2F_1\left (-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^2}{a}\right )}{(c x)^{3/2} \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 208, normalized size = 0.7 \begin{align*}{\frac{2}{5\,c} \left ( 12\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}-6\,\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 ){a}^{2}+{b}^{2}{x}^{4}-4\,ab{x}^{2}-5\,{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{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{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{c x}}{c^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.90356, size = 49, normalized size = 0.17 \begin{align*} \frac{a^{\frac{3}{2}} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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