Optimal. Leaf size=126 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{2 a^{5/4} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{\sqrt{c x}}{a c \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.0700747, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {290, 329, 220} \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{2 a^{5/4} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{\sqrt{c x}}{a c \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c x} \left (a+b x^2\right )^{3/2}} \, dx &=\frac{\sqrt{c x}}{a c \sqrt{a+b x^2}}+\frac{\int \frac{1}{\sqrt{c x} \sqrt{a+b x^2}} \, dx}{2 a}\\ &=\frac{\sqrt{c x}}{a c \sqrt{a+b x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{a c}\\ &=\frac{\sqrt{c x}}{a c \sqrt{a+b x^2}}+\frac{\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 )}{2 a^{5/4} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0184135, size = 59, normalized size = 0.47 \[ \frac{x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right )+x}{a \sqrt{c x} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 114, normalized size = 0.9 \begin{align*}{\frac{1}{2\,ab} \left ({\it EllipticF} \left ( \sqrt{{ \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{ \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{ \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}\sqrt{-ab}+2\,bx \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{1}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{c x}}\,{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{b x^{2} + a} \sqrt{c x}}{b^{2} c x^{5} + 2 \, a b c x^{3} + a^{2} c x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.11185, size = 44, normalized size = 0.35 \begin{align*} \frac{\sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{c} \Gamma \left (\frac{5}{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{1}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{c x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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