Optimal. Leaf size=157 \[ \frac{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{12 a^{9/4} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{5 \sqrt{c x}}{6 a^2 c \sqrt{a+b x^2}}+\frac{\sqrt{c x}}{3 a c \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0882105, antiderivative size = 157, 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 = {290, 329, 220} \[ \frac{5 \sqrt{c x}}{6 a^2 c \sqrt{a+b x^2}}+\frac{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{\sqrt{c x}}{3 a c \left (a+b x^2\right )^{3/2}} \]
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 )^{5/2}} \, dx &=\frac{\sqrt{c x}}{3 a c \left (a+b x^2\right )^{3/2}}+\frac{5 \int \frac{1}{\sqrt{c x} \left (a+b x^2\right )^{3/2}} \, dx}{6 a}\\ &=\frac{\sqrt{c x}}{3 a c \left (a+b x^2\right )^{3/2}}+\frac{5 \sqrt{c x}}{6 a^2 c \sqrt{a+b x^2}}+\frac{5 \int \frac{1}{\sqrt{c x} \sqrt{a+b x^2}} \, dx}{12 a^2}\\ &=\frac{\sqrt{c x}}{3 a c \left (a+b x^2\right )^{3/2}}+\frac{5 \sqrt{c x}}{6 a^2 c \sqrt{a+b x^2}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{6 a^2 c}\\ &=\frac{\sqrt{c x}}{3 a c \left (a+b x^2\right )^{3/2}}+\frac{5 \sqrt{c x}}{6 a^2 c \sqrt{a+b x^2}}+\frac{5 \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 )}{12 a^{9/4} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0352485, size = 79, normalized size = 0.5 \[ \frac{5 x \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right )+7 a x+5 b x^3}{6 a^2 \sqrt{c x} \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 216, normalized size = 1.4 \begin{align*}{\frac{1}{12\,{a}^{2}b} \left ( 5\,\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 ) \sqrt{-ab}{x}^{2}b+5\,\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 ) \sqrt{-ab}a+10\,{b}^{2}{x}^{3}+14\,abx \right ){\frac{1}{\sqrt{cx}}} \left ( b{x}^{2}+a \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{1}{{\left (b x^{2} + a\right )}^{\frac{5}{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^{3} c x^{7} + 3 \, a b^{2} c x^{5} + 3 \, a^{2} b c x^{3} + a^{3} c x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 8.71567, size = 44, normalized size = 0.28 \begin{align*} \frac{\sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{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{5}{2}} \sqrt{c x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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