Optimal. Leaf size=302 \[ -\frac{\sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{4 a^{7/4} b^{3/4} \sqrt{a+b x^2}}+\frac{\sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{2 a^{7/4} b^{3/4} \sqrt{a+b x^2}}+\frac{(c x)^{3/2}}{2 a^2 c \sqrt{a+b x^2}}-\frac{\sqrt{c x} \sqrt{a+b x^2}}{2 a^2 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{(c x)^{3/2}}{3 a c \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.223984, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {290, 329, 305, 220, 1196} \[ -\frac{\sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{4 a^{7/4} b^{3/4} \sqrt{a+b x^2}}+\frac{\sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{2 a^{7/4} b^{3/4} \sqrt{a+b x^2}}+\frac{(c x)^{3/2}}{2 a^2 c \sqrt{a+b x^2}}-\frac{\sqrt{c x} \sqrt{a+b x^2}}{2 a^2 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{(c x)^{3/2}}{3 a c \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\sqrt{c x}}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac{(c x)^{3/2}}{3 a c \left (a+b x^2\right )^{3/2}}+\frac{\int \frac{\sqrt{c x}}{\left (a+b x^2\right )^{3/2}} \, dx}{2 a}\\ &=\frac{(c x)^{3/2}}{3 a c \left (a+b x^2\right )^{3/2}}+\frac{(c x)^{3/2}}{2 a^2 c \sqrt{a+b x^2}}-\frac{\int \frac{\sqrt{c x}}{\sqrt{a+b x^2}} \, dx}{4 a^2}\\ &=\frac{(c x)^{3/2}}{3 a c \left (a+b x^2\right )^{3/2}}+\frac{(c x)^{3/2}}{2 a^2 c \sqrt{a+b x^2}}-\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{2 a^2 c}\\ &=\frac{(c x)^{3/2}}{3 a c \left (a+b x^2\right )^{3/2}}+\frac{(c x)^{3/2}}{2 a^2 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 )}{2 a^{3/2} \sqrt{b}}+\frac{\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 )}{2 a^{3/2} \sqrt{b}}\\ &=\frac{(c x)^{3/2}}{3 a c \left (a+b x^2\right )^{3/2}}+\frac{(c x)^{3/2}}{2 a^2 c \sqrt{a+b x^2}}-\frac{\sqrt{c x} \sqrt{a+b x^2}}{2 a^2 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{\sqrt{c} \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 )}{2 a^{7/4} b^{3/4} \sqrt{a+b x^2}}-\frac{\sqrt{c} \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 )}{4 a^{7/4} b^{3/4} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0122114, size = 59, normalized size = 0.2 \[ \frac{2 x \sqrt{c x} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{3}{4},\frac{5}{2};\frac{7}{4};-\frac{b x^2}{a}\right )}{3 a^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 382, normalized size = 1.3 \begin{align*} -{\frac{1}{12\,{a}^{2}bx} \left ( 6\,\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 ){x}^{2}ab-3\,\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}^{2}ab+6\,\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}-3\,\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}-6\,{b}^{2}{x}^{4}-10\,ab{x}^{2} \right ) \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{\sqrt{c x}}{{\left (b x^{2} + a\right )}^{\frac{5}{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{b x^{2} + a} \sqrt{c x}}{b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.78339, size = 44, normalized size = 0.15 \begin{align*} \frac{\sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{2}} \Gamma \left (\frac{7}{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{\sqrt{c x}}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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