Optimal. Leaf size=185 \[ -\frac{5 b^{3/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{4 a^{13/4} c^{5/2} \sqrt{a+b x^2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 c (c x)^{3/2}}+\frac{3}{2 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}+\frac{1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.110333, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {290, 325, 329, 220} \[ -\frac{5 b^{3/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{4 a^{13/4} c^{5/2} \sqrt{a+b x^2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 c (c x)^{3/2}}+\frac{3}{2 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}+\frac{1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx &=\frac{1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac{3 \int \frac{1}{(c x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx}{2 a}\\ &=\frac{1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac{3}{2 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}+\frac{15 \int \frac{1}{(c x)^{5/2} \sqrt{a+b x^2}} \, dx}{4 a^2}\\ &=\frac{1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac{3}{2 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 c (c x)^{3/2}}-\frac{(5 b) \int \frac{1}{\sqrt{c x} \sqrt{a+b x^2}} \, dx}{4 a^3 c^2}\\ &=\frac{1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac{3}{2 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 c (c x)^{3/2}}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{2 a^3 c^3}\\ &=\frac{1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac{3}{2 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 c (c x)^{3/2}}-\frac{5 b^{3/4} \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 )}{4 a^{13/4} c^{5/2} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.013029, size = 59, normalized size = 0.32 \[ -\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{3}{4},\frac{5}{2};\frac{1}{4};-\frac{b x^2}{a}\right )}{3 a^2 (c x)^{5/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 227, normalized size = 1.2 \begin{align*} -{\frac{1}{12\,x{c}^{2}{a}^{3}} \left ( 15\,\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+15\,\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}xa+30\,{b}^{2}{x}^{4}+42\,ab{x}^{2}+8\,{a}^{2} \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}} \left (c x\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} c^{3} x^{9} + 3 \, a b^{2} c^{3} x^{7} + 3 \, a^{2} b c^{3} x^{5} + a^{3} c^{3} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 85.2115, size = 48, normalized size = 0.26 \begin{align*} \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{5}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{2}} c^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{1}{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}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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