Optimal. Leaf size=329 \[ -\frac{4 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}-\frac{8 a^3 c^2 \sqrt{c x} \sqrt{a+b x^2}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{8 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c} \]
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Rubi [A] time = 0.255973, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {279, 321, 329, 305, 220, 1196} \[ -\frac{8 a^3 c^2 \sqrt{c x} \sqrt{a+b x^2}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{4 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^{13/4} c^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int (c x)^{5/2} \left (a+b x^2\right )^{3/2} \, dx &=\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac{1}{13} (6 a) \int (c x)^{5/2} \sqrt{a+b x^2} \, dx\\ &=\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac{1}{39} \left (4 a^2\right ) \int \frac{(c x)^{5/2}}{\sqrt{a+b x^2}} \, dx\\ &=\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}-\frac{\left (4 a^3 c^2\right ) \int \frac{\sqrt{c x}}{\sqrt{a+b x^2}} \, dx}{65 b}\\ &=\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}-\frac{\left (8 a^3 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{65 b}\\ &=\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}-\frac{\left (8 a^{7/2} c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{65 b^{3/2}}+\frac{\left (8 a^{7/2} c^2\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 )}{65 b^{3/2}}\\ &=\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c}-\frac{8 a^3 c^2 \sqrt{c x} \sqrt{a+b x^2}}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 (c x)^{7/2} \left (a+b x^2\right )^{3/2}}{13 c}+\frac{8 a^{13/4} c^{5/2} \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 )}{65 b^{7/4} \sqrt{a+b x^2}}-\frac{4 a^{13/4} c^{5/2} \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 )}{65 b^{7/4} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0621499, size = 89, normalized size = 0.27 \[ \frac{2 c (c x)^{3/2} \sqrt{a+b x^2} \left (\left (a+b x^2\right )^2 \sqrt{\frac{b x^2}{a}+1}-a^2 \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )\right )}{13 b \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 232, normalized size = 0.7 \begin{align*} -{\frac{2\,{c}^{2}}{195\,{b}^{2}x}\sqrt{cx} \left ( -15\,{x}^{8}{b}^{4}-40\,{x}^{6}a{b}^{3}+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}^{4}-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}^{4}-29\,{x}^{4}{a}^{2}{b}^{2}-4\,{x}^{2}{a}^{3}b \right ){\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{\left (b x^{2} + a\right )}^{\frac{3}{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 ({\left (b c^{2} x^{4} + a c^{2} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{c x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 91.753, size = 46, normalized size = 0.14 \begin{align*} \frac{a^{\frac{3}{2}} c^{\frac{5}{2}} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{11}{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{\left (b x^{2} + a\right )}^{\frac{3}{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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