Optimal. Leaf size=255 \[ \frac{2 a^{9/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}-\frac{4 a^{9/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}+\frac{4 a^2 x \sqrt{a+c x^4}}{15 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac{2}{15} a x^3 \sqrt{a+c x^4} \]
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Rubi [A] time = 0.0861974, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 305, 220, 1196} \[ \frac{2 a^{9/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}-\frac{4 a^{9/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}+\frac{4 a^2 x \sqrt{a+c x^4}}{15 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac{2}{15} a x^3 \sqrt{a+c x^4} \]
Antiderivative was successfully verified.
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Rule 279
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int x^2 \left (a+c x^4\right )^{3/2} \, dx &=\frac{1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac{1}{3} (2 a) \int x^2 \sqrt{a+c x^4} \, dx\\ &=\frac{2}{15} a x^3 \sqrt{a+c x^4}+\frac{1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac{1}{15} \left (4 a^2\right ) \int \frac{x^2}{\sqrt{a+c x^4}} \, dx\\ &=\frac{2}{15} a x^3 \sqrt{a+c x^4}+\frac{1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac{\left (4 a^{5/2}\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{15 \sqrt{c}}-\frac{\left (4 a^{5/2}\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx}{15 \sqrt{c}}\\ &=\frac{2}{15} a x^3 \sqrt{a+c x^4}+\frac{4 a^2 x \sqrt{a+c x^4}}{15 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{1}{9} x^3 \left (a+c x^4\right )^{3/2}-\frac{4 a^{9/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}+\frac{2 a^{9/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0105692, size = 52, normalized size = 0.2 \[ \frac{a x^3 \sqrt{a+c x^4} \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right )}{3 \sqrt{\frac{c x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 128, normalized size = 0.5 \begin{align*}{\frac{c{x}^{7}}{9}\sqrt{c{x}^{4}+a}}+{\frac{11\,a{x}^{3}}{45}\sqrt{c{x}^{4}+a}}+{{\frac{4\,i}{15}}{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}{\frac{1}{\sqrt{c}}}} \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 (c x^{4} + a\right )}^{\frac{3}{2}} x^{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 (c x^{6} + a x^{2}\right )} \sqrt{c x^{4} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.2457, size = 39, normalized size = 0.15 \begin{align*} \frac{a^{\frac{3}{2}} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \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{\left (c x^{4} + a\right )}^{\frac{3}{2}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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