Optimal. Leaf size=122 \[ \frac{2 a^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{7 \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{2}{7} a x \sqrt{a+c x^4}+\frac{1}{7} x \left (a+c x^4\right )^{3/2} \]
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Rubi [A] time = 0.0275611, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {195, 220} \[ \frac{2 a^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{2}{7} a x \sqrt{a+c x^4}+\frac{1}{7} x \left (a+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 220
Rubi steps
\begin{align*} \int \left (a+c x^4\right )^{3/2} \, dx &=\frac{1}{7} x \left (a+c x^4\right )^{3/2}+\frac{1}{7} (6 a) \int \sqrt{a+c x^4} \, dx\\ &=\frac{2}{7} a x \sqrt{a+c x^4}+\frac{1}{7} x \left (a+c x^4\right )^{3/2}+\frac{1}{7} \left (4 a^2\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx\\ &=\frac{2}{7} a x \sqrt{a+c x^4}+\frac{1}{7} x \left (a+c x^4\right )^{3/2}+\frac{2 a^{7/4} \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 )}{7 \sqrt [4]{c} \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0048849, size = 47, normalized size = 0.39 \[ \frac{a x \sqrt{a+c x^4} \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^4}{a}\right )}{\sqrt{\frac{c x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.005, size = 103, normalized size = 0.8 \begin{align*}{\frac{c{x}^{5}}{7}\sqrt{c{x}^{4}+a}}+{\frac{3\,ax}{7}\sqrt{c{x}^{4}+a}}+{\frac{4\,{a}^{2}}{7}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+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 (c x^{4} + a\right )}^{\frac{3}{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^{4} + a\right )}^{\frac{3}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.800272, size = 37, normalized size = 0.3 \begin{align*} \frac{a^{\frac{3}{2}} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \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{\left (c x^{4} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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