Optimal. Leaf size=148 \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{77 c^{5/4} \sqrt{a+c x^4}}+\frac{4 a^2 x \sqrt{a+c x^4}}{77 c}+\frac{1}{11} x^5 \left (a+c x^4\right )^{3/2}+\frac{6}{77} a x^5 \sqrt{a+c x^4} \]
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Rubi [A] time = 0.0478541, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {279, 321, 220} \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{77 c^{5/4} \sqrt{a+c x^4}}+\frac{4 a^2 x \sqrt{a+c x^4}}{77 c}+\frac{1}{11} x^5 \left (a+c x^4\right )^{3/2}+\frac{6}{77} a x^5 \sqrt{a+c x^4} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 220
Rubi steps
\begin{align*} \int x^4 \left (a+c x^4\right )^{3/2} \, dx &=\frac{1}{11} x^5 \left (a+c x^4\right )^{3/2}+\frac{1}{11} (6 a) \int x^4 \sqrt{a+c x^4} \, dx\\ &=\frac{6}{77} a x^5 \sqrt{a+c x^4}+\frac{1}{11} x^5 \left (a+c x^4\right )^{3/2}+\frac{1}{77} \left (12 a^2\right ) \int \frac{x^4}{\sqrt{a+c x^4}} \, dx\\ &=\frac{4 a^2 x \sqrt{a+c x^4}}{77 c}+\frac{6}{77} a x^5 \sqrt{a+c x^4}+\frac{1}{11} x^5 \left (a+c x^4\right )^{3/2}-\frac{\left (4 a^3\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx}{77 c}\\ &=\frac{4 a^2 x \sqrt{a+c x^4}}{77 c}+\frac{6}{77} a x^5 \sqrt{a+c x^4}+\frac{1}{11} x^5 \left (a+c x^4\right )^{3/2}-\frac{2 a^{11/4} \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 )}{77 c^{5/4} \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0573863, size = 67, normalized size = 0.45 \[ \frac{x \sqrt{a+c x^4} \left (\left (a+c x^4\right )^2-\frac{a^2 \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^4}{a}\right )}{\sqrt{\frac{c x^4}{a}+1}}\right )}{11 c} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 126, normalized size = 0.9 \begin{align*}{\frac{c{x}^{9}}{11}\sqrt{c{x}^{4}+a}}+{\frac{13\,a{x}^{5}}{77}\sqrt{c{x}^{4}+a}}+{\frac{4\,{a}^{2}x}{77\,c}\sqrt{c{x}^{4}+a}}-{\frac{4\,{a}^{3}}{77\,c}\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}} x^{4}\,{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^{8} + a x^{4}\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 = 1.07325, size = 39, normalized size = 0.26 \begin{align*} \frac{a^{\frac{3}{2}} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{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^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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