Optimal. Leaf size=124 \[ \frac{2 a^{3/4} c^{3/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{3 \sqrt{a+c x^4}}-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3}+\frac{2}{3} c x \sqrt{a+c x^4} \]
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Rubi [A] time = 0.0300392, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {277, 195, 220} \[ \frac{2 a^{3/4} c^{3/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+c x^4}}-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3}+\frac{2}{3} c x \sqrt{a+c x^4} \]
Antiderivative was successfully verified.
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Rule 277
Rule 195
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+c x^4\right )^{3/2}}{x^4} \, dx &=-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3}+(2 c) \int \sqrt{a+c x^4} \, dx\\ &=\frac{2}{3} c x \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3}+\frac{1}{3} (4 a c) \int \frac{1}{\sqrt{a+c x^4}} \, dx\\ &=\frac{2}{3} c x \sqrt{a+c x^4}-\frac{\left (a+c x^4\right )^{3/2}}{3 x^3}+\frac{2 a^{3/4} c^{3/4} \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 )}{3 \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0085894, size = 52, normalized size = 0.42 \[ -\frac{a \sqrt{a+c x^4} \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{c x^4}{a}\right )}{3 x^3 \sqrt{\frac{c x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 102, normalized size = 0.8 \begin{align*} -{\frac{a}{3\,{x}^{3}}\sqrt{c{x}^{4}+a}}+{\frac{cx}{3}\sqrt{c{x}^{4}+a}}+{\frac{4\,ac}{3}\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 \frac{{\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 (\frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.03264, size = 42, normalized size = 0.34 \begin{align*} \frac{a^{\frac{3}{2}} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \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{{\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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