Optimal. Leaf size=252 \[ \frac{6 \sqrt [4]{a} c^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{5 \sqrt{a+c x^4}}+\frac{12 c^{3/2} x \sqrt{a+c x^4}}{5 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{12 \sqrt [4]{a} c^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+c x^4}}-\frac{6 c \sqrt{a+c x^4}}{5 x}-\frac{\left (a+c x^4\right )^{3/2}}{5 x^5} \]
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Rubi [A] time = 0.0855572, antiderivative size = 252, 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 = {277, 305, 220, 1196} \[ \frac{12 c^{3/2} x \sqrt{a+c x^4}}{5 \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{6 \sqrt [4]{a} c^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+c x^4}}-\frac{12 \sqrt [4]{a} c^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+c x^4}}-\frac{6 c \sqrt{a+c x^4}}{5 x}-\frac{\left (a+c x^4\right )^{3/2}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 277
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a+c x^4\right )^{3/2}}{x^6} \, dx &=-\frac{\left (a+c x^4\right )^{3/2}}{5 x^5}+\frac{1}{5} (6 c) \int \frac{\sqrt{a+c x^4}}{x^2} \, dx\\ &=-\frac{6 c \sqrt{a+c x^4}}{5 x}-\frac{\left (a+c x^4\right )^{3/2}}{5 x^5}+\frac{1}{5} \left (12 c^2\right ) \int \frac{x^2}{\sqrt{a+c x^4}} \, dx\\ &=-\frac{6 c \sqrt{a+c x^4}}{5 x}-\frac{\left (a+c x^4\right )^{3/2}}{5 x^5}+\frac{1}{5} \left (12 \sqrt{a} c^{3/2}\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx-\frac{1}{5} \left (12 \sqrt{a} c^{3/2}\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx\\ &=-\frac{6 c \sqrt{a+c x^4}}{5 x}+\frac{12 c^{3/2} x \sqrt{a+c x^4}}{5 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\left (a+c x^4\right )^{3/2}}{5 x^5}-\frac{12 \sqrt [4]{a} c^{5/4} \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 )}{5 \sqrt{a+c x^4}}+\frac{6 \sqrt [4]{a} c^{5/4} \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 )}{5 \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.009079, size = 52, normalized size = 0.21 \[ -\frac{a \sqrt{a+c x^4} \, _2F_1\left (-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{c x^4}{a}\right )}{5 x^5 \sqrt{\frac{c x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 128, normalized size = 0.5 \begin{align*} -{\frac{a}{5\,{x}^{5}}\sqrt{c{x}^{4}+a}}-{\frac{7\,c}{5\,x}\sqrt{c{x}^{4}+a}}+{{\frac{12\,i}{5}}{c}^{{\frac{3}{2}}}\sqrt{a}\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}}}} \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^{6}}\,{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^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.47777, size = 46, normalized size = 0.18 \begin{align*} \frac{a^{\frac{3}{2}} \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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