Optimal. Leaf size=126 \[ \frac{2 c^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{7 \sqrt [4]{a} \sqrt{a+c x^4}}-\frac{2 c \sqrt{a+c x^4}}{7 x^3}-\frac{\left (a+c x^4\right )^{3/2}}{7 x^7} \]
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Rubi [A] time = 0.0322656, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {277, 220} \[ \frac{2 c^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{a} \sqrt{a+c x^4}}-\frac{2 c \sqrt{a+c x^4}}{7 x^3}-\frac{\left (a+c x^4\right )^{3/2}}{7 x^7} \]
Antiderivative was successfully verified.
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Rule 277
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+c x^4\right )^{3/2}}{x^8} \, dx &=-\frac{\left (a+c x^4\right )^{3/2}}{7 x^7}+\frac{1}{7} (6 c) \int \frac{\sqrt{a+c x^4}}{x^4} \, dx\\ &=-\frac{2 c \sqrt{a+c x^4}}{7 x^3}-\frac{\left (a+c x^4\right )^{3/2}}{7 x^7}+\frac{1}{7} \left (4 c^2\right ) \int \frac{1}{\sqrt{a+c x^4}} \, dx\\ &=-\frac{2 c \sqrt{a+c x^4}}{7 x^3}-\frac{\left (a+c x^4\right )^{3/2}}{7 x^7}+\frac{2 c^{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]{a} \sqrt{a+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0108799, size = 52, normalized size = 0.41 \[ -\frac{a \sqrt{a+c x^4} \, _2F_1\left (-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{c x^4}{a}\right )}{7 x^7 \sqrt{\frac{c x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.014, size = 105, normalized size = 0.8 \begin{align*} -{\frac{a}{7\,{x}^{7}}\sqrt{c{x}^{4}+a}}-{\frac{3\,c}{7\,{x}^{3}}\sqrt{c{x}^{4}+a}}+{\frac{4\,{c}^{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 \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{x^{8}}\,{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^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.73654, size = 46, normalized size = 0.37 \begin{align*} \frac{a^{\frac{3}{2}} \Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, - \frac{3}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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