Optimal. Leaf size=251 \[ \frac {6 a^{5/4} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}-\frac {12 a^{5/4} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}-\frac {\left (a+c x^4\right )^{3/2}}{x}+\frac {6}{5} c x^3 \sqrt {a+c x^4}+\frac {12 a \sqrt {c} x \sqrt {a+c x^4}}{5 \left (\sqrt {a}+\sqrt {c} x^2\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {277, 279, 305, 220, 1196} \[ \frac {6 a^{5/4} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}-\frac {12 a^{5/4} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}+\frac {6}{5} c x^3 \sqrt {a+c x^4}+\frac {12 a \sqrt {c} x \sqrt {a+c x^4}}{5 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (a+c x^4\right )^{3/2}}{x} \]
Antiderivative was successfully verified.
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Rule 220
Rule 277
Rule 279
Rule 305
Rule 1196
Rubi steps
\begin {align*} \int \frac {\left (a+c x^4\right )^{3/2}}{x^2} \, dx &=-\frac {\left (a+c x^4\right )^{3/2}}{x}+(6 c) \int x^2 \sqrt {a+c x^4} \, dx\\ &=\frac {6}{5} c x^3 \sqrt {a+c x^4}-\frac {\left (a+c x^4\right )^{3/2}}{x}+\frac {1}{5} (12 a c) \int \frac {x^2}{\sqrt {a+c x^4}} \, dx\\ &=\frac {6}{5} c x^3 \sqrt {a+c x^4}-\frac {\left (a+c x^4\right )^{3/2}}{x}+\frac {1}{5} \left (12 a^{3/2} \sqrt {c}\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx-\frac {1}{5} \left (12 a^{3/2} \sqrt {c}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx\\ &=\frac {6}{5} c x^3 \sqrt {a+c x^4}+\frac {12 a \sqrt {c} x \sqrt {a+c x^4}}{5 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (a+c x^4\right )^{3/2}}{x}-\frac {12 a^{5/4} \sqrt [4]{c} \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 a^{5/4} \sqrt [4]{c} \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*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.20 \[ -\frac {a \sqrt {a+c x^4} \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};\frac {3}{4};-\frac {c x^4}{a}\right )}{x \sqrt {\frac {c x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 128, normalized size = 0.51 \[ \frac {\sqrt {c \,x^{4}+a}\, c \,x^{3}}{5}+\frac {12 i \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )\right ) a^{\frac {3}{2}} \sqrt {c}}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {\sqrt {c \,x^{4}+a}\, a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 40, normalized size = 0.16 \[ \frac {{\left (c\,x^4+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {5}{4};\ -\frac {1}{4};\ -\frac {a}{c\,x^4}\right )}{5\,x\,{\left (\frac {a}{c\,x^4}+1\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.65, size = 41, normalized size = 0.16 \[ \frac {a^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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