Optimal. Leaf size=126 \[ -\frac {2 a^{3/4} b^{3/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a+\frac {b}{x^4}}}-\frac {2 b \sqrt {a+\frac {b}{x^4}}}{3 x}+\frac {1}{3} x^3 \left (a+\frac {b}{x^4}\right )^{3/2} \]
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Rubi [A] time = 0.06, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {335, 277, 195, 220} \[ -\frac {2 a^{3/4} b^{3/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a+\frac {b}{x^4}}}+\frac {1}{3} x^3 \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {2 b \sqrt {a+\frac {b}{x^4}}}{3 x} \]
Antiderivative was successfully verified.
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Rule 195
Rule 220
Rule 277
Rule 335
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^4}\right )^{3/2} x^2 \, dx &=-\operatorname {Subst}\left (\int \frac {\left (a+b x^4\right )^{3/2}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \left (a+\frac {b}{x^4}\right )^{3/2} x^3-(2 b) \operatorname {Subst}\left (\int \sqrt {a+b x^4} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 b \sqrt {a+\frac {b}{x^4}}}{3 x}+\frac {1}{3} \left (a+\frac {b}{x^4}\right )^{3/2} x^3-\frac {1}{3} (4 a b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 b \sqrt {a+\frac {b}{x^4}}}{3 x}+\frac {1}{3} \left (a+\frac {b}{x^4}\right )^{3/2} x^3-\frac {2 a^{3/4} b^{3/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt {a+\frac {b}{x^4}}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.41 \[ -\frac {b \sqrt {a+\frac {b}{x^4}} \, _2F_1\left (-\frac {3}{2},-\frac {3}{4};\frac {1}{4};-\frac {a x^4}{b}\right )}{3 x \sqrt {\frac {a x^4}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x^{4} + b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{x^{2}}, x\right ) \]
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 \[ \int {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 138, normalized size = 1.10 \[ \frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{2} x^{8}+4 \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, a b \,x^{3} \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )-\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{2}\right ) x^{3}}{3 \left (a \,x^{4}+b \right )^{2} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}} \]
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 \[ \int {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\left (a+\frac {b}{x^4}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.61, size = 44, normalized size = 0.35 \[ - \frac {a^{\frac {3}{2}} x^{3} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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