Optimal. Leaf size=63 \[ \frac {1}{4} x^4 \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {3}{4} b \sqrt {a+\frac {b}{x^4}}+\frac {3}{4} \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 63, 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 = {266, 47, 50, 63, 208} \[ \frac {1}{4} x^4 \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {3}{4} b \sqrt {a+\frac {b}{x^4}}+\frac {3}{4} \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^4}\right )^{3/2} x^3 \, dx &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^4}\right )\right )\\ &=\frac {1}{4} \left (a+\frac {b}{x^4}\right )^{3/2} x^4-\frac {1}{8} (3 b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x^4}\right )\\ &=-\frac {3}{4} b \sqrt {a+\frac {b}{x^4}}+\frac {1}{4} \left (a+\frac {b}{x^4}\right )^{3/2} x^4-\frac {1}{8} (3 a b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right )\\ &=-\frac {3}{4} b \sqrt {a+\frac {b}{x^4}}+\frac {1}{4} \left (a+\frac {b}{x^4}\right )^{3/2} x^4-\frac {1}{4} (3 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^4}}\right )\\ &=-\frac {3}{4} b \sqrt {a+\frac {b}{x^4}}+\frac {1}{4} \left (a+\frac {b}{x^4}\right )^{3/2} x^4+\frac {3}{4} \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 49, normalized size = 0.78 \[ -\frac {b \sqrt {a+\frac {b}{x^4}} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {a x^4}{b}\right )}{2 \sqrt {\frac {a x^4}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 129, normalized size = 2.05 \[ \left [\frac {3}{8} \, \sqrt {a} b \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) + \frac {1}{4} \, {\left (a x^{4} - 2 \, b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}, -\frac {3}{4} \, \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) + \frac {1}{4} \, {\left (a x^{4} - 2 \, b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 78, normalized size = 1.24 \[ \frac {1}{4} \, \sqrt {a x^{4} + b} a x^{2} - \frac {3}{8} \, \sqrt {a} b \log \left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2}\right ) + \frac {\sqrt {a} b^{2}}{{\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 82, normalized size = 1.30 \[ \frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} \left (\sqrt {a \,x^{4}+b}\, a \,x^{4}+3 \sqrt {a}\, b \,x^{2} \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )-2 \sqrt {a \,x^{4}+b}\, b \right ) x^{4}}{4 \left (a \,x^{4}+b \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.02, size = 66, normalized size = 1.05 \[ \frac {1}{4} \, \sqrt {a + \frac {b}{x^{4}}} a x^{4} - \frac {3}{8} \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right ) - \frac {1}{2} \, \sqrt {a + \frac {b}{x^{4}}} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.74, size = 48, normalized size = 0.76 \[ \frac {a\,x^4\,\sqrt {a+\frac {b}{x^4}}}{4}-\frac {b\,\sqrt {a+\frac {b}{x^4}}}{2}+\frac {3\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.28, size = 95, normalized size = 1.51 \[ \frac {3 \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{4} + \frac {a^{2} x^{6}}{4 \sqrt {b} \sqrt {\frac {a x^{4}}{b} + 1}} - \frac {a \sqrt {b} x^{2}}{4 \sqrt {\frac {a x^{4}}{b} + 1}} - \frac {b^{\frac {3}{2}}}{2 x^{2} \sqrt {\frac {a x^{4}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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