Optimal. Leaf size=80 \[ \frac {5}{2} a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )+\frac {1}{2} x^2 \left (a+\frac {b}{x^2}\right )^{5/2}-\frac {5}{6} b \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {5}{2} a b \sqrt {a+\frac {b}{x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {266, 47, 50, 63, 208} \[ \frac {5}{2} a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )+\frac {1}{2} x^2 \left (a+\frac {b}{x^2}\right )^{5/2}-\frac {5}{6} b \left (a+\frac {b}{x^2}\right )^{3/2}-\frac {5}{2} a b \sqrt {a+\frac {b}{x^2}} \]
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^2}\right )^{5/2} x \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{2} \left (a+\frac {b}{x^2}\right )^{5/2} x^2-\frac {1}{4} (5 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {5}{6} b \left (a+\frac {b}{x^2}\right )^{3/2}+\frac {1}{2} \left (a+\frac {b}{x^2}\right )^{5/2} x^2-\frac {1}{4} (5 a b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {5}{2} a b \sqrt {a+\frac {b}{x^2}}-\frac {5}{6} b \left (a+\frac {b}{x^2}\right )^{3/2}+\frac {1}{2} \left (a+\frac {b}{x^2}\right )^{5/2} x^2-\frac {1}{4} \left (5 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {5}{2} a b \sqrt {a+\frac {b}{x^2}}-\frac {5}{6} b \left (a+\frac {b}{x^2}\right )^{3/2}+\frac {1}{2} \left (a+\frac {b}{x^2}\right )^{5/2} x^2-\frac {1}{2} \left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )\\ &=-\frac {5}{2} a b \sqrt {a+\frac {b}{x^2}}-\frac {5}{6} b \left (a+\frac {b}{x^2}\right )^{3/2}+\frac {1}{2} \left (a+\frac {b}{x^2}\right )^{5/2} x^2+\frac {5}{2} a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 54, normalized size = 0.68 \[ -\frac {b^2 \sqrt {a+\frac {b}{x^2}} \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};-\frac {a x^2}{b}\right )}{3 x^2 \sqrt {\frac {a x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 170, normalized size = 2.12 \[ \left [\frac {15 \, a^{\frac {3}{2}} b x^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + 2 \, {\left (3 \, a^{2} x^{4} - 14 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{12 \, x^{2}}, -\frac {15 \, \sqrt {-a} a b x^{2} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) - {\left (3 \, a^{2} x^{4} - 14 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{6 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.50, size = 142, normalized size = 1.78 \[ \frac {1}{2} \, \sqrt {a x^{2} + b} a^{2} x \mathrm {sgn}\relax (x) - \frac {5}{4} \, a^{\frac {3}{2}} b \log \left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\relax (x) + \frac {2 \, {\left (9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{4} a^{\frac {3}{2}} b^{2} \mathrm {sgn}\relax (x) - 12 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} a^{\frac {3}{2}} b^{3} \mathrm {sgn}\relax (x) + 7 \, a^{\frac {3}{2}} b^{4} \mathrm {sgn}\relax (x)\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 149, normalized size = 1.86 \[ -\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} \left (-15 a^{2} b^{3} x^{3} \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )-15 \sqrt {a \,x^{2}+b}\, a^{\frac {5}{2}} b^{2} x^{4}-10 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{\frac {5}{2}} b \,x^{4}-8 \left (a \,x^{2}+b \right )^{\frac {5}{2}} a^{\frac {5}{2}} x^{4}+8 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a^{\frac {3}{2}} x^{2}+2 \left (a \,x^{2}+b \right )^{\frac {7}{2}} \sqrt {a}\, b \right ) x^{2}}{6 \left (a \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {a}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.87, size = 81, normalized size = 1.01 \[ \frac {1}{2} \, \sqrt {a + \frac {b}{x^{2}}} a^{2} x^{2} - \frac {5}{4} \, a^{\frac {3}{2}} b \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right ) - \frac {1}{3} \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b - 2 \, \sqrt {a + \frac {b}{x^{2}}} a b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.75, size = 66, normalized size = 0.82 \[ \frac {a^2\,x^2\,\sqrt {a+\frac {b}{x^2}}}{2}-\frac {b\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}{3}-2\,a\,b\,\sqrt {a+\frac {b}{x^2}}-\frac {a^{3/2}\,b\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x^2}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.00, size = 112, normalized size = 1.40 \[ \frac {a^{\frac {5}{2}} x^{2} \sqrt {1 + \frac {b}{a x^{2}}}}{2} - \frac {7 a^{\frac {3}{2}} b \sqrt {1 + \frac {b}{a x^{2}}}}{3} - \frac {5 a^{\frac {3}{2}} b \log {\left (\frac {b}{a x^{2}} \right )}}{4} + \frac {5 a^{\frac {3}{2}} b \log {\left (\sqrt {1 + \frac {b}{a x^{2}}} + 1 \right )}}{2} - \frac {\sqrt {a} b^{2} \sqrt {1 + \frac {b}{a x^{2}}}}{3 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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