Optimal. Leaf size=86 \[ -\frac {15}{8} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )+x \left (a+\frac {b}{x^2}\right )^{5/2}-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}-\frac {15 a b \sqrt {a+\frac {b}{x^2}}}{8 x} \]
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Rubi [A] time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {242, 277, 195, 217, 206} \[ -\frac {15}{8} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )+x \left (a+\frac {b}{x^2}\right )^{5/2}-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}-\frac {15 a b \sqrt {a+\frac {b}{x^2}}}{8 x} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 242
Rule 277
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^{5/2} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{5/2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\left (a+\frac {b}{x^2}\right )^{5/2} x-(5 b) \operatorname {Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}+\left (a+\frac {b}{x^2}\right )^{5/2} x-\frac {1}{4} (15 a b) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {15 a b \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}+\left (a+\frac {b}{x^2}\right )^{5/2} x-\frac {1}{8} \left (15 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {15 a b \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}+\left (a+\frac {b}{x^2}\right )^{5/2} x-\frac {1}{8} \left (15 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^2}} x}\right )\\ &=-\frac {15 a b \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {5 b \left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}+\left (a+\frac {b}{x^2}\right )^{5/2} x-\frac {15}{8} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 49, normalized size = 0.57 \[ -\frac {a^2 x^5 \left (a+\frac {b}{x^2}\right )^{5/2} \left (a x^2+b\right ) \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};\frac {a x^2}{b}+1\right )}{7 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 172, normalized size = 2.00 \[ \left [\frac {15 \, a^{2} \sqrt {b} x^{3} \log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \, {\left (8 \, a^{2} x^{4} - 9 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{16 \, x^{3}}, \frac {15 \, a^{2} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (8 \, a^{2} x^{4} - 9 \, a b x^{2} - 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{8 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 96, normalized size = 1.12 \[ \frac {\frac {15 \, a^{3} b \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b}} + 8 \, \sqrt {a x^{2} + b} a^{3} \mathrm {sgn}\relax (x) - \frac {9 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{3} b \mathrm {sgn}\relax (x) - 7 \, \sqrt {a x^{2} + b} a^{3} b^{2} \mathrm {sgn}\relax (x)}{a^{2} x^{4}}}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 144, normalized size = 1.67 \[ -\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} \left (15 a^{2} b^{\frac {5}{2}} x^{4} \ln \left (\frac {2 b +2 \sqrt {a \,x^{2}+b}\, \sqrt {b}}{x}\right )-15 \sqrt {a \,x^{2}+b}\, a^{2} b^{2} x^{4}-5 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{2} b \,x^{4}-3 \left (a \,x^{2}+b \right )^{\frac {5}{2}} a^{2} x^{4}+3 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a \,x^{2}+2 \left (a \,x^{2}+b \right )^{\frac {7}{2}} b \right ) x}{8 \left (a \,x^{2}+b \right )^{\frac {5}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.90, size = 130, normalized size = 1.51 \[ \sqrt {a + \frac {b}{x^{2}}} a^{2} x + \frac {15}{16} \, a^{2} \sqrt {b} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right ) - \frac {9 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{2} b x^{3} - 7 \, \sqrt {a + \frac {b}{x^{2}}} a^{2} b^{2} x}{8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{2} x^{4} - 2 \, {\left (a + \frac {b}{x^{2}}\right )} b x^{2} + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 36, normalized size = 0.42 \[ \frac {x\,{\left (a\,x^2+b\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b}{a\,x^2}\right )}{{\left (\frac {b}{a}+x^2\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.44, size = 117, normalized size = 1.36 \[ \frac {a^{\frac {5}{2}} x}{\sqrt {1 + \frac {b}{a x^{2}}}} - \frac {a^{\frac {3}{2}} b}{8 x \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {11 \sqrt {a} b^{2}}{8 x^{3} \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {15 a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{8} - \frac {b^{3}}{4 \sqrt {a} x^{5} \sqrt {1 + \frac {b}{a x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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