Optimal. Leaf size=88 \[ -\frac {5}{2} a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )-\frac {5 b^2 \sqrt {a+\frac {b}{x^2}}}{2 x}+\frac {5}{3} b x \left (a+\frac {b}{x^2}\right )^{3/2}+\frac {1}{3} x^3 \left (a+\frac {b}{x^2}\right )^{5/2} \]
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Rubi [A] time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {335, 277, 195, 217, 206} \[ -\frac {5 b^2 \sqrt {a+\frac {b}{x^2}}}{2 x}-\frac {5}{2} a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )+\frac {1}{3} x^3 \left (a+\frac {b}{x^2}\right )^{5/2}+\frac {5}{3} b x \left (a+\frac {b}{x^2}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 277
Rule 335
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^{5/2} x^2 \, dx &=-\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{5/2}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \left (a+\frac {b}{x^2}\right )^{5/2} x^3-\frac {1}{3} (5 b) \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{3} b \left (a+\frac {b}{x^2}\right )^{3/2} x+\frac {1}{3} \left (a+\frac {b}{x^2}\right )^{5/2} x^3-\left (5 b^2\right ) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5 b^2 \sqrt {a+\frac {b}{x^2}}}{2 x}+\frac {5}{3} b \left (a+\frac {b}{x^2}\right )^{3/2} x+\frac {1}{3} \left (a+\frac {b}{x^2}\right )^{5/2} x^3-\frac {1}{2} \left (5 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5 b^2 \sqrt {a+\frac {b}{x^2}}}{2 x}+\frac {5}{3} b \left (a+\frac {b}{x^2}\right )^{3/2} x+\frac {1}{3} \left (a+\frac {b}{x^2}\right )^{5/2} x^3-\frac {1}{2} \left (5 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^2}} x}\right )\\ &=-\frac {5 b^2 \sqrt {a+\frac {b}{x^2}}}{2 x}+\frac {5}{3} b \left (a+\frac {b}{x^2}\right )^{3/2} x+\frac {1}{3} \left (a+\frac {b}{x^2}\right )^{5/2} x^3-\frac {5}{2} a b^{3/2} \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 = 47, normalized size = 0.53 \[ \frac {a x^5 \left (a+\frac {b}{x^2}\right )^{5/2} \left (a x^2+b\right ) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {a x^2}{b}+1\right )}{7 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 165, normalized size = 1.88 \[ \left [\frac {15 \, a b^{\frac {3}{2}} x \log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \, {\left (2 \, a^{2} x^{4} + 14 \, a b x^{2} - 3 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{12 \, x}, \frac {15 \, a \sqrt {-b} b x \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + {\left (2 \, a^{2} x^{4} + 14 \, a b x^{2} - 3 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{6 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 90, normalized size = 1.02 \[ \frac {\frac {15 \, a^{2} b^{2} \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b}} + 2 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{2} \mathrm {sgn}\relax (x) + 12 \, \sqrt {a x^{2} + b} a^{2} b \mathrm {sgn}\relax (x) - \frac {3 \, \sqrt {a x^{2} + b} a b^{2} \mathrm {sgn}\relax (x)}{x^{2}}}{6 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 122, normalized size = 1.39 \[ -\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} \left (15 a \,b^{\frac {5}{2}} x^{2} \ln \left (\frac {2 b +2 \sqrt {a \,x^{2}+b}\, \sqrt {b}}{x}\right )-15 \sqrt {a \,x^{2}+b}\, a \,b^{2} x^{2}-5 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a b \,x^{2}-3 \left (a \,x^{2}+b \right )^{\frac {5}{2}} a \,x^{2}+3 \left (a \,x^{2}+b \right )^{\frac {7}{2}}\right ) x^{3}}{6 \left (a \,x^{2}+b \right )^{\frac {5}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.80, size = 105, normalized size = 1.19 \[ \frac {1}{3} \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a x^{3} + 2 \, \sqrt {a + \frac {b}{x^{2}}} a b x - \frac {\sqrt {a + \frac {b}{x^{2}}} a b^{2} x}{2 \, {\left ({\left (a + \frac {b}{x^{2}}\right )} x^{2} - b\right )}} + \frac {5}{4} \, a b^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right ) \]
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 \[ \int x^2\,{\left (a+\frac {b}{x^2}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.00, size = 112, normalized size = 1.27 \[ \frac {a^{2} \sqrt {b} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3} + \frac {7 a b^{\frac {3}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3} + \frac {5 a b^{\frac {3}{2}} \log {\left (\frac {a x^{2}}{b} \right )}}{4} - \frac {5 a b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a x^{2}}{b} + 1} + 1 \right )}}{2} - \frac {b^{\frac {5}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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