Optimal. Leaf size=71 \[ -\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{8 \sqrt {b}}-\frac {3 a \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{4 x} \]
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Rubi [A] time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {335, 195, 217, 206} \[ -\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{8 \sqrt {b}}-\frac {3 a \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{4 x} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 335
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^2} \, dx &=-\operatorname {Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}-\frac {1}{4} (3 a) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 a \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}-\frac {1}{8} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 a \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}-\frac {1}{8} \left (3 a^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 {3 a \sqrt {a+\frac {b}{x^2}}}{8 x}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{4 x}-\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{8 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 85, normalized size = 1.20 \[ -\frac {\sqrt {a+\frac {b}{x^2}} \left (3 a^2 x^4 \sqrt {\frac {a x^2}{b}+1} \tanh ^{-1}\left (\sqrt {\frac {a x^2}{b}+1}\right )+5 a^2 x^4+7 a b x^2+2 b^2\right )}{8 x^3 \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 163, normalized size = 2.30 \[ \left [\frac {3 \, a^{2} \sqrt {b} x^{3} \log \left (-\frac {a x^{2} - 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \, {\left (5 \, a b x^{2} + 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{16 \, b x^{3}}, \frac {3 \, a^{2} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) - {\left (5 \, a b x^{2} + 2 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{8 \, b x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 76, normalized size = 1.07 \[ \frac {\frac {3 \, a^{3} \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b}} - \frac {5 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{3} \mathrm {sgn}\relax (x) - 3 \, \sqrt {a x^{2} + b} a^{3} b \mathrm {sgn}\relax (x)}{a^{2} x^{4}}}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 125, normalized size = 1.76 \[ -\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} \left (3 a^{2} b^{\frac {3}{2}} x^{4} \ln \left (\frac {2 b +2 \sqrt {a \,x^{2}+b}\, \sqrt {b}}{x}\right )-3 \sqrt {a \,x^{2}+b}\, a^{2} b \,x^{4}-\left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{2} x^{4}+\left (a \,x^{2}+b \right )^{\frac {5}{2}} a \,x^{2}+2 \left (a \,x^{2}+b \right )^{\frac {5}{2}} b \right )}{8 \left (a \,x^{2}+b \right )^{\frac {3}{2}} b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.94, size = 113, normalized size = 1.59 \[ \frac {3 \, a^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{16 \, \sqrt {b}} - \frac {5 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{2} x^{3} - 3 \, \sqrt {a + \frac {b}{x^{2}}} a^{2} b x}{8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{2} x^{4} - 2 \, {\left (a + \frac {b}{x^{2}}\right )} b x^{2} + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 39, normalized size = 0.55 \[ -\frac {{\left (a\,x^2+b\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {b}{a\,x^2}\right )}{x\,{\left (\frac {b}{a}+x^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.65, size = 71, normalized size = 1.00 \[ - \frac {5 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{2}}}}{8 x} - \frac {\sqrt {a} b \sqrt {1 + \frac {b}{a x^{2}}}}{4 x^{3}} - \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{8 \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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