Optimal. Leaf size=63 \[ -\frac {\left (a+b x^2\right )^{3/2}}{x}+\frac {3}{2} b x \sqrt {a+b x^2}+\frac {3}{2} a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {277, 195, 217, 206} \[ -\frac {\left (a+b x^2\right )^{3/2}}{x}+\frac {3}{2} b x \sqrt {a+b x^2}+\frac {3}{2} a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 277
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{x^2} \, dx &=-\frac {\left (a+b x^2\right )^{3/2}}{x}+(3 b) \int \sqrt {a+b x^2} \, dx\\ &=\frac {3}{2} b x \sqrt {a+b x^2}-\frac {\left (a+b x^2\right )^{3/2}}{x}+\frac {1}{2} (3 a b) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {3}{2} b x \sqrt {a+b x^2}-\frac {\left (a+b x^2\right )^{3/2}}{x}+\frac {1}{2} (3 a b) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {3}{2} b x \sqrt {a+b x^2}-\frac {\left (a+b x^2\right )^{3/2}}{x}+\frac {3}{2} a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.79 \[ -\frac {a \sqrt {a+b x^2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {b x^2}{a}\right )}{x \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 112, normalized size = 1.78 \[ \left [\frac {3 \, a \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, \sqrt {b x^{2} + a} {\left (b x^{2} - 2 \, a\right )}}{4 \, x}, -\frac {3 \, a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - \sqrt {b x^{2} + a} {\left (b x^{2} - 2 \, a\right )}}{2 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 73, normalized size = 1.16 \[ \frac {1}{2} \, \sqrt {b x^{2} + a} b x - \frac {3}{4} \, a \sqrt {b} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, a^{2} \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 69, normalized size = 1.10 \[ \frac {3 a \sqrt {b}\, \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2}+\frac {3 \sqrt {b \,x^{2}+a}\, b x}{2}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} b x}{a}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 43, normalized size = 0.68 \[ \frac {3}{2} \, \sqrt {b x^{2} + a} b x + \frac {3}{2} \, a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.15, size = 40, normalized size = 0.63 \[ -\frac {{\left (b\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b\,x^2}{a}\right )}{x\,{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.37, size = 88, normalized size = 1.40 \[ - \frac {a^{\frac {3}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} b x}{2 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2} + \frac {b^{2} x^{3}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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