Optimal. Leaf size=41 \[ \frac {1}{a \sqrt {a+b x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 41, 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 = {266, 51, 63, 208} \[ \frac {1}{a \sqrt {a+b x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {1}{a \sqrt {a+b x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {1}{a \sqrt {a+b x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{a b}\\ &=\frac {1}{a \sqrt {a+b x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 33, normalized size = 0.80 \[ \frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x^2}{a}+1\right )}{a \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 126, normalized size = 3.07 \[ \left [\frac {{\left (b x^{2} + a\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, \sqrt {b x^{2} + a} a}{2 \, {\left (a^{2} b x^{2} + a^{3}\right )}}, \frac {{\left (b x^{2} + a\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + \sqrt {b x^{2} + a} a}{a^{2} b x^{2} + a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 39, normalized size = 0.95 \[ \frac {\arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {1}{\sqrt {b x^{2} + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 43, normalized size = 1.05 \[ -\frac {\ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{a^{\frac {3}{2}}}+\frac {1}{\sqrt {b \,x^{2}+a}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 31, normalized size = 0.76 \[ -\frac {\operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {3}{2}}} + \frac {1}{\sqrt {b x^{2} + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.78, size = 33, normalized size = 0.80 \[ \frac {1}{a\,\sqrt {b\,x^2+a}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.73, size = 184, normalized size = 4.49 \[ \frac {2 a^{3} \sqrt {1 + \frac {b x^{2}}{a}}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{3} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{2} b x^{2} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{2} b x^{2} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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