Optimal. Leaf size=50 \[ -\frac {1}{b \sqrt {a+b x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {261, 444, 63, 208} \[ -\frac {1}{b \sqrt {a+b x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 261
Rule 444
Rubi steps
\begin {align*} \int \left (\frac {x}{\left (a+b x^2\right )^{3/2}}+\frac {x}{\left (1+x^2\right ) \sqrt {a+b x^2}}\right ) \, dx &=\int \frac {x}{\left (a+b x^2\right )^{3/2}} \, dx+\int \frac {x}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx\\ &=-\frac {1}{b \sqrt {a+b x^2}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=-\frac {1}{b \sqrt {a+b x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=-\frac {1}{b \sqrt {a+b x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 71, normalized size = 1.42 \[ \frac {b \sqrt {a-b} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a-b}}\right )+a-b}{b (b-a) \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 268, normalized size = 5.36 \[ \left [\frac {{\left (b^{2} x^{2} + a b\right )} \sqrt {a - b} \log \left (\frac {b^{2} x^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2}\right )} x^{2} - 4 \, {\left (b x^{2} + 2 \, a - b\right )} \sqrt {b x^{2} + a} \sqrt {a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{x^{4} + 2 \, x^{2} + 1}\right ) - 4 \, \sqrt {b x^{2} + a} {\left (a - b\right )}}{4 \, {\left (a^{2} b - a b^{2} + {\left (a b^{2} - b^{3}\right )} x^{2}\right )}}, -\frac {{\left (b^{2} x^{2} + a b\right )} \sqrt {-a + b} \arctan \left (-\frac {{\left (b x^{2} + 2 \, a - b\right )} \sqrt {b x^{2} + a} \sqrt {-a + b}}{2 \, {\left ({\left (a b - b^{2}\right )} x^{2} + a^{2} - a b\right )}}\right ) + 2 \, \sqrt {b x^{2} + a} {\left (a - b\right )}}{2 \, {\left (a^{2} b - a b^{2} + {\left (a b^{2} - b^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 41, normalized size = 0.82 \[ \frac {\arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a + b}}\right )}{\sqrt {-a + b}} - \frac {1}{\sqrt {b x^{2} + a} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 42, normalized size = 0.84 \[ \frac {\arctan \left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}-\frac {1}{\sqrt {b \,x^{2}+a}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.76, size = 42, normalized size = 0.84 \[ -\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {1}{b\,\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.71, size = 49, normalized size = 0.98 \[ \begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} + \frac {\operatorname {atan}{\left (\frac {\sqrt {a + b x^{2}}}{\sqrt {- a + b}} \right )}}{\sqrt {- a + b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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