Optimal. Leaf size=68 \[ -\frac{1}{b \sqrt{a+b x^2}}-\frac{1}{3 b \left (a+b x^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]
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Rubi [A] time = 0.0417059, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.085, Rules used = {261, 444, 63, 208} \[ -\frac{1}{b \sqrt{a+b x^2}}-\frac{1}{3 b \left (a+b x^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]
Antiderivative was successfully verified.
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Rule 261
Rule 444
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (\frac{x}{\left (a+b x^2\right )^{5/2}}+\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 )^{5/2}} \, 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}{3 b \left (a+b x^2\right )^{3/2}}-\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}{3 b \left (a+b x^2\right )^{3/2}}-\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}{3 b \left (a+b x^2\right )^{3/2}}-\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*}
Mathematica [A] time = 0.190112, size = 63, normalized size = 0.93 \[ \frac{-3 a-3 b x^2-1}{3 b \left (a+b x^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 56, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\arctan \left ({\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{b-a}}}} \right ){\frac{1}{\sqrt{b-a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.64651, size = 799, normalized size = 11.75 \begin{align*} \left [\frac{3 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} 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 \,{\left (3 \,{\left (a b - b^{2}\right )} x^{2} + 3 \, a^{2} -{\left (3 \, a + 1\right )} b + a\right )} \sqrt{b x^{2} + a}}{12 \,{\left ({\left (a b^{3} - b^{4}\right )} x^{4} + a^{3} b - a^{2} b^{2} + 2 \,{\left (a^{2} b^{2} - a b^{3}\right )} x^{2}\right )}}, -\frac{3 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} 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 \,{\left (3 \,{\left (a b - b^{2}\right )} x^{2} + 3 \, a^{2} -{\left (3 \, a + 1\right )} b + a\right )} \sqrt{b x^{2} + a}}{6 \,{\left ({\left (a b^{3} - b^{4}\right )} x^{4} + a^{3} b - a^{2} b^{2} + 2 \,{\left (a^{2} b^{2} - a b^{3}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.15144, size = 97, normalized size = 1.43 \begin{align*} \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} + \begin{cases} - \frac{1}{3 a b \sqrt{a + b x^{2}} + 3 b^{2} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases} + \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{- a + b}} \right )}}{\sqrt{- a + b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16293, size = 74, normalized size = 1.09 \begin{align*} \frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a + b}}\right )}{\sqrt{-a + b}} - \frac{1}{\sqrt{b x^{2} + a} b} - \frac{1}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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