Optimal. Leaf size=69 \[ \frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{2 d}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{2 \sqrt{a} d} \]
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Rubi [A] time = 0.0539107, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1591, 242, 47, 63, 208} \[ \frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{2 d}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{2 \sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 1591
Rule 242
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x \sqrt{a+\frac{b}{c+d x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{a+\frac{b}{x}} \, dx,x,c+d x^2\right )}{2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,\frac{1}{c+d x^2}\right )}{2 d}\\ &=\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{2 d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{c+d x^2}\right )}{4 d}\\ &=\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{c+d x^2}}\right )}{2 d}\\ &=\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{2 d}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{2 \sqrt{a} d}\\ \end{align*}
Mathematica [A] time = 0.0870507, size = 77, normalized size = 1.12 \[ \frac{\sqrt{a} \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}+b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{2 \sqrt{a} d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 180, normalized size = 2.6 \begin{align*}{\frac{d{x}^{2}+c}{4\,d}\sqrt{{\frac{ad{x}^{2}+ac+b}{d{x}^{2}+c}}} \left ( b\ln \left ({\frac{1}{2} \left ( 2\,a{d}^{2}{x}^{2}+2\,acd+2\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{a{d}^{2}}+bd \right ){\frac{1}{\sqrt{a{d}^{2}}}}} \right ) d+2\,\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}\sqrt{a{d}^{2}} \right ){\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( ad{x}^{2}+ac+b \right ) }}}{\frac{1}{\sqrt{a{d}^{2}}}}} \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 [A] time = 1.41242, size = 608, normalized size = 8.81 \begin{align*} \left [\frac{\sqrt{a} b \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \,{\left (2 \, a d^{2} x^{4} +{\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt{a} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \,{\left (a d x^{2} + a c\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, a d}, -\frac{\sqrt{-a} b \arctan \left (\frac{{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt{-a} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{2 \,{\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \,{\left (a d x^{2} + a c\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, a d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\frac{a c + a d x^{2} + b}{c + d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.3464, size = 171, normalized size = 2.48 \begin{align*} -\frac{1}{4} \,{\left (\frac{b \log \left ({\left | -2 \, a^{\frac{3}{2}} c d - 2 \,{\left (\sqrt{a d^{2}} x^{2} - \sqrt{a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a{\left | d \right |} - \sqrt{a} b d \right |}\right )}{\sqrt{a}{\left | d \right |}} - \frac{2 \, \sqrt{a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}}{d}\right )} \mathrm{sgn}\left (d x^{2} + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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