Optimal. Leaf size=56 \[ \frac{2 \sqrt{x}}{b \sqrt{b x+c x^2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.0216995, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {666, 660, 207} \[ \frac{2 \sqrt{x}}{b \sqrt{b x+c x^2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 666
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac{2 \sqrt{x}}{b \sqrt{b x+c x^2}}+\frac{\int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{b}\\ &=\frac{2 \sqrt{x}}{b \sqrt{b x+c x^2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{b}\\ &=\frac{2 \sqrt{x}}{b \sqrt{b x+c x^2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0088132, size = 37, normalized size = 0.66 \[ \frac{2 \sqrt{x} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{c x}{b}+1\right )}{b \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 51, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{{b}^{3/2}\sqrt{x} \left ( cx+b \right ) } \left ({\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}-\sqrt{b} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0726, size = 359, normalized size = 6.41 \begin{align*} \left [\frac{{\left (c x^{2} + b x\right )} \sqrt{b} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 2 \, \sqrt{c x^{2} + b x} b \sqrt{x}}{b^{2} c x^{2} + b^{3} x}, \frac{2 \,{\left ({\left (c x^{2} + b x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) + \sqrt{c x^{2} + b x} b \sqrt{x}\right )}}{b^{2} c x^{2} + b^{3} x}\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 \frac{\sqrt{x}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23874, size = 90, normalized size = 1.61 \begin{align*} \frac{2 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{2 \,{\left (\sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b}\right )}}{\sqrt{-b} b^{\frac{3}{2}}} + \frac{2}{\sqrt{c x + b} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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