Optimal. Leaf size=56 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}-\frac{\sqrt{b x+c x^2}}{b x^{3/2}} \]
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Rubi [A] time = 0.0220336, 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 = {672, 660, 207} \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}-\frac{\sqrt{b x+c x^2}}{b x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 672
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx &=-\frac{\sqrt{b x+c x^2}}{b x^{3/2}}-\frac{c \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{2 b}\\ &=-\frac{\sqrt{b x+c x^2}}{b x^{3/2}}-\frac{c \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{b}\\ &=-\frac{\sqrt{b x+c x^2}}{b x^{3/2}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0583343, size = 63, normalized size = 1.12 \[ \frac{2 c \sqrt{x (b+c x)} \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{c x}{b}+1}\right )}{2 \sqrt{\frac{c x}{b}+1}}-\frac{b}{2 c x}\right )}{b^2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.196, size = 52, normalized size = 0.9 \begin{align*}{\sqrt{x \left ( cx+b \right ) } \left ({\it Artanh} \left ({\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ) xc-\sqrt{cx+b}\sqrt{b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}} \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{1}{\sqrt{c x^{2} + b x} x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93033, size = 315, normalized size = 5.62 \begin{align*} \left [\frac{\sqrt{b} c x^{2} \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}}{2 \, b^{2} x^{2}}, -\frac{\sqrt{-b} c x^{2} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) + \sqrt{c x^{2} + b x} b \sqrt{x}}{b^{2} x^{2}}\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{1}{x^{\frac{3}{2}} \sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25034, size = 59, normalized size = 1.05 \begin{align*} -c{\left (\frac{\arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{\sqrt{c x + b}}{b c x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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