Optimal. Leaf size=54 \[ -\frac{\sqrt{b x+c x^2}}{x^{3/2}}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0207981, antiderivative size = 54, 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 = {662, 660, 207} \[ -\frac{\sqrt{b x+c x^2}}{x^{3/2}}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 662
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{\sqrt{b x+c x^2}}{x^{5/2}} \, dx &=-\frac{\sqrt{b x+c x^2}}{x^{3/2}}+\frac{1}{2} c \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx\\ &=-\frac{\sqrt{b x+c x^2}}{x^{3/2}}+c \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )\\ &=-\frac{\sqrt{b x+c x^2}}{x^{3/2}}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0408061, size = 51, normalized size = 0.94 \[ -\frac{c x \sqrt{\frac{c x}{b}+1} \tanh ^{-1}\left (\sqrt{\frac{c x}{b}+1}\right )+b+c x}{\sqrt{x} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.183, size = 53, normalized size = 1. \begin{align*}{ \left ( -{\it Artanh} \left ({\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ) xc-\sqrt{cx+b}\sqrt{b} \right ) \sqrt{x \left ( cx+b \right ) }{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{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{\sqrt{c x^{2} + b x}}{x^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02776, size = 308, normalized size = 5.7 \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 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 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{\sqrt{x \left (b + c x\right )}}{x^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21015, size = 51, normalized size = 0.94 \begin{align*} c{\left (\frac{\arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{c x + b}}{c x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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