Optimal. Leaf size=42 \[ \sqrt{b x+c x^2}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.019189, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {664, 620, 206} \[ \sqrt{b x+c x^2}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 664
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{b x+c x^2}}{x} \, dx &=\sqrt{b x+c x^2}+\frac{1}{2} b \int \frac{1}{\sqrt{b x+c x^2}} \, dx\\ &=\sqrt{b x+c x^2}+b \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )\\ &=\sqrt{b x+c x^2}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0988608, size = 66, normalized size = 1.57 \[ \sqrt{x (b+c x)} \left (\frac{b^{3/2} \sqrt{\frac{c x}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{c} \sqrt{x} (b+c x)}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 43, normalized size = 1. \begin{align*} \sqrt{c{x}^{2}+bx}+{\frac{b}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.89567, size = 232, normalized size = 5.52 \begin{align*} \left [\frac{b \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \, \sqrt{c x^{2} + b x} c}{2 \, c}, -\frac{b \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) - \sqrt{c x^{2} + b x} c}{c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x \left (b + c x\right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.46597, size = 65, normalized size = 1.55 \begin{align*} -\frac{b \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, \sqrt{c}} + \sqrt{c x^{2} + b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]