Optimal. Leaf size=40 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right )}{\sqrt{b} \sqrt{c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0172882, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1979, 620, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right )}{\sqrt{b} \sqrt{c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1979
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c \left (a x+b x^2\right )}} \, dx &=\int \frac{1}{\sqrt{a c x+b c x^2}} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{1}{1-b c x^2} \, dx,x,\frac{x}{\sqrt{a c x+b c x^2}}\right )\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right )}{\sqrt{b} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0040466, size = 58, normalized size = 1.45 \[ \frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{c x (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 37, normalized size = 0.9 \begin{align*}{\ln \left ({ \left ({\frac{ac}{2}}+bcx \right ){\frac{1}{\sqrt{bc}}}}+\sqrt{bc{x}^{2}+acx} \right ){\frac{1}{\sqrt{bc}}}} \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 = 1.46868, size = 197, normalized size = 4.92 \begin{align*} \left [\frac{\sqrt{b c} \log \left (2 \, b c x + a c + 2 \, \sqrt{b c x^{2} + a c x} \sqrt{b c}\right )}{b c}, -\frac{2 \, \sqrt{-b c} \arctan \left (\frac{\sqrt{b c x^{2} + a c x} \sqrt{-b c}}{b c x}\right )}{b 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{1}{\sqrt{c \left (a x + b x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21455, size = 68, normalized size = 1.7 \begin{align*} -\frac{\sqrt{b c} \log \left ({\left | -2 \,{\left (\sqrt{b c} x - \sqrt{b c x^{2} + a c x}\right )} b - \sqrt{b c} a \right |}\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]