Optimal. Leaf size=28 \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{c (a+b x)^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0123486, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 29} \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{c (a+b x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 247
Rule 15
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c (a+b x)^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{c x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,a+b x\right )}{b \sqrt{c (a+b x)^2}}\\ &=\frac{(a+b x) \log (a+b x)}{b \sqrt{c (a+b x)^2}}\\ \end{align*}
Mathematica [A] time = 0.006314, size = 28, normalized size = 1. \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{c (a+b x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ) \ln \left ( bx+a \right ) }{b}{\frac{1}{\sqrt{c \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.969728, size = 24, normalized size = 0.86 \begin{align*} \sqrt{\frac{1}{b^{2} c}} \log \left (x + \frac{a}{b}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.29266, size = 92, normalized size = 3.29 \begin{align*} \frac{\sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c} \log \left (b x + a\right )}{b^{2} c x + a b c} \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 + b x\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12227, size = 46, normalized size = 1.64 \begin{align*} \frac{\log \left ({\left | b x + a \right |} \sqrt{{\left | c \right |}}{\left | \mathrm{sgn}\left (b x + a\right ) \right |}\right )}{b \sqrt{c} \mathrm{sgn}\left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]