Optimal. Leaf size=62 \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{a \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0166675, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 43} \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{a \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{a b+b^2 x}{x} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (b^2+\frac{a b}{x}\right ) \, dx}{a b+b^2 x}\\ &=\frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{a \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.0095251, size = 27, normalized size = 0.44 \[ \frac{\sqrt{(a+b x)^2} (a \log (x)+b x)}{a+b x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.328, size = 19, normalized size = 0.3 \begin{align*}{\it csgn} \left ( bx+a \right ) \left ( bx+a+a\ln \left ( bx \right ) \right ) \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 = 1.86431, size = 22, normalized size = 0.35 \begin{align*} b x + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.264238, size = 7, normalized size = 0.11 \begin{align*} a \log{\left (x \right )} + b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21377, size = 28, normalized size = 0.45 \begin{align*} b x \mathrm{sgn}\left (b x + a\right ) + a \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]