Optimal. Leaf size=26 \[ -\frac{(a+b x)^2}{2 a x \sqrt{c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0038607, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 37} \[ -\frac{(a+b x)^2}{2 a x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 37
Rubi steps
\begin{align*} \int \frac{a+b x}{x^2 \sqrt{c x^2}} \, dx &=\frac{x \int \frac{a+b x}{x^3} \, dx}{\sqrt{c x^2}}\\ &=-\frac{(a+b x)^2}{2 a x \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0059155, size = 23, normalized size = 0.88 \[ \frac{c x (-a-2 b x)}{2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 19, normalized size = 0.7 \begin{align*} -{\frac{2\,bx+a}{2\,x}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05018, size = 26, normalized size = 1. \begin{align*} -\frac{b}{\sqrt{c} x} - \frac{a}{2 \, \sqrt{c} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69186, size = 51, normalized size = 1.96 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )}}{2 \, c x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.505369, size = 31, normalized size = 1.19 \begin{align*} - \frac{a}{2 \sqrt{c} x \sqrt{x^{2}}} - \frac{b}{\sqrt{c} \sqrt{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]