Optimal. Leaf size=47 \[ -\frac{a^2}{\sqrt{c x^2}}+\frac{2 a b x \log (x)}{\sqrt{c x^2}}+\frac{b^2 x^2}{\sqrt{c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0105107, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 43} \[ -\frac{a^2}{\sqrt{c x^2}}+\frac{2 a b x \log (x)}{\sqrt{c x^2}}+\frac{b^2 x^2}{\sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x \sqrt{c x^2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^2} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (b^2+\frac{a^2}{x^2}+\frac{2 a b}{x}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{a^2}{\sqrt{c x^2}}+\frac{b^2 x^2}{\sqrt{c x^2}}+\frac{2 a b x \log (x)}{\sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0093845, size = 34, normalized size = 0.72 \[ \frac{c x^2 \left (-a^2+2 a b x \log (x)+b^2 x^2\right )}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 29, normalized size = 0.6 \begin{align*}{(2\,ab\ln \left ( x \right ) x+{b}^{2}{x}^{2}-{a}^{2}){\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01363, size = 47, normalized size = 1. \begin{align*} \frac{2 \, a b \log \left (x\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2}} b^{2}}{c} - \frac{a^{2}}{\sqrt{c} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5568, size = 73, normalized size = 1.55 \begin{align*} \frac{{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt{c x^{2}}}{c x^{2}} \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{\left (a + b x\right )^{2}}{x \sqrt{c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.0755, size = 88, normalized size = 1.87 \begin{align*} \frac{\sqrt{c x^{2}} b^{2}}{c} - \frac{2 \,{\left (a b \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right ) - \frac{a^{2} \sqrt{c}}{\sqrt{c} x - \sqrt{c x^{2}}}\right )}}{\sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]