Optimal. Leaf size=52 \[ \frac{a^2 x \log (x)}{\sqrt{c x^2}}+\frac{2 a b x^2}{\sqrt{c x^2}}+\frac{b^2 x^3}{2 \sqrt{c x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0264642, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a^2 x \log (x)}{\sqrt{c x^2}}+\frac{2 a b x^2}{\sqrt{c x^2}}+\frac{b^2 x^3}{2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2/Sqrt[c*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \sqrt{c x^{2}} \log{\left (x \right )}}{c x} + \frac{2 a b \sqrt{c x^{2}}}{c} + \frac{b^{2} \sqrt{c x^{2}} \int x\, dx}{c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2/(c*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0109473, size = 32, normalized size = 0.62 \[ \frac{x \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2/Sqrt[c*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 31, normalized size = 0.6 \[{\frac{x \left ({b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( x \right ) +4\,abx \right ) }{2}{\frac{1}{\sqrt{c{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2/(c*x^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.34248, size = 47, normalized size = 0.9 \[ \frac{b^{2} x^{2}}{2 \, \sqrt{c}} + \frac{a^{2} \log \left (x\right )}{\sqrt{c}} + \frac{2 \, \sqrt{c x^{2}} a b}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/sqrt(c*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.211219, size = 47, normalized size = 0.9 \[ \frac{{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right )\right )} \sqrt{c x^{2}}}{2 \, c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/sqrt(c*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{2}}{\sqrt{c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2/(c*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.216858, size = 68, normalized size = 1.31 \[ -\frac{a^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} + \frac{1}{2} \, \sqrt{c x^{2}}{\left (\frac{b^{2} x}{c} + \frac{4 \, a b}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/sqrt(c*x^2),x, algorithm="giac")
[Out]