Optimal. Leaf size=43 \[ \frac{a x}{b^2 \sqrt{c x^2} (a+b x)}+\frac{x \log (a+b x)}{b^2 \sqrt{c x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0383983, antiderivative size = 43, 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 \[ \frac{a x}{b^2 \sqrt{c x^2} (a+b x)}+\frac{x \log (a+b x)}{b^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(Sqrt[c*x^2]*(a + b*x)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.1177, size = 42, normalized size = 0.98 \[ \frac{a \sqrt{c x^{2}}}{b^{2} c x \left (a + b x\right )} + \frac{\sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{2} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x+a)**2/(c*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0163927, size = 35, normalized size = 0.81 \[ \frac{x ((a+b x) \log (a+b x)+a)}{b^2 \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(Sqrt[c*x^2]*(a + b*x)^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.004, size = 39, normalized size = 0.9 \[{\frac{x \left ( b\ln \left ( bx+a \right ) x+a\ln \left ( bx+a \right ) +a \right ) }{ \left ( bx+a \right ){b}^{2}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x+a)^2/(c*x^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(c*x^2)*(b*x + a)^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.210533, size = 54, normalized size = 1.26 \[ \frac{\sqrt{c x^{2}}{\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )}}{b^{3} c x^{2} + a b^{2} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(c*x^2)*(b*x + a)^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{c x^{2}} \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x+a)**2/(c*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{c x^{2}}{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(c*x^2)*(b*x + a)^2),x, algorithm="giac")
[Out]