3.57 \(\int \frac{(a+b x)^2}{x^2} \, dx\)

Optimal. Leaf size=20 \[ -\frac{a^2}{x}+2 a b \log (x)+b^2 x \]

[Out]

-(a^2/x) + b^2*x + 2*a*b*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0220168, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^2}{x}+2 a b \log (x)+b^2 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/x^2,x]

[Out]

-(a^2/x) + b^2*x + 2*a*b*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{x} + 2 a b \log{\left (x \right )} + \int b^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2/x**2,x)

[Out]

-a**2/x + 2*a*b*log(x) + Integral(b**2, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00155256, size = 20, normalized size = 1. \[ -\frac{a^2}{x}+2 a b \log (x)+b^2 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^2/x^2,x]

[Out]

-(a^2/x) + b^2*x + 2*a*b*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 21, normalized size = 1.1 \[ -{\frac{{a}^{2}}{x}}+{b}^{2}x+2\,ab\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/x^2,x)

[Out]

-a^2/x+b^2*x+2*a*b*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.33993, size = 27, normalized size = 1.35 \[ b^{2} x + 2 \, a b \log \left (x\right ) - \frac{a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/x^2,x, algorithm="maxima")

[Out]

b^2*x + 2*a*b*log(x) - a^2/x

_______________________________________________________________________________________

Fricas [A]  time = 0.19319, size = 32, normalized size = 1.6 \[ \frac{b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/x^2,x, algorithm="fricas")

[Out]

(b^2*x^2 + 2*a*b*x*log(x) - a^2)/x

_______________________________________________________________________________________

Sympy [A]  time = 1.06526, size = 17, normalized size = 0.85 \[ - \frac{a^{2}}{x} + 2 a b \log{\left (x \right )} + b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2/x**2,x)

[Out]

-a**2/x + 2*a*b*log(x) + b**2*x

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210489, size = 28, normalized size = 1.4 \[ b^{2} x + 2 \, a b{\rm ln}\left ({\left | x \right |}\right ) - \frac{a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/x^2,x, algorithm="giac")

[Out]

b^2*x + 2*a*b*ln(abs(x)) - a^2/x