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

Optimal. Leaf size=62 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{b^2}-\frac{a (a+b x) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

Sqrt[a^2 + 2*a*b*x + b^2*x^2]/b^2 - (a*(a + b*x)*Log[a + b*x])/(b^2*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0616258, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{b^2}-\frac{a (a+b x) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

Sqrt[a^2 + 2*a*b*x + b^2*x^2]/b^2 - (a*(a + b*x)*Log[a + b*x])/(b^2*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.1324, size = 60, normalized size = 0.97 \[ - \frac{a \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/((b*x+a)**2)**(1/2),x)

[Out]

-a*(a + b*x)*log(a + b*x)/(b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + sqrt(a**2 +
2*a*b*x + b**2*x**2)/b**2

_______________________________________________________________________________________

Mathematica [A]  time = 0.0165735, size = 33, normalized size = 0.53 \[ \frac{(a+b x) (b x-a \log (a+b x))}{b^2 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((a + b*x)*(b*x - a*Log[a + b*x]))/(b^2*Sqrt[(a + b*x)^2])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 33, normalized size = 0.5 \[ -{\frac{ \left ( bx+a \right ) \left ( a\ln \left ( bx+a \right ) -bx \right ) }{{b}^{2}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/((b*x+a)^2)^(1/2),x)

[Out]

-(b*x+a)*(a*ln(b*x+a)-b*x)/((b*x+a)^2)^(1/2)/b^2

_______________________________________________________________________________________

Maxima [A]  time = 0.699345, size = 57, normalized size = 0.92 \[ -\frac{a \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(b^(-2))*log(x + a/b)/b + sqrt(b^2*x^2 + 2*a*b*x + a^2)/b^2

_______________________________________________________________________________________

Fricas [A]  time = 0.218901, size = 23, normalized size = 0.37 \[ \frac{b x - a \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x - a*log(b*x + a))/b^2

_______________________________________________________________________________________

Sympy [A]  time = 1.11825, size = 14, normalized size = 0.23 \[ - \frac{a \log{\left (a + b x \right )}}{b^{2}} + \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((b*x+a)**2)**(1/2),x)

[Out]

-a*log(a + b*x)/b**2 + x/b

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205998, size = 42, normalized size = 0.68 \[ \frac{x{\rm sign}\left (b x + a\right )}{b} - \frac{a{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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