3.1579 \(\int \frac{d+e x}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=69 \[ \frac{(a+b x) (b d-a e) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.079359, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{(a+b x) (b d-a e) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^2 + ((b*d - a*e)*(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 = 11.0295, size = 66, normalized size = 0.96 \[ \frac{e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{2}} - \frac{\left (a + b x\right ) \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0311113, size = 40, normalized size = 0.58 \[ \frac{(a+b x) ((b d-a e) \log (a+b x)+b e x)}{b^2 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 45, normalized size = 0.7 \[ -{\frac{ \left ( bx+a \right ) \left ( \ln \left ( bx+a \right ) ae-\ln \left ( bx+a \right ) bd-bex \right ) }{{b}^{2}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/((b*x+a)^2)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.688302, size = 80, normalized size = 1.16 \[ \sqrt{\frac{1}{b^{2}}} d \log \left (x + \frac{a}{b}\right ) - \frac{a \sqrt{\frac{1}{b^{2}}} e \log \left (x + \frac{a}{b}\right )}{b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.204243, size = 32, normalized size = 0.46 \[ \frac{b e x +{\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.27463, size = 20, normalized size = 0.29 \[ \frac{e x}{b} - \frac{\left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*x/b - (a*e - b*d)*log(a + b*x)/b**2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212786, size = 62, normalized size = 0.9 \[ \frac{x e{\rm sign}\left (b x + a\right )}{b} + \frac{{\left (b d{\rm sign}\left (b x + a\right ) - a e{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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