3.1959 \(\int (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b} \]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(3*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.0142613, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(3*b)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.48482, size = 12, normalized size = 0.44 \[ \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((a + b*x)**2)**(3/2)/(3*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0146536, size = 18, normalized size = 0.67 \[ \frac{\left ((a+b x)^2\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^2)^(3/2)/(3*b)

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 38, normalized size = 1.4 \[{\frac{x \left ({b}^{2}{x}^{2}+3\,abx+3\,{a}^{2} \right ) }{3\,bx+3\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x*(b^2*x^2+3*a*b*x+3*a^2)*((b*x+a)^2)^(1/2)/(b*x+a)

_______________________________________________________________________________________

Maxima [A]  time = 0.708212, size = 19, normalized size = 0.7 \[ \frac{{\left ({\left (b x + a\right )}^{2}\right )}^{\frac{3}{2}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((b*x + a)^2)^(3/2)/b

_______________________________________________________________________________________

Fricas [A]  time = 0.283892, size = 27, normalized size = 1. \[ \frac{1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b^2*x^3 + a*b*x^2 + a^2*x

_______________________________________________________________________________________

Sympy [A]  time = 0.155478, size = 19, normalized size = 0.7 \[ a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x + a*b*x**2 + b**2*x**3/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278746, size = 24, normalized size = 0.89 \[ \frac{{\left (b x + a\right )}^{3}{\rm sign}\left (b x + a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x + a)^3*sign(b*x + a)/b