3.153 \(\int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=32 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \]

[Out]

((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(4*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.0198559, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(4*b)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.90449, size = 31, normalized size = 0.97 \[ \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(8*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0172957, size = 23, normalized size = 0.72 \[ \frac{(a+b x) \left ((a+b x)^2\right )^{3/2}}{4 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((a + b*x)*((a + b*x)^2)^(3/2))/(4*b)

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 49, normalized size = 1.5 \[{\frac{x \left ({b}^{3}{x}^{3}+4\,a{b}^{2}{x}^{2}+6\,{a}^{2}bx+4\,{a}^{3} \right ) }{4\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/4*x*(b^3*x^3+4*a*b^2*x^2+6*a^2*b*x+4*a^3)*((b*x+a)^2)^(3/2)/(b*x+a)^3

_______________________________________________________________________________________

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((b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.233485, size = 42, normalized size = 1.31 \[ \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")

[Out]

1/4*b^3*x^4 + a*b^2*x^3 + 3/2*a^2*b*x^2 + a^3*x

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a**2 + 2*a*b*x + b**2*x**2)**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.209472, size = 93, normalized size = 2.91 \[ \frac{1}{4} \, b^{3} x^{4}{\rm sign}\left (b x + a\right ) + a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + a^{3} x{\rm sign}\left (b x + a\right ) + \frac{a^{4}{\rm sign}\left (b x + a\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")

[Out]

1/4*b^3*x^4*sign(b*x + a) + a*b^2*x^3*sign(b*x + a) + 3/2*a^2*b*x^2*sign(b*x + a
) + a^3*x*sign(b*x + a) + 1/4*a^4*sign(b*x + a)/b