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

Optimal. Leaf size=61 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}-\frac{a (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^2} \]

[Out]

-(a*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b^2) + (a^2 + 2*a*b*x + b^2*x^2)
^(3/2)/(3*b^2)

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Rubi [A]  time = 0.0561733, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}-\frac{a (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b^2) + (a^2 + 2*a*b*x + b^2*x^2)
^(3/2)/(3*b^2)

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Rubi in Sympy [A]  time = 8.19547, size = 60, normalized size = 0.98 \[ - \frac{a \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 b^{2}} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(4*b**2) + (a**2 + 2*a*b*x + b
**2*x**2)**(3/2)/(3*b**2)

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Mathematica [A]  time = 0.0136479, size = 33, normalized size = 0.54 \[ \frac{x^2 \sqrt{(a+b x)^2} (3 a+2 b x)}{6 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[(a + b*x)^2]*(3*a + 2*b*x))/(6*(a + b*x))

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Maple [A]  time = 0.004, size = 30, normalized size = 0.5 \[{\frac{{x}^{2} \left ( 2\,bx+3\,a \right ) }{6\,bx+6\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*((b*x+a)^2)^(1/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222649, size = 18, normalized size = 0.3 \[ \frac{1}{3} \, b x^{3} + \frac{1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b*x^3 + 1/2*a*x^2

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Sympy [A]  time = 0.140569, size = 12, normalized size = 0.2 \[ \frac{a x^{2}}{2} + \frac{b x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**2/2 + b*x**3/3

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GIAC/XCAS [A]  time = 0.208473, size = 53, normalized size = 0.87 \[ \frac{1}{3} \, b x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a x^{2}{\rm sign}\left (b x + a\right ) - \frac{a^{3}{\rm sign}\left (b x + a\right )}{6 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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