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

Optimal. Leaf size=71 \[ \frac{b x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0764171, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.13858, size = 58, normalized size = 0.82 \[ \frac{a x^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 \left (a + b x\right )} + \frac{x^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(12*(a + b*x)) + x**3*sqrt(a**2 + 2*a*b*
x + b**2*x**2)/4

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Mathematica [A]  time = 0.0145099, size = 33, normalized size = 0.46 \[ \frac{x^3 \sqrt{(a+b x)^2} (4 a+3 b x)}{12 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*Sqrt[(a + b*x)^2]*(4*a + 3*b*x))/(12*(a + b*x))

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Maple [A]  time = 0.003, size = 30, normalized size = 0.4 \[{\frac{{x}^{3} \left ( 3\,bx+4\,a \right ) }{12\,bx+12\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*((b*x+a)^2)^(1/2),x)

[Out]

1/12*x^3*(3*b*x+4*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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22712, size = 18, normalized size = 0.25 \[ \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b*x^4 + 1/3*a*x^3

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Sympy [A]  time = 0.140304, size = 12, normalized size = 0.17 \[ \frac{a x^{3}}{3} + \frac{b x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**3/3 + b*x**4/4

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GIAC/XCAS [A]  time = 0.209526, size = 53, normalized size = 0.75 \[ \frac{1}{4} \, b x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, a x^{3}{\rm sign}\left (b x + a\right ) + \frac{a^{4}{\rm sign}\left (b x + a\right )}{12 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b*x^4*sign(b*x + a) + 1/3*a*x^3*sign(b*x + a) + 1/12*a^4*sign(b*x + a)/b^3