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

Optimal. Leaf size=96 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^3}-\frac{2 a \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^3}+\frac{a^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^3} \]

[Out]

(a^2*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(4*b^3) - (2*a*(a^2 + 2*a*b*x +
b^2*x^2)^(5/2))/(5*b^3) + ((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(6*b^3)

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Rubi [A]  time = 0.100227, antiderivative size = 107, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3}-\frac{2 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{5 b^3}+\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{4 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b^3) - (2*a*(a + b*x)^4*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(5*b^3) + ((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(6*b^3)

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Rubi in Sympy [A]  time = 13.0657, size = 99, normalized size = 1.03 \[ \frac{a^{2} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{24 b^{3}} - \frac{a \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{15 b^{3}} + \frac{x^{2} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(24*b**3) - a*(a**2 + 2*a
*b*x + b**2*x**2)**(5/2)/(15*b**3) + x**2*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x
**2)**(3/2)/(12*b)

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Mathematica [A]  time = 0.0234839, size = 55, normalized size = 0.57 \[ \frac{x^3 \sqrt{(a+b x)^2} \left (20 a^3+45 a^2 b x+36 a b^2 x^2+10 b^3 x^3\right )}{60 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*Sqrt[(a + b*x)^2]*(20*a^3 + 45*a^2*b*x + 36*a*b^2*x^2 + 10*b^3*x^3))/(60*(a
+ b*x))

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Maple [A]  time = 0.009, size = 52, normalized size = 0.5 \[{\frac{{x}^{3} \left ( 10\,{b}^{3}{x}^{3}+36\,a{b}^{2}{x}^{2}+45\,{a}^{2}bx+20\,{a}^{3} \right ) }{60\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*x^3*(10*b^3*x^3+36*a*b^2*x^2+45*a^2*b*x+20*a^3)*((b*x+a)^2)^(3/2)/(b*x+a)^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232582, size = 47, normalized size = 0.49 \[ \frac{1}{6} \, b^{3} x^{6} + \frac{3}{5} \, a b^{2} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{3} \, a^{3} x^{3} \]

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^2,x, algorithm="fricas")

[Out]

1/6*b^3*x^6 + 3/5*a*b^2*x^5 + 3/4*a^2*b*x^4 + 1/3*a^3*x^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208774, size = 99, normalized size = 1.03 \[ \frac{1}{6} \, b^{3} x^{6}{\rm sign}\left (b x + a\right ) + \frac{3}{5} \, a b^{2} x^{5}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, a^{2} b x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, a^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{a^{6}{\rm sign}\left (b x + a\right )}{60 \, b^{3}} \]

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^2,x, algorithm="giac")

[Out]

1/6*b^3*x^6*sign(b*x + a) + 3/5*a*b^2*x^5*sign(b*x + a) + 3/4*a^2*b*x^4*sign(b*x
+ a) + 1/3*a^3*x^3*sign(b*x + a) + 1/60*a^6*sign(b*x + a)/b^3