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

Optimal. Leaf size=61 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}-\frac{a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0651242, 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 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}-\frac{a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.21996, size = 60, normalized size = 0.98 \[ - \frac{a \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 b^{2}} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(8*b**2) + (a**2 + 2*a*b*x
+ b**2*x**2)**(5/2)/(5*b**2)

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Mathematica [A]  time = 0.0236704, size = 55, normalized size = 0.9 \[ \frac{x^2 \sqrt{(a+b x)^2} \left (10 a^3+20 a^2 b x+15 a b^2 x^2+4 b^3 x^3\right )}{20 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[(a + b*x)^2]*(10*a^3 + 20*a^2*b*x + 15*a*b^2*x^2 + 4*b^3*x^3))/(20*(a
+ b*x))

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Maple [A]  time = 0.005, size = 52, normalized size = 0.9 \[{\frac{{x}^{2} \left ( 4\,{b}^{3}{x}^{3}+15\,a{b}^{2}{x}^{2}+20\,{a}^{2}bx+10\,{a}^{3} \right ) }{20\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*x^2*(4*b^3*x^3+15*a*b^2*x^2+20*a^2*b*x+10*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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237009, size = 46, normalized size = 0.75 \[ \frac{1}{5} \, b^{3} x^{5} + \frac{3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac{1}{2} \, a^{3} x^{2} \]

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,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.209737, size = 97, normalized size = 1.59 \[ \frac{1}{5} \, b^{3} x^{5}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, a b^{2} x^{4}{\rm sign}\left (b x + a\right ) + a^{2} b x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{3} x^{2}{\rm sign}\left (b x + a\right ) - \frac{a^{5}{\rm sign}\left (b x + a\right )}{20 \, b^{2}} \]

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,x, algorithm="giac")

[Out]

1/5*b^3*x^5*sign(b*x + a) + 3/4*a*b^2*x^4*sign(b*x + a) + a^2*b*x^3*sign(b*x + a
) + 1/2*a^3*x^2*sign(b*x + a) - 1/20*a^5*sign(b*x + a)/b^2