3.513 \(\int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{2} a^2 A x^2+\frac{1}{4} b x^4 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]

[Out]

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

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Rubi [A]  time = 0.0827226, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{1}{2} a^2 A x^2+\frac{1}{4} b x^4 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]

Antiderivative was successfully verified.

[In]  Int[x*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.5845, size = 53, normalized size = 0.96 \[ \frac{B \left (a + b x\right )^{5}}{5 b^{3}} - \frac{a \left (a + b x\right )^{3} \left (A b - B a\right )}{3 b^{3}} + \frac{\left (a + b x\right )^{4} \left (A b - 2 B a\right )}{4 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(B*x+A)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Mathematica [A]  time = 0.0176058, size = 50, normalized size = 0.91 \[ \frac{1}{60} x^2 \left (10 a^2 (3 A+2 B x)+10 a b x (4 A+3 B x)+3 b^2 x^2 (5 A+4 B x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(x^2*(10*a^2*(3*A + 2*B*x) + 10*a*b*x*(4*A + 3*B*x) + 3*b^2*x^2*(5*A + 4*B*x)))/
60

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Maple [A]  time = 0.001, size = 52, normalized size = 1. \[{\frac{{b}^{2}B{x}^{5}}{5}}+{\frac{ \left ({b}^{2}A+2\,abB \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,abA+{a}^{2}B \right ){x}^{3}}{3}}+{\frac{{a}^{2}A{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

1/5*b^2*B*x^5+1/4*(A*b^2+2*B*a*b)*x^4+1/3*(2*A*a*b+B*a^2)*x^3+1/2*a^2*A*x^2

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Maxima [A]  time = 0.67617, size = 69, normalized size = 1.25 \[ \frac{1}{5} \, B b^{2} x^{5} + \frac{1}{2} \, A a^{2} x^{2} + \frac{1}{4} \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)*x,x, algorithm="maxima")

[Out]

1/5*B*b^2*x^5 + 1/2*A*a^2*x^2 + 1/4*(2*B*a*b + A*b^2)*x^4 + 1/3*(B*a^2 + 2*A*a*b
)*x^3

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Fricas [A]  time = 0.2568, size = 1, normalized size = 0.02 \[ \frac{1}{5} x^{5} b^{2} B + \frac{1}{2} x^{4} b a B + \frac{1}{4} x^{4} b^{2} A + \frac{1}{3} x^{3} a^{2} B + \frac{2}{3} x^{3} b a A + \frac{1}{2} x^{2} a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.110088, size = 54, normalized size = 0.98 \[ \frac{A a^{2} x^{2}}{2} + \frac{B b^{2} x^{5}}{5} + x^{4} \left (\frac{A b^{2}}{4} + \frac{B a b}{2}\right ) + x^{3} \left (\frac{2 A a b}{3} + \frac{B a^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(B*x+A)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

A*a**2*x**2/2 + B*b**2*x**5/5 + x**4*(A*b**2/4 + B*a*b/2) + x**3*(2*A*a*b/3 + B*
a**2/3)

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GIAC/XCAS [A]  time = 0.267137, size = 72, normalized size = 1.31 \[ \frac{1}{5} \, B b^{2} x^{5} + \frac{1}{2} \, B a b x^{4} + \frac{1}{4} \, A b^{2} x^{4} + \frac{1}{3} \, B a^{2} x^{3} + \frac{2}{3} \, A a b x^{3} + \frac{1}{2} \, A a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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