3.52 \(\int x (a+b x) (A+B x) \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{3} x^3 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{4} b B x^4 \]

[Out]

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

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Rubi [A]  time = 0.0518347, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{1}{3} x^3 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{4} b B x^4 \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)*(A + B*x),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a \int x\, dx + \frac{B b x^{4}}{4} + x^{3} \left (\frac{A b}{3} + \frac{B a}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*Integral(x, x) + B*b*x**4/4 + x**3*(A*b/3 + B*a/3)

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Mathematica [A]  time = 0.00755864, size = 29, normalized size = 0.88 \[ \frac{1}{12} x^2 (a (6 A+4 B x)+b x (4 A+3 B x)) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(b*x*(4*A + 3*B*x) + a*(6*A + 4*B*x)))/12

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Maple [A]  time = 0.002, size = 28, normalized size = 0.9 \[{\frac{aA{x}^{2}}{2}}+{\frac{ \left ( Ab+Ba \right ){x}^{3}}{3}}+{\frac{bB{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)*(B*x+A),x)

[Out]

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

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Maxima [A]  time = 1.33478, size = 36, normalized size = 1.09 \[ \frac{1}{4} \, B b x^{4} + \frac{1}{2} \, A a x^{2} + \frac{1}{3} \,{\left (B a + A b\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.181589, size = 1, normalized size = 0.03 \[ \frac{1}{4} x^{4} b B + \frac{1}{3} x^{3} a B + \frac{1}{3} x^{3} b A + \frac{1}{2} x^{2} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.08546, size = 29, normalized size = 0.88 \[ \frac{A a x^{2}}{2} + \frac{B b x^{4}}{4} + x^{3} \left (\frac{A b}{3} + \frac{B a}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)*(B*x+A),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.230383, size = 39, normalized size = 1.18 \[ \frac{1}{4} \, B b x^{4} + \frac{1}{3} \, B a x^{3} + \frac{1}{3} \, A b x^{3} + \frac{1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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