3.51 \(\int x^2 (a+b x) (A+B x) \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{4} x^4 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{5} b B x^5 \]

[Out]

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

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Rubi [A]  time = 0.0629032, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{1}{4} x^4 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{5} b B x^5 \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)*(A + B*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.0678, size = 29, normalized size = 0.88 \[ \frac{A a x^{3}}{3} + \frac{B b x^{5}}{5} + x^{4} \left (\frac{A b}{4} + \frac{B a}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00689147, size = 33, normalized size = 1. \[ \frac{1}{4} x^4 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{5} b B x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 28, normalized size = 0.9 \[{\frac{aA{x}^{3}}{3}}+{\frac{ \left ( Ab+Ba \right ){x}^{4}}{4}}+{\frac{bB{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)*(B*x+A),x)

[Out]

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

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Maxima [A]  time = 1.35067, size = 36, normalized size = 1.09 \[ \frac{1}{5} \, B b x^{5} + \frac{1}{3} \, A a x^{3} + \frac{1}{4} \,{\left (B a + A b\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.17843, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} b B + \frac{1}{4} x^{4} a B + \frac{1}{4} x^{4} b A + \frac{1}{3} x^{3} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.076649, size = 29, normalized size = 0.88 \[ \frac{A a x^{3}}{3} + \frac{B b x^{5}}{5} + x^{4} \left (\frac{A b}{4} + \frac{B a}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.279738, size = 39, normalized size = 1.18 \[ \frac{1}{5} \, B b x^{5} + \frac{1}{4} \, B a x^{4} + \frac{1}{4} \, A b x^{4} + \frac{1}{3} \, A a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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