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

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 = 0.022522, 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, Rules used = {76} \[ \frac{1}{4} x^4 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{5} b B x^5 \]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int x^2 (a+b x) (A+B x) \, dx &=\int \left (a A x^2+(A b+a B) x^3+b B x^4\right ) \, dx\\ &=\frac{1}{3} a A x^3+\frac{1}{4} (A b+a B) x^4+\frac{1}{5} b B x^5\\ \end{align*}

Mathematica [A]  time = 0.0045146, 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., size = 28, normalized size = 0.9 \begin{align*}{\frac{aA{x}^{3}}{3}}+{\frac{ \left ( Ab+Ba \right ){x}^{4}}{4}}+{\frac{bB{x}^{5}}{5}} \end{align*}

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 = 1.12032, size = 36, normalized size = 1.09 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(B*x+A),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 = 1.68377, size = 74, normalized size = 2.24 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(B*x+A),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 = 0.069797, size = 29, normalized size = 0.88 \begin{align*} \frac{A a x^{3}}{3} + \frac{B b x^{5}}{5} + x^{4} \left (\frac{A b}{4} + \frac{B a}{4}\right ) \end{align*}

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 [A]  time = 1.23243, size = 39, normalized size = 1.18 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)*(B*x+A),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