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

Optimal. Leaf size=55 \[ \frac{1}{3} a^2 A x^3+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{4} a x^4 (a B+2 A b)+\frac{1}{6} b^2 B x^6 \]

[Out]

(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^4)/4 + (b*(A*b + 2*a*B)*x^5)/5 + (b^2*B*x^6)/6

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Rubi [A]  time = 0.0287897, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ \frac{1}{3} a^2 A x^3+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{4} a x^4 (a B+2 A b)+\frac{1}{6} b^2 B x^6 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^2*(A + B*x),x]

[Out]

(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^4)/4 + (b*(A*b + 2*a*B)*x^5)/5 + (b^2*B*x^6)/6

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)^2 (A+B x) \, dx &=\int \left (a^2 A x^2+a (2 A b+a B) x^3+b (A b+2 a B) x^4+b^2 B x^5\right ) \, dx\\ &=\frac{1}{3} a^2 A x^3+\frac{1}{4} a (2 A b+a B) x^4+\frac{1}{5} b (A b+2 a B) x^5+\frac{1}{6} b^2 B x^6\\ \end{align*}

Mathematica [A]  time = 0.0105954, size = 50, normalized size = 0.91 \[ \frac{1}{60} x^3 \left (5 a^2 (4 A+3 B x)+6 a b x (5 A+4 B x)+2 b^2 x^2 (6 A+5 B x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(5*a^2*(4*A + 3*B*x) + 6*a*b*x*(5*A + 4*B*x) + 2*b^2*x^2*(6*A + 5*B*x)))/60

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Maple [A]  time = 0., size = 52, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{6}}{6}}+{\frac{ \left ({b}^{2}A+2\,abB \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,abA+{a}^{2}B \right ){x}^{4}}{4}}+{\frac{{a}^{2}A{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*B*x^6+1/5*(A*b^2+2*B*a*b)*x^5+1/4*(2*A*a*b+B*a^2)*x^4+1/3*a^2*A*x^3

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Maxima [A]  time = 1.04979, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{3} \, A a^{2} x^{3} + \frac{1}{5} \,{\left (2 \, B a b + A b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{2} + 2 \, 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)^2*(B*x+A),x, algorithm="maxima")

[Out]

1/6*B*b^2*x^6 + 1/3*A*a^2*x^3 + 1/5*(2*B*a*b + A*b^2)*x^5 + 1/4*(B*a^2 + 2*A*a*b)*x^4

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Fricas [A]  time = 1.75352, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{6} x^{6} b^{2} B + \frac{2}{5} x^{5} b a B + \frac{1}{5} x^{5} b^{2} A + \frac{1}{4} x^{4} a^{2} B + \frac{1}{2} x^{4} b a A + \frac{1}{3} x^{3} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*b^2*B + 2/5*x^5*b*a*B + 1/5*x^5*b^2*A + 1/4*x^4*a^2*B + 1/2*x^4*b*a*A + 1/3*x^3*a^2*A

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Sympy [A]  time = 0.078262, size = 54, normalized size = 0.98 \begin{align*} \frac{A a^{2} x^{3}}{3} + \frac{B b^{2} x^{6}}{6} + x^{5} \left (\frac{A b^{2}}{5} + \frac{2 B a b}{5}\right ) + x^{4} \left (\frac{A a b}{2} + \frac{B a^{2}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*x**3/3 + B*b**2*x**6/6 + x**5*(A*b**2/5 + 2*B*a*b/5) + x**4*(A*a*b/2 + B*a**2/4)

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Giac [A]  time = 1.21344, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{6} \, B b^{2} x^{6} + \frac{2}{5} \, B a b x^{5} + \frac{1}{5} \, A b^{2} x^{5} + \frac{1}{4} \, B a^{2} x^{4} + \frac{1}{2} \, A a b x^{4} + \frac{1}{3} \, A a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*b^2*x^6 + 2/5*B*a*b*x^5 + 1/5*A*b^2*x^5 + 1/4*B*a^2*x^4 + 1/2*A*a*b*x^4 + 1/3*A*a^2*x^3