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

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 = 0.0265814, antiderivative size = 55, 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} \[ \frac{1}{2} a^2 A x^2+\frac{1}{4} b x^4 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^2*(A + B*x),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

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

Mathematica [A]  time = 0.0092655, 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)^2*(A + B*x),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. \begin{align*}{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)^2*(B*x+A),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 = 2.02775, size = 69, normalized size = 1.25 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^2*(B*x+A),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 = 1.74787, size = 128, normalized size = 2.33 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^2*(B*x+A),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 = 0.075575, size = 54, normalized size = 0.98 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**2*(B*x+A),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 [A]  time = 1.20159, size = 72, normalized size = 1.31 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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