3.1015 \(\int (a+b x) (A+B x) \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{2} x^2 (a B+A b)+a A x+\frac{1}{3} b B x^3 \]

[Out]

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

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Rubi [A]  time = 0.0182654, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{1}{2} x^2 (a B+A b)+a A x+\frac{1}{3} b B x^3 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(A + B*x),x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.004805, size = 28, normalized size = 1. \[ \frac{1}{2} x^2 (a B+A b)+a A x+\frac{1}{3} b B x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(A + B*x),x]

[Out]

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

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Maple [A]  time = 0.003, size = 25, normalized size = 0.9 \begin{align*} aAx+{\frac{ \left ( Ab+Ba \right ){x}^{2}}{2}}+{\frac{bB{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07243, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{3} \, B b x^{3} + A a x + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.52043, size = 66, normalized size = 2.36 \begin{align*} \frac{1}{3} x^{3} b B + \frac{1}{2} x^{2} a B + \frac{1}{2} x^{2} b A + x a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.057172, size = 26, normalized size = 0.93 \begin{align*} A a x + \frac{B b x^{3}}{3} + x^{2} \left (\frac{A b}{2} + \frac{B a}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22534, size = 35, normalized size = 1.25 \begin{align*} \frac{1}{3} \, B b x^{3} + \frac{1}{2} \, B a x^{2} + \frac{1}{2} \, A b x^{2} + A a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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