3.1239 \(\int (a+b x) (c+d x) \, dx\)

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

[Out]

a*c*x + ((b*c + a*d)*x^2)/2 + (b*d*x^3)/3

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Rubi [A]  time = 0.015441, 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} \[ \frac{1}{2} x^2 (a d+b c)+a c x+\frac{1}{3} b d x^3 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(c + d*x),x]

[Out]

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

Mathematica [A]  time = 0.0039147, size = 28, normalized size = 1. \[ \frac{1}{2} x^2 (a d+b c)+a c x+\frac{1}{3} b d x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(c + d*x),x]

[Out]

a*c*x + ((b*c + a*d)*x^2)/2 + (b*d*x^3)/3

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Maple [A]  time = 0.001, size = 25, normalized size = 0.9 \begin{align*} acx+{\frac{ \left ( ad+bc \right ){x}^{2}}{2}}+{\frac{bd{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(d*x+c),x)

[Out]

a*c*x+1/2*(a*d+b*c)*x^2+1/3*b*d*x^3

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Maxima [A]  time = 0.970086, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{3} \, b d x^{3} + a c x + \frac{1}{2} \,{\left (b c + a d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c),x, algorithm="maxima")

[Out]

1/3*b*d*x^3 + a*c*x + 1/2*(b*c + a*d)*x^2

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Fricas [A]  time = 1.73416, size = 66, normalized size = 2.36 \begin{align*} \frac{1}{3} x^{3} d b + \frac{1}{2} x^{2} c b + \frac{1}{2} x^{2} d a + x c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c),x, algorithm="fricas")

[Out]

1/3*x^3*d*b + 1/2*x^2*c*b + 1/2*x^2*d*a + x*c*a

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Sympy [A]  time = 0.055744, size = 26, normalized size = 0.93 \begin{align*} a c x + \frac{b d x^{3}}{3} + x^{2} \left (\frac{a d}{2} + \frac{b c}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c),x)

[Out]

a*c*x + b*d*x**3/3 + x**2*(a*d/2 + b*c/2)

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Giac [A]  time = 1.0495, size = 35, normalized size = 1.25 \begin{align*} \frac{1}{3} \, b d x^{3} + \frac{1}{2} \, b c x^{2} + \frac{1}{2} \, a d x^{2} + a c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c),x, algorithm="giac")

[Out]

1/3*b*d*x^3 + 1/2*b*c*x^2 + 1/2*a*d*x^2 + a*c*x