3.1249 \(\int (a+b x) (c+d x)^2 \, dx\)

Optimal. Leaf size=38 \[ \frac{b (c+d x)^4}{4 d^2}-\frac{(c+d x)^3 (b c-a d)}{3 d^2} \]

[Out]

-((b*c - a*d)*(c + d*x)^3)/(3*d^2) + (b*(c + d*x)^4)/(4*d^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0273636, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{b (c+d x)^4}{4 d^2}-\frac{(c+d x)^3 (b c-a d)}{3 d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(c + d*x)^2,x]

[Out]

-((b*c - a*d)*(c + d*x)^3)/(3*d^2) + (b*(c + d*x)^4)/(4*d^2)

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)^2 \, dx &=\int \left (\frac{(-b c+a d) (c+d x)^2}{d}+\frac{b (c+d x)^3}{d}\right ) \, dx\\ &=-\frac{(b c-a d) (c+d x)^3}{3 d^2}+\frac{b (c+d x)^4}{4 d^2}\\ \end{align*}

Mathematica [A]  time = 0.0097692, size = 47, normalized size = 1.24 \[ \frac{1}{12} x \left (4 d x^2 (a d+2 b c)+6 c x (2 a d+b c)+12 a c^2+3 b d^2 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(c + d*x)^2,x]

[Out]

(x*(12*a*c^2 + 6*c*(b*c + 2*a*d)*x + 4*d*(2*b*c + a*d)*x^2 + 3*b*d^2*x^3))/12

________________________________________________________________________________________

Maple [A]  time = 0., size = 49, normalized size = 1.3 \begin{align*}{\frac{b{d}^{2}{x}^{4}}{4}}+{\frac{ \left ( a{d}^{2}+2\,bcd \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,acd+b{c}^{2} \right ){x}^{2}}{2}}+a{c}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(d*x+c)^2,x)

[Out]

1/4*b*d^2*x^4+1/3*(a*d^2+2*b*c*d)*x^3+1/2*(2*a*c*d+b*c^2)*x^2+a*c^2*x

________________________________________________________________________________________

Maxima [A]  time = 0.948592, size = 65, normalized size = 1.71 \begin{align*} \frac{1}{4} \, b d^{2} x^{4} + a c^{2} x + \frac{1}{3} \,{\left (2 \, b c d + a d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b c^{2} + 2 \, a c d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^2,x, algorithm="maxima")

[Out]

1/4*b*d^2*x^4 + a*c^2*x + 1/3*(2*b*c*d + a*d^2)*x^3 + 1/2*(b*c^2 + 2*a*c*d)*x^2

________________________________________________________________________________________

Fricas [A]  time = 1.78969, size = 115, normalized size = 3.03 \begin{align*} \frac{1}{4} x^{4} d^{2} b + \frac{2}{3} x^{3} d c b + \frac{1}{3} x^{3} d^{2} a + \frac{1}{2} x^{2} c^{2} b + x^{2} d c a + x c^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^2,x, algorithm="fricas")

[Out]

1/4*x^4*d^2*b + 2/3*x^3*d*c*b + 1/3*x^3*d^2*a + 1/2*x^2*c^2*b + x^2*d*c*a + x*c^2*a

________________________________________________________________________________________

Sympy [A]  time = 0.067188, size = 49, normalized size = 1.29 \begin{align*} a c^{2} x + \frac{b d^{2} x^{4}}{4} + x^{3} \left (\frac{a d^{2}}{3} + \frac{2 b c d}{3}\right ) + x^{2} \left (a c d + \frac{b c^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)**2,x)

[Out]

a*c**2*x + b*d**2*x**4/4 + x**3*(a*d**2/3 + 2*b*c*d/3) + x**2*(a*c*d + b*c**2/2)

________________________________________________________________________________________

Giac [A]  time = 1.05866, size = 66, normalized size = 1.74 \begin{align*} \frac{1}{4} \, b d^{2} x^{4} + \frac{2}{3} \, b c d x^{3} + \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{2} \, b c^{2} x^{2} + a c d x^{2} + a c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^2,x, algorithm="giac")

[Out]

1/4*b*d^2*x^4 + 2/3*b*c*d*x^3 + 1/3*a*d^2*x^3 + 1/2*b*c^2*x^2 + a*c*d*x^2 + a*c^2*x