### 3.1761 $$\int (a+b x) (a c+(b c+a d) x+b d x^2) \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0284002, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.08, Rules used = {626, 43} $\frac{(a+b x)^3 (b c-a d)}{3 b^2}+\frac{d (a+b x)^4}{4 b^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2),x]

[Out]

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

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

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

Mathematica [A]  time = 0.0074121, size = 46, normalized size = 1.21 $\frac{1}{12} x \left (6 a^2 (2 c+d x)+4 a b x (3 c+2 d x)+b^2 x^2 (4 c+3 d x)\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2),x]

[Out]

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

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Maple [A]  time = 0.04, size = 55, normalized size = 1.5 \begin{align*}{\frac{{b}^{2}d{x}^{4}}{4}}+{\frac{ \left ( abd+b \left ( ad+bc \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( a \left ( ad+bc \right ) +abc \right ){x}^{2}}{2}}+{a}^{2}cx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06322, size = 65, normalized size = 1.71 \begin{align*} \frac{1}{4} \, b^{2} d x^{4} + a^{2} c x + \frac{1}{3} \,{\left (b^{2} c + 2 \, a b d\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.58811, size = 115, normalized size = 3.03 \begin{align*} \frac{1}{4} x^{4} d b^{2} + \frac{1}{3} x^{3} c b^{2} + \frac{2}{3} x^{3} d b a + x^{2} c b a + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.07156, size = 49, normalized size = 1.29 \begin{align*} a^{2} c x + \frac{b^{2} d x^{4}}{4} + x^{3} \left (\frac{2 a b d}{3} + \frac{b^{2} c}{3}\right ) + x^{2} \left (\frac{a^{2} d}{2} + a b c\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20138, size = 66, normalized size = 1.74 \begin{align*} \frac{1}{4} \, b^{2} d x^{4} + \frac{1}{3} \, b^{2} c x^{3} + \frac{2}{3} \, a b d x^{3} + a b c x^{2} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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