\(\int (a+b x) (A+B x) (d+e x) \, dx\) [1014]

Optimal result

Integrand size = 16, antiderivative size = 56 \[ \int (a+b x) (A+B x) (d+e x) \, dx=a A d x+\frac {1}{2} (A b d+a B d+a A e) x^2+\frac {1}{3} (b B d+A b e+a B e) x^3+\frac {1}{4} b B e x^4 \]

[Out]

a*A*d*x+1/2*(A*a*e+A*b*d+B*a*d)*x^2+1/3*(A*b*e+B*a*e+B*b*d)*x^3+1/4*b*B*e*x^4

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int (a+b x) (A+B x) (d+e x) \, dx=\frac {1}{3} x^3 (a B e+A b e+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} b B e x^4 \]

[In]

Int[(a + b*x)*(A + B*x)*(d + e*x),x]

[Out]

a*A*d*x + ((A*b*d + a*B*d + a*A*e)*x^2)/2 + ((b*B*d + A*b*e + a*B*e)*x^3)/3 + (b*B*e*x^4)/4

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (a A d+(A b d+a B d+a A e) x+(b B d+A b e+a B e) x^2+b B e x^3\right ) \, dx \\ & = a A d x+\frac {1}{2} (A b d+a B d+a A e) x^2+\frac {1}{3} (b B d+A b e+a B e) x^3+\frac {1}{4} b B e x^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95 \[ \int (a+b x) (A+B x) (d+e x) \, dx=\frac {1}{12} x \left (12 a A d+6 (A b d+a B d+a A e) x+4 (b B d+A b e+a B e) x^2+3 b B e x^3\right ) \]

[In]

Integrate[(a + b*x)*(A + B*x)*(d + e*x),x]

[Out]

(x*(12*a*A*d + 6*(A*b*d + a*B*d + a*A*e)*x + 4*(b*B*d + A*b*e + a*B*e)*x^2 + 3*b*B*e*x^3))/12

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {b B e \,x^{4}}{4}+\frac {\left (\left (A b +B a \right ) e +B b d \right ) x^{3}}{3}+\frac {\left (A a e +\left (A b +B a \right ) d \right ) x^{2}}{2}+a A d x\)	\(53\)
norman	\(\frac {b B e \,x^{4}}{4}+\left (\frac {1}{3} A b e +\frac {1}{3} B a e +\frac {1}{3} B b d \right ) x^{3}+\left (\frac {1}{2} A a e +\frac {1}{2} A b d +\frac {1}{2} B a d \right ) x^{2}+a A d x\)	\(55\)
gosper	\(\frac {1}{4} b B e \,x^{4}+\frac {1}{3} x^{3} A b e +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} B b d +\frac {1}{2} x^{2} A a e +\frac {1}{2} x^{2} A b d +\frac {1}{2} x^{2} B a d +a A d x\)	\(63\)
risch	\(\frac {1}{4} b B e \,x^{4}+\frac {1}{3} x^{3} A b e +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} B b d +\frac {1}{2} x^{2} A a e +\frac {1}{2} x^{2} A b d +\frac {1}{2} x^{2} B a d +a A d x\)	\(63\)
parallelrisch	\(\frac {1}{4} b B e \,x^{4}+\frac {1}{3} x^{3} A b e +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} B b d +\frac {1}{2} x^{2} A a e +\frac {1}{2} x^{2} A b d +\frac {1}{2} x^{2} B a d +a A d x\)	\(63\)

[In]

int((b*x+a)*(B*x+A)*(e*x+d),x,method=_RETURNVERBOSE)

[Out]

1/4*b*B*e*x^4+1/3*((A*b+B*a)*e+B*b*d)*x^3+1/2*(A*a*e+(A*b+B*a)*d)*x^2+a*A*d*x

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.11 \[ \int (a+b x) (A+B x) (d+e x) \, dx=\frac {1}{4} x^{4} e b B + \frac {1}{3} x^{3} d b B + \frac {1}{3} x^{3} e a B + \frac {1}{3} x^{3} e b A + \frac {1}{2} x^{2} d a B + \frac {1}{2} x^{2} d b A + \frac {1}{2} x^{2} e a A + x d a A \]

[In]

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

[Out]

1/4*x^4*e*b*B + 1/3*x^3*d*b*B + 1/3*x^3*e*a*B + 1/3*x^3*e*b*A + 1/2*x^2*d*a*B + 1/2*x^2*d*b*A + 1/2*x^2*e*a*A
+ x*d*a*A

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12 \[ \int (a+b x) (A+B x) (d+e x) \, dx=A a d x + \frac {B b e x^{4}}{4} + x^{3} \left (\frac {A b e}{3} + \frac {B a e}{3} + \frac {B b d}{3}\right ) + x^{2} \left (\frac {A a e}{2} + \frac {A b d}{2} + \frac {B a d}{2}\right ) \]

[In]

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

[Out]

A*a*d*x + B*b*e*x**4/4 + x**3*(A*b*e/3 + B*a*e/3 + B*b*d/3) + x**2*(A*a*e/2 + A*b*d/2 + B*a*d/2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93 \[ \int (a+b x) (A+B x) (d+e x) \, dx=\frac {1}{4} \, B b e x^{4} + A a d x + \frac {1}{3} \, {\left (B b d + {\left (B a + A b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a e + {\left (B a + A b\right )} d\right )} x^{2} \]

[In]

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

[Out]

1/4*B*b*e*x^4 + A*a*d*x + 1/3*(B*b*d + (B*a + A*b)*e)*x^3 + 1/2*(A*a*e + (B*a + A*b)*d)*x^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.11 \[ \int (a+b x) (A+B x) (d+e x) \, dx=\frac {1}{4} \, B b e x^{4} + \frac {1}{3} \, B b d x^{3} + \frac {1}{3} \, B a e x^{3} + \frac {1}{3} \, A b e x^{3} + \frac {1}{2} \, B a d x^{2} + \frac {1}{2} \, A b d x^{2} + \frac {1}{2} \, A a e x^{2} + A a d x \]

[In]

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

[Out]

1/4*B*b*e*x^4 + 1/3*B*b*d*x^3 + 1/3*B*a*e*x^3 + 1/3*A*b*e*x^3 + 1/2*B*a*d*x^2 + 1/2*A*b*d*x^2 + 1/2*A*a*e*x^2
+ A*a*d*x

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int (a+b x) (A+B x) (d+e x) \, dx=\frac {B\,b\,e\,x^4}{4}+\left (\frac {A\,b\,e}{3}+\frac {B\,a\,e}{3}+\frac {B\,b\,d}{3}\right )\,x^3+\left (\frac {A\,a\,e}{2}+\frac {A\,b\,d}{2}+\frac {B\,a\,d}{2}\right )\,x^2+A\,a\,d\,x \]

[In]

int((A + B*x)*(a + b*x)*(d + e*x),x)

[Out]

x^2*((A*a*e)/2 + (A*b*d)/2 + (B*a*d)/2) + x^3*((A*b*e)/3 + (B*a*e)/3 + (B*b*d)/3) + (B*b*e*x^4)/4 + A*a*d*x