3.2309 \(\int (A+B x) (d+e x) (a+b x+c x^2) \, dx\)

Optimal. Leaf size=80 \[ \frac{1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac{1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac{1}{4} x^4 (A c e+b B e+B c d)+\frac{1}{5} B c e x^5 \]

[Out]

a*A*d*x + ((A*b*d + a*B*d + a*A*e)*x^2)/2 + ((b*B*d + A*c*d + A*b*e + a*B*e)*x^3)/3 + ((B*c*d + b*B*e + A*c*e)
*x^4)/4 + (B*c*e*x^5)/5

________________________________________________________________________________________

Rubi [A]  time = 0.0814684, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {771} \[ \frac{1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac{1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac{1}{4} x^4 (A c e+b B e+B c d)+\frac{1}{5} B c e x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*A*d*x + ((A*b*d + a*B*d + a*A*e)*x^2)/2 + ((b*B*d + A*c*d + A*b*e + a*B*e)*x^3)/3 + ((B*c*d + b*B*e + A*c*e)
*x^4)/4 + (B*c*e*x^5)/5

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (A+B x) (d+e x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a A d+(A b d+a B d+a A e) x+(b B d+A c d+A b e+a B e) x^2+(B c d+b B e+A c e) x^3+B c e x^4\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 c d+A b e+a B e) x^3+\frac{1}{4} (B c d+b B e+A c e) x^4+\frac{1}{5} B c e x^5\\ \end{align*}

Mathematica [A]  time = 0.0314289, size = 80, normalized size = 1. \[ \frac{1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac{1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac{1}{4} x^4 (A c e+b B e+B c d)+\frac{1}{5} B c e x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*A*d*x + ((A*b*d + a*B*d + a*A*e)*x^2)/2 + ((b*B*d + A*c*d + A*b*e + a*B*e)*x^3)/3 + ((B*c*d + b*B*e + A*c*e)
*x^4)/4 + (B*c*e*x^5)/5

________________________________________________________________________________________

Maple [A]  time = 0., size = 76, normalized size = 1. \begin{align*}{\frac{Bce{x}^{5}}{5}}+{\frac{ \left ( c \left ( Ae+Bd \right ) +bBe \right ){x}^{4}}{4}}+{\frac{ \left ( Acd+b \left ( Ae+Bd \right ) +aBe \right ){x}^{3}}{3}}+{\frac{ \left ( Abd+a \left ( Ae+Bd \right ) \right ){x}^{2}}{2}}+aAdx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*c*e*x^5+1/4*(c*(A*e+B*d)+b*B*e)*x^4+1/3*(A*c*d+b*(A*e+B*d)+a*B*e)*x^3+1/2*(A*b*d+a*(A*e+B*d))*x^2+a*A*d*
x

________________________________________________________________________________________

Maxima [A]  time = 1.38863, size = 103, normalized size = 1.29 \begin{align*} \frac{1}{5} \, B c e x^{5} + \frac{1}{4} \,{\left (B c d +{\left (B b + A c\right )} e\right )} x^{4} + A a d x + \frac{1}{3} \,{\left ({\left (B b + A c\right )} 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*c*e*x^5 + 1/4*(B*c*d + (B*b + A*c)*e)*x^4 + A*a*d*x + 1/3*((B*b + A*c)*d + (B*a + A*b)*e)*x^3 + 1/2*(A*a
*e + (B*a + A*b)*d)*x^2

________________________________________________________________________________________

Fricas [A]  time = 1.16278, size = 250, normalized size = 3.12 \begin{align*} \frac{1}{5} x^{5} e c B + \frac{1}{4} x^{4} d c B + \frac{1}{4} x^{4} e b B + \frac{1}{4} x^{4} e c A + \frac{1}{3} x^{3} d b B + \frac{1}{3} x^{3} e a B + \frac{1}{3} x^{3} d c A + \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*e*c*B + 1/4*x^4*d*c*B + 1/4*x^4*e*b*B + 1/4*x^4*e*c*A + 1/3*x^3*d*b*B + 1/3*x^3*e*a*B + 1/3*x^3*d*c*A
+ 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]  time = 0.070365, size = 94, normalized size = 1.18 \begin{align*} A a d x + \frac{B c e x^{5}}{5} + x^{4} \left (\frac{A c e}{4} + \frac{B b e}{4} + \frac{B c d}{4}\right ) + x^{3} \left (\frac{A b e}{3} + \frac{A c d}{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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*d*x + B*c*e*x**5/5 + x**4*(A*c*e/4 + B*b*e/4 + B*c*d/4) + x**3*(A*b*e/3 + A*c*d/3 + B*a*e/3 + B*b*d/3) + x
**2*(A*a*e/2 + A*b*d/2 + B*a*d/2)

________________________________________________________________________________________

Giac [A]  time = 1.12722, size = 135, normalized size = 1.69 \begin{align*} \frac{1}{5} \, B c x^{5} e + \frac{1}{4} \, B c d x^{4} + \frac{1}{4} \, B b x^{4} e + \frac{1}{4} \, A c x^{4} e + \frac{1}{3} \, B b d x^{3} + \frac{1}{3} \, A c d x^{3} + \frac{1}{3} \, B a x^{3} e + \frac{1}{3} \, A b x^{3} e + \frac{1}{2} \, B a d x^{2} + \frac{1}{2} \, A b d x^{2} + \frac{1}{2} \, A a x^{2} e + A a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*c*x^5*e + 1/4*B*c*d*x^4 + 1/4*B*b*x^4*e + 1/4*A*c*x^4*e + 1/3*B*b*d*x^3 + 1/3*A*c*d*x^3 + 1/3*B*a*x^3*e
+ 1/3*A*b*x^3*e + 1/2*B*a*d*x^2 + 1/2*A*b*d*x^2 + 1/2*A*a*x^2*e + A*a*d*x