3.1013 \(\int (a+b x) (A+B x) (d+e x)^2 \, dx\)

Optimal. Leaf size=77 \[ -\frac{(d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3}+\frac{(d+e x)^3 (b d-a e) (B d-A e)}{3 e^3}+\frac{b B (d+e x)^5}{5 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0710015, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ -\frac{(d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3}+\frac{(d+e x)^3 (b d-a e) (B d-A e)}{3 e^3}+\frac{b B (d+e x)^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 77

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

Mathematica [A]  time = 0.0328729, size = 96, normalized size = 1.25 \[ \frac{1}{3} x^3 \left (a A e^2+2 a B d e+2 A b d e+b B d^2\right )+\frac{1}{4} e x^4 (a B e+A b e+2 b B d)+\frac{1}{2} d x^2 (2 a A e+a B d+A b d)+a A d^2 x+\frac{1}{5} b B e^2 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 94, normalized size = 1.2 \begin{align*}{\frac{bB{e}^{2}{x}^{5}}{5}}+{\frac{ \left ( \left ( Ab+Ba \right ){e}^{2}+2\,bBde \right ){x}^{4}}{4}}+{\frac{ \left ( aA{e}^{2}+2\, \left ( Ab+Ba \right ) de+bB{d}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,aAde+ \left ( Ab+Ba \right ){d}^{2} \right ){x}^{2}}{2}}+aA{d}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08316, size = 126, normalized size = 1.64 \begin{align*} \frac{1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac{1}{4} \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b d^{2} + A a e^{2} + 2 \,{\left (B a + A b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a d e +{\left (B a + A b\right )} d^{2}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.49302, size = 277, normalized size = 3.6 \begin{align*} \frac{1}{5} x^{5} e^{2} b B + \frac{1}{2} x^{4} e d b B + \frac{1}{4} x^{4} e^{2} a B + \frac{1}{4} x^{4} e^{2} b A + \frac{1}{3} x^{3} d^{2} b B + \frac{2}{3} x^{3} e d a B + \frac{2}{3} x^{3} e d b A + \frac{1}{3} x^{3} e^{2} a A + \frac{1}{2} x^{2} d^{2} a B + \frac{1}{2} x^{2} d^{2} b A + x^{2} e d a A + x d^{2} a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.082934, size = 116, normalized size = 1.51 \begin{align*} A a d^{2} x + \frac{B b e^{2} x^{5}}{5} + x^{4} \left (\frac{A b e^{2}}{4} + \frac{B a e^{2}}{4} + \frac{B b d e}{2}\right ) + x^{3} \left (\frac{A a e^{2}}{3} + \frac{2 A b d e}{3} + \frac{2 B a d e}{3} + \frac{B b d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac{A b d^{2}}{2} + \frac{B a d^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.7954, size = 153, normalized size = 1.99 \begin{align*} \frac{1}{5} \, B b x^{5} e^{2} + \frac{1}{2} \, B b d x^{4} e + \frac{1}{3} \, B b d^{2} x^{3} + \frac{1}{4} \, B a x^{4} e^{2} + \frac{1}{4} \, A b x^{4} e^{2} + \frac{2}{3} \, B a d x^{3} e + \frac{2}{3} \, A b d x^{3} e + \frac{1}{2} \, B a d^{2} x^{2} + \frac{1}{2} \, A b d^{2} x^{2} + \frac{1}{3} \, A a x^{3} e^{2} + A a d x^{2} e + A a d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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