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]

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

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Rubi [A]  time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.83, size = 113, normalized size = 1.47 \[ \frac {1}{5} x^{5} e^{2} b B + \frac {1}{2} x^{4} e d b B + \frac {1}{4} x^{4} e^{2} a B + \frac {1}{4} x^{4} e^{2} b A + \frac {1}{3} x^{3} d^{2} b B + \frac {2}{3} x^{3} e d a B + \frac {2}{3} x^{3} e d b A + \frac {1}{3} x^{3} e^{2} a A + \frac {1}{2} x^{2} d^{2} a B + \frac {1}{2} x^{2} d^{2} b A + x^{2} e d a A + x d^{2} a A \]

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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giac [A]  time = 1.13, size = 113, normalized size = 1.47 \[ \frac {1}{5} \, B b x^{5} e^{2} + \frac {1}{2} \, B b d x^{4} e + \frac {1}{3} \, B b d^{2} x^{3} + \frac {1}{4} \, B a x^{4} e^{2} + \frac {1}{4} \, A b x^{4} e^{2} + \frac {2}{3} \, B a d x^{3} e + \frac {2}{3} \, A b d x^{3} e + \frac {1}{2} \, B a d^{2} x^{2} + \frac {1}{2} \, A b d^{2} x^{2} + \frac {1}{3} \, A a x^{3} e^{2} + A a d x^{2} e + A a d^{2} x \]

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

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maple [A]  time = 0.00, size = 94, normalized size = 1.22 \[ \frac {B b \,e^{2} x^{5}}{5}+A a \,d^{2} x +\frac {\left (2 B b d e +\left (A b +B a \right ) e^{2}\right ) x^{4}}{4}+\frac {\left (A a \,e^{2}+B b \,d^{2}+2 \left (A b +B a \right ) d e \right ) x^{3}}{3}+\frac {\left (2 A a d e +\left (A b +B a \right ) d^{2}\right ) x^{2}}{2} \]

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 = 0.54, size = 93, normalized size = 1.21 \[ \frac {1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac {1}{4} \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{2} + A a e^{2} + 2 \, {\left (B a + A b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a d e + {\left (B a + A b\right )} d^{2}\right )} x^{2} \]

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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mupad [B]  time = 0.04, size = 98, normalized size = 1.27 \[ x^3\,\left (\frac {A\,a\,e^2}{3}+\frac {B\,b\,d^2}{3}+\frac {2\,A\,b\,d\,e}{3}+\frac {2\,B\,a\,d\,e}{3}\right )+x^2\,\left (\frac {A\,b\,d^2}{2}+\frac {B\,a\,d^2}{2}+A\,a\,d\,e\right )+x^4\,\left (\frac {A\,b\,e^2}{4}+\frac {B\,a\,e^2}{4}+\frac {B\,b\,d\,e}{2}\right )+A\,a\,d^2\,x+\frac {B\,b\,e^2\,x^5}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.08, size = 116, normalized size = 1.51 \[ A a d^{2} x + \frac {B b e^{2} x^{5}}{5} + x^{4} \left (\frac {A b e^{2}}{4} + \frac {B a e^{2}}{4} + \frac {B b d e}{2}\right ) + x^{3} \left (\frac {A a e^{2}}{3} + \frac {2 A b d e}{3} + \frac {2 B a d e}{3} + \frac {B b d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac {A b d^{2}}{2} + \frac {B a d^{2}}{2}\right ) \]

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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