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

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)

_______________________________________________________________________________________

Rubi [A]  time = 0.166841, 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 \[ -\frac{(d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3}+\frac{(d+e x)^3 (b d-a e) (B d-A e)}{3 e^3}+\frac{b B (d+e x)^5}{5 e^3} \]

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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.7773, size = 68, normalized size = 0.88 \[ \frac{B b \left (d + e x\right )^{5}}{5 e^{3}} + \frac{\left (d + e x\right )^{4} \left (A b e + B a e - 2 B b d\right )}{4 e^{3}} + \frac{\left (d + e x\right )^{3} \left (A e - B d\right ) \left (a e - b d\right )}{3 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)*(e*x+d)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0520696, 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

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

Maxima [A]  time = 1.32366, size = 126, normalized size = 1.64 \[ \frac{1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac{1}{4} \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b d^{2} + A a e^{2} + 2 \,{\left (B a + A b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a d e +{\left (B a + A b\right )} d^{2}\right )} x^{2} \]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.178681, size = 1, normalized size = 0.01 \[ \frac{1}{5} x^{5} e^{2} b B + \frac{1}{2} x^{4} e d b B + \frac{1}{4} x^{4} e^{2} a B + \frac{1}{4} x^{4} e^{2} b A + \frac{1}{3} x^{3} d^{2} b B + \frac{2}{3} x^{3} e d a B + \frac{2}{3} x^{3} e d b A + \frac{1}{3} x^{3} e^{2} a A + \frac{1}{2} x^{2} d^{2} a B + \frac{1}{2} x^{2} d^{2} b A + x^{2} e d a A + x d^{2} a A \]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.13114, 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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21537, size = 153, normalized size = 1.99 \[ \frac{1}{5} \, B b x^{5} e^{2} + \frac{1}{2} \, B b d x^{4} e + \frac{1}{3} \, B b d^{2} x^{3} + \frac{1}{4} \, B a x^{4} e^{2} + \frac{1}{4} \, A b x^{4} e^{2} + \frac{2}{3} \, B a d x^{3} e + \frac{2}{3} \, A b d x^{3} e + \frac{1}{2} \, B a d^{2} x^{2} + \frac{1}{2} \, A b d^{2} x^{2} + \frac{1}{3} \, A a x^{3} e^{2} + A a d x^{2} e + A a d^{2} x \]

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