3.223 \(\int (d+e x)^2 \left (b x+c x^2\right ) \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{4} e x^4 (b e+2 c d)+\frac{1}{3} d x^3 (2 b e+c d)+\frac{1}{2} b d^2 x^2+\frac{1}{5} c e^2 x^5 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.113674, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{1}{4} e x^4 (b e+2 c d)+\frac{1}{3} d x^3 (2 b e+c d)+\frac{1}{2} b d^2 x^2+\frac{1}{5} c e^2 x^5 \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2*(b*x + c*x^2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ b d^{2} \int x\, dx + \frac{c e^{2} x^{5}}{5} + \frac{d x^{3} \left (2 b e + c d\right )}{3} + \frac{e x^{4} \left (b e + 2 c d\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2*(c*x**2+b*x),x)

[Out]

b*d**2*Integral(x, x) + c*e**2*x**5/5 + d*x**3*(2*b*e + c*d)/3 + e*x**4*(b*e + 2
*c*d)/4

_______________________________________________________________________________________

Mathematica [A]  time = 0.0190143, size = 49, normalized size = 0.89 \[ \frac{1}{60} x^2 \left (15 e x^2 (b e+2 c d)+20 d x (2 b e+c d)+30 b d^2+12 c e^2 x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2*(b*x + c*x^2),x]

[Out]

(x^2*(30*b*d^2 + 20*d*(c*d + 2*b*e)*x + 15*e*(2*c*d + b*e)*x^2 + 12*c*e^2*x^3))/
60

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 52, normalized size = 1. \[{\frac{c{e}^{2}{x}^{5}}{5}}+{\frac{ \left ({e}^{2}b+2\,dec \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,bde+c{d}^{2} \right ){x}^{3}}{3}}+{\frac{b{d}^{2}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2*(c*x^2+b*x),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.716565, size = 69, normalized size = 1.25 \[ \frac{1}{5} \, c e^{2} x^{5} + \frac{1}{2} \, b d^{2} x^{2} + \frac{1}{4} \,{\left (2 \, c d e + b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{2} + 2 \, b d e\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(e*x + d)^2,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.215561, size = 1, normalized size = 0.02 \[ \frac{1}{5} x^{5} e^{2} c + \frac{1}{2} x^{4} e d c + \frac{1}{4} x^{4} e^{2} b + \frac{1}{3} x^{3} d^{2} c + \frac{2}{3} x^{3} e d b + \frac{1}{2} x^{2} d^{2} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(e*x + d)^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.112424, size = 54, normalized size = 0.98 \[ \frac{b d^{2} x^{2}}{2} + \frac{c e^{2} x^{5}}{5} + x^{4} \left (\frac{b e^{2}}{4} + \frac{c d e}{2}\right ) + x^{3} \left (\frac{2 b d e}{3} + \frac{c d^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2*(c*x**2+b*x),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.206213, size = 72, normalized size = 1.31 \[ \frac{1}{5} \, c x^{5} e^{2} + \frac{1}{2} \, c d x^{4} e + \frac{1}{3} \, c d^{2} x^{3} + \frac{1}{4} \, b x^{4} e^{2} + \frac{2}{3} \, b d x^{3} e + \frac{1}{2} \, b d^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(e*x + d)^2,x, algorithm="giac")

[Out]

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