3.2110 \(\int (d+e x) \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=96 \[ a^2 d x+\frac{1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6 \]

[Out]

a^2*d*x + (a*(2*b*d + a*e)*x^2)/2 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + ((2*b*
c*d + b^2*e + 2*a*c*e)*x^4)/4 + (c*(c*d + 2*b*e)*x^5)/5 + (c^2*e*x^6)/6

_______________________________________________________________________________________

Rubi [A]  time = 0.210046, antiderivative size = 96, 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 \[ a^2 d x+\frac{1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6 \]

Antiderivative was successfully verified.

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

[Out]

a^2*d*x + (a*(2*b*d + a*e)*x^2)/2 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + ((2*b*
c*d + b^2*e + 2*a*c*e)*x^4)/4 + (c*(c*d + 2*b*e)*x^5)/5 + (c^2*e*x^6)/6

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} \int d\, dx + a \left (a e + 2 b d\right ) \int x\, dx + \frac{c^{2} e x^{6}}{6} + \frac{c x^{5} \left (2 b e + c d\right )}{5} + x^{4} \left (\frac{a c e}{2} + \frac{b^{2} e}{4} + \frac{b c d}{2}\right ) + x^{3} \left (\frac{2 a b e}{3} + \frac{2 a c d}{3} + \frac{b^{2} d}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*Integral(d, x) + a*(a*e + 2*b*d)*Integral(x, x) + c**2*e*x**6/6 + c*x**5*(2
*b*e + c*d)/5 + x**4*(a*c*e/2 + b**2*e/4 + b*c*d/2) + x**3*(2*a*b*e/3 + 2*a*c*d/
3 + b**2*d/3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.039563, size = 96, normalized size = 1. \[ a^2 d x+\frac{1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{2} a x^2 (a e+2 b d)+\frac{1}{5} c x^5 (2 b e+c d)+\frac{1}{6} c^2 e x^6 \]

Antiderivative was successfully verified.

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

[Out]

a^2*d*x + (a*(2*b*d + a*e)*x^2)/2 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + ((2*b*
c*d + b^2*e + 2*a*c*e)*x^4)/4 + (c*(c*d + 2*b*e)*x^5)/5 + (c^2*e*x^6)/6

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 91, normalized size = 1. \[{\frac{{c}^{2}e{x}^{6}}{6}}+{\frac{ \left ( 2\,bce+{c}^{2}d \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,bcd+e \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( 2\,ac+{b}^{2} \right ) +2\,bea \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{2}+2\,bda \right ){x}^{2}}{2}}+{a}^{2}dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x+a)^2,x)

[Out]

1/6*c^2*e*x^6+1/5*(2*b*c*e+c^2*d)*x^5+1/4*(2*b*c*d+e*(2*a*c+b^2))*x^4+1/3*(d*(2*
a*c+b^2)+2*b*e*a)*x^3+1/2*(a^2*e+2*a*b*d)*x^2+a^2*d*x

_______________________________________________________________________________________

Maxima [A]  time = 0.810993, size = 122, normalized size = 1.27 \[ \frac{1}{6} \, c^{2} e x^{6} + \frac{1}{5} \,{\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac{1}{4} \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} x^{4} + a^{2} d x + \frac{1}{3} \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d + a^{2} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*c^2*e*x^6 + 1/5*(c^2*d + 2*b*c*e)*x^5 + 1/4*(2*b*c*d + (b^2 + 2*a*c)*e)*x^4
+ a^2*d*x + 1/3*(2*a*b*e + (b^2 + 2*a*c)*d)*x^3 + 1/2*(2*a*b*d + a^2*e)*x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.182866, size = 1, normalized size = 0.01 \[ \frac{1}{6} x^{6} e c^{2} + \frac{1}{5} x^{5} d c^{2} + \frac{2}{5} x^{5} e c b + \frac{1}{2} x^{4} d c b + \frac{1}{4} x^{4} e b^{2} + \frac{1}{2} x^{4} e c a + \frac{1}{3} x^{3} d b^{2} + \frac{2}{3} x^{3} d c a + \frac{2}{3} x^{3} e b a + x^{2} d b a + \frac{1}{2} x^{2} e a^{2} + x d a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^6*e*c^2 + 1/5*x^5*d*c^2 + 2/5*x^5*e*c*b + 1/2*x^4*d*c*b + 1/4*x^4*e*b^2 +
1/2*x^4*e*c*a + 1/3*x^3*d*b^2 + 2/3*x^3*d*c*a + 2/3*x^3*e*b*a + x^2*d*b*a + 1/2*
x^2*e*a^2 + x*d*a^2

_______________________________________________________________________________________

Sympy [A]  time = 0.15036, size = 100, normalized size = 1.04 \[ a^{2} d x + \frac{c^{2} e x^{6}}{6} + x^{5} \left (\frac{2 b c e}{5} + \frac{c^{2} d}{5}\right ) + x^{4} \left (\frac{a c e}{2} + \frac{b^{2} e}{4} + \frac{b c d}{2}\right ) + x^{3} \left (\frac{2 a b e}{3} + \frac{2 a c d}{3} + \frac{b^{2} d}{3}\right ) + x^{2} \left (\frac{a^{2} e}{2} + a b d\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*d*x + c**2*e*x**6/6 + x**5*(2*b*c*e/5 + c**2*d/5) + x**4*(a*c*e/2 + b**2*e/
4 + b*c*d/2) + x**3*(2*a*b*e/3 + 2*a*c*d/3 + b**2*d/3) + x**2*(a**2*e/2 + a*b*d)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.202601, size = 142, normalized size = 1.48 \[ \frac{1}{6} \, c^{2} x^{6} e + \frac{1}{5} \, c^{2} d x^{5} + \frac{2}{5} \, b c x^{5} e + \frac{1}{2} \, b c d x^{4} + \frac{1}{4} \, b^{2} x^{4} e + \frac{1}{2} \, a c x^{4} e + \frac{1}{3} \, b^{2} d x^{3} + \frac{2}{3} \, a c d x^{3} + \frac{2}{3} \, a b x^{3} e + a b d x^{2} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*c^2*x^6*e + 1/5*c^2*d*x^5 + 2/5*b*c*x^5*e + 1/2*b*c*d*x^4 + 1/4*b^2*x^4*e +
1/2*a*c*x^4*e + 1/3*b^2*d*x^3 + 2/3*a*c*d*x^3 + 2/3*a*b*x^3*e + a*b*d*x^2 + 1/2*
a^2*x^2*e + a^2*d*x