3.2353 \(\int x (d+e x)^2 \left (a+b x+c x^2\right ) \, dx\)

Optimal. Leaf size=78 \[ \frac{1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+\frac{1}{2} a d^2 x^2+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{6} c e^2 x^6 \]

[Out]

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

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Rubi [A]  time = 0.177791, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+\frac{1}{2} a d^2 x^2+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{6} c e^2 x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0485798, size = 70, normalized size = 0.9 \[ \frac{1}{60} x^2 \left (15 x^2 \left (e (a e+2 b d)+c d^2\right )+20 d x (2 a e+b d)+30 a d^2+12 e x^3 (b e+2 c d)+10 c e^2 x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(30*a*d^2 + 20*d*(b*d + 2*a*e)*x + 15*(c*d^2 + e*(2*b*d + a*e))*x^2 + 12*e*
(2*c*d + b*e)*x^3 + 10*c*e^2*x^4))/60

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Maple [A]  time = 0.001, size = 73, normalized size = 0.9 \[{\frac{c{e}^{2}{x}^{6}}{6}}+{\frac{ \left ( b{e}^{2}+2\,cde \right ){x}^{5}}{5}}+{\frac{ \left ({e}^{2}a+2\,bde+c{d}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,dea+{d}^{2}b \right ){x}^{3}}{3}}+{\frac{a{d}^{2}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.694414, size = 97, normalized size = 1.24 \[ \frac{1}{6} \, c e^{2} x^{6} + \frac{1}{5} \,{\left (2 \, c d e + b e^{2}\right )} x^{5} + \frac{1}{2} \, a d^{2} x^{2} + \frac{1}{4} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (b d^{2} + 2 \, a d e\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.241617, size = 1, normalized size = 0.01 \[ \frac{1}{6} x^{6} e^{2} c + \frac{2}{5} x^{5} e d c + \frac{1}{5} x^{5} e^{2} b + \frac{1}{4} x^{4} d^{2} c + \frac{1}{2} x^{4} e d b + \frac{1}{4} x^{4} e^{2} a + \frac{1}{3} x^{3} d^{2} b + \frac{2}{3} x^{3} e d a + \frac{1}{2} x^{2} d^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.125473, size = 80, normalized size = 1.03 \[ \frac{a d^{2} x^{2}}{2} + \frac{c e^{2} x^{6}}{6} + x^{5} \left (\frac{b e^{2}}{5} + \frac{2 c d e}{5}\right ) + x^{4} \left (\frac{a e^{2}}{4} + \frac{b d e}{2} + \frac{c d^{2}}{4}\right ) + x^{3} \left (\frac{2 a d e}{3} + \frac{b d^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.30759, size = 107, normalized size = 1.37 \[ \frac{1}{6} \, c x^{6} e^{2} + \frac{2}{5} \, c d x^{5} e + \frac{1}{4} \, c d^{2} x^{4} + \frac{1}{5} \, b x^{5} e^{2} + \frac{1}{2} \, b d x^{4} e + \frac{1}{3} \, b d^{2} x^{3} + \frac{1}{4} \, a x^{4} e^{2} + \frac{2}{3} \, a d x^{3} e + \frac{1}{2} \, a d^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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