3.2351 \(\int x (d+e x)^4 \left (a+b x+c x^2\right ) \, dx\)

Optimal. Leaf size=103 \[ \frac{(d+e x)^6 \left (3 c d^2-e (2 b d-a e)\right )}{6 e^4}-\frac{d (d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac{(d+e x)^7 (3 c d-b e)}{7 e^4}+\frac{c (d+e x)^8}{8 e^4} \]

[Out]

-(d*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5)/(5*e^4) + ((3*c*d^2 - e*(2*b*d - a*e))*
(d + e*x)^6)/(6*e^4) - ((3*c*d - b*e)*(d + e*x)^7)/(7*e^4) + (c*(d + e*x)^8)/(8*
e^4)

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Rubi [A]  time = 0.297688, antiderivative size = 103, 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 \[ \frac{(d+e x)^6 \left (3 c d^2-e (2 b d-a e)\right )}{6 e^4}-\frac{d (d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac{(d+e x)^7 (3 c d-b e)}{7 e^4}+\frac{c (d+e x)^8}{8 e^4} \]

Antiderivative was successfully verified.

[In]  Int[x*(d + e*x)^4*(a + b*x + c*x^2),x]

[Out]

-(d*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5)/(5*e^4) + ((3*c*d^2 - e*(2*b*d - a*e))*
(d + e*x)^6)/(6*e^4) - ((3*c*d - b*e)*(d + e*x)^7)/(7*e^4) + (c*(d + e*x)^8)/(8*
e^4)

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Rubi in Sympy [A]  time = 30.1, size = 92, normalized size = 0.89 \[ \frac{c \left (d + e x\right )^{8}}{8 e^{4}} - \frac{d \left (d + e x\right )^{5} \left (a e^{2} - b d e + c d^{2}\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{7} \left (b e - 3 c d\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{6} \left (a e^{2} - 2 b d e + 3 c d^{2}\right )}{6 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(e*x+d)**4*(c*x**2+b*x+a),x)

[Out]

c*(d + e*x)**8/(8*e**4) - d*(d + e*x)**5*(a*e**2 - b*d*e + c*d**2)/(5*e**4) + (d
+ e*x)**7*(b*e - 3*c*d)/(7*e**4) + (d + e*x)**6*(a*e**2 - 2*b*d*e + 3*c*d**2)/(
6*e**4)

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Mathematica [A]  time = 0.066183, size = 140, normalized size = 1.36 \[ \frac{1}{6} e^2 x^6 \left (a e^2+4 b d e+6 c d^2\right )+\frac{2}{5} d e x^5 \left (2 a e^2+3 b d e+2 c d^2\right )+\frac{1}{4} d^2 x^4 \left (6 a e^2+4 b d e+c d^2\right )+\frac{1}{3} d^3 x^3 (4 a e+b d)+\frac{1}{2} a d^4 x^2+\frac{1}{7} e^3 x^7 (b e+4 c d)+\frac{1}{8} c e^4 x^8 \]

Antiderivative was successfully verified.

[In]  Integrate[x*(d + e*x)^4*(a + b*x + c*x^2),x]

[Out]

(a*d^4*x^2)/2 + (d^3*(b*d + 4*a*e)*x^3)/3 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^4
)/4 + (2*d*e*(2*c*d^2 + 3*b*d*e + 2*a*e^2)*x^5)/5 + (e^2*(6*c*d^2 + 4*b*d*e + a*
e^2)*x^6)/6 + (e^3*(4*c*d + b*e)*x^7)/7 + (c*e^4*x^8)/8

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Maple [A]  time = 0.003, size = 139, normalized size = 1.4 \[{\frac{{e}^{4}c{x}^{8}}{8}}+{\frac{ \left ({e}^{4}b+4\,d{e}^{3}c \right ){x}^{7}}{7}}+{\frac{ \left ({e}^{4}a+4\,d{e}^{3}b+6\,{d}^{2}{e}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,d{e}^{3}a+6\,{d}^{2}{e}^{2}b+4\,{d}^{3}ec \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}a+4\,{d}^{3}eb+{d}^{4}c \right ){x}^{4}}{4}}+{\frac{ \left ( 4\,{d}^{3}ea+{d}^{4}b \right ){x}^{3}}{3}}+{\frac{{d}^{4}a{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x+d)^4*(c*x^2+b*x+a),x)

[Out]

1/8*e^4*c*x^8+1/7*(b*e^4+4*c*d*e^3)*x^7+1/6*(a*e^4+4*b*d*e^3+6*c*d^2*e^2)*x^6+1/
5*(4*a*d*e^3+6*b*d^2*e^2+4*c*d^3*e)*x^5+1/4*(6*a*d^2*e^2+4*b*d^3*e+c*d^4)*x^4+1/
3*(4*a*d^3*e+b*d^4)*x^3+1/2*d^4*a*x^2

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Maxima [A]  time = 0.70312, size = 186, normalized size = 1.81 \[ \frac{1}{8} \, c e^{4} x^{8} + \frac{1}{7} \,{\left (4 \, c d e^{3} + b e^{4}\right )} x^{7} + \frac{1}{2} \, a d^{4} x^{2} + \frac{1}{6} \,{\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{6} + \frac{2}{5} \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (b d^{4} + 4 \, a d^{3} e\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^4*x,x, algorithm="maxima")

[Out]

1/8*c*e^4*x^8 + 1/7*(4*c*d*e^3 + b*e^4)*x^7 + 1/2*a*d^4*x^2 + 1/6*(6*c*d^2*e^2 +
4*b*d*e^3 + a*e^4)*x^6 + 2/5*(2*c*d^3*e + 3*b*d^2*e^2 + 2*a*d*e^3)*x^5 + 1/4*(c
*d^4 + 4*b*d^3*e + 6*a*d^2*e^2)*x^4 + 1/3*(b*d^4 + 4*a*d^3*e)*x^3

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Fricas [A]  time = 0.23951, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} e^{4} c + \frac{4}{7} x^{7} e^{3} d c + \frac{1}{7} x^{7} e^{4} b + x^{6} e^{2} d^{2} c + \frac{2}{3} x^{6} e^{3} d b + \frac{1}{6} x^{6} e^{4} a + \frac{4}{5} x^{5} e d^{3} c + \frac{6}{5} x^{5} e^{2} d^{2} b + \frac{4}{5} x^{5} e^{3} d a + \frac{1}{4} x^{4} d^{4} c + x^{4} e d^{3} b + \frac{3}{2} x^{4} e^{2} d^{2} a + \frac{1}{3} x^{3} d^{4} b + \frac{4}{3} x^{3} e d^{3} a + \frac{1}{2} x^{2} d^{4} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^4*x,x, algorithm="fricas")

[Out]

1/8*x^8*e^4*c + 4/7*x^7*e^3*d*c + 1/7*x^7*e^4*b + x^6*e^2*d^2*c + 2/3*x^6*e^3*d*
b + 1/6*x^6*e^4*a + 4/5*x^5*e*d^3*c + 6/5*x^5*e^2*d^2*b + 4/5*x^5*e^3*d*a + 1/4*
x^4*d^4*c + x^4*e*d^3*b + 3/2*x^4*e^2*d^2*a + 1/3*x^3*d^4*b + 4/3*x^3*e*d^3*a +
1/2*x^2*d^4*a

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Sympy [A]  time = 0.166879, size = 153, normalized size = 1.49 \[ \frac{a d^{4} x^{2}}{2} + \frac{c e^{4} x^{8}}{8} + x^{7} \left (\frac{b e^{4}}{7} + \frac{4 c d e^{3}}{7}\right ) + x^{6} \left (\frac{a e^{4}}{6} + \frac{2 b d e^{3}}{3} + c d^{2} e^{2}\right ) + x^{5} \left (\frac{4 a d e^{3}}{5} + \frac{6 b d^{2} e^{2}}{5} + \frac{4 c d^{3} e}{5}\right ) + x^{4} \left (\frac{3 a d^{2} e^{2}}{2} + b d^{3} e + \frac{c d^{4}}{4}\right ) + x^{3} \left (\frac{4 a d^{3} e}{3} + \frac{b d^{4}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(e*x+d)**4*(c*x**2+b*x+a),x)

[Out]

a*d**4*x**2/2 + c*e**4*x**8/8 + x**7*(b*e**4/7 + 4*c*d*e**3/7) + x**6*(a*e**4/6
+ 2*b*d*e**3/3 + c*d**2*e**2) + x**5*(4*a*d*e**3/5 + 6*b*d**2*e**2/5 + 4*c*d**3*
e/5) + x**4*(3*a*d**2*e**2/2 + b*d**3*e + c*d**4/4) + x**3*(4*a*d**3*e/3 + b*d**
4/3)

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GIAC/XCAS [A]  time = 0.261396, size = 193, normalized size = 1.87 \[ \frac{1}{8} \, c x^{8} e^{4} + \frac{4}{7} \, c d x^{7} e^{3} + c d^{2} x^{6} e^{2} + \frac{4}{5} \, c d^{3} x^{5} e + \frac{1}{4} \, c d^{4} x^{4} + \frac{1}{7} \, b x^{7} e^{4} + \frac{2}{3} \, b d x^{6} e^{3} + \frac{6}{5} \, b d^{2} x^{5} e^{2} + b d^{3} x^{4} e + \frac{1}{3} \, b d^{4} x^{3} + \frac{1}{6} \, a x^{6} e^{4} + \frac{4}{5} \, a d x^{5} e^{3} + \frac{3}{2} \, a d^{2} x^{4} e^{2} + \frac{4}{3} \, a d^{3} x^{3} e + \frac{1}{2} \, a d^{4} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^4*x,x, algorithm="giac")

[Out]

1/8*c*x^8*e^4 + 4/7*c*d*x^7*e^3 + c*d^2*x^6*e^2 + 4/5*c*d^3*x^5*e + 1/4*c*d^4*x^
4 + 1/7*b*x^7*e^4 + 2/3*b*d*x^6*e^3 + 6/5*b*d^2*x^5*e^2 + b*d^3*x^4*e + 1/3*b*d^
4*x^3 + 1/6*a*x^6*e^4 + 4/5*a*d*x^5*e^3 + 3/2*a*d^2*x^4*e^2 + 4/3*a*d^3*x^3*e +
1/2*a*d^4*x^2