3.2149 \(\int (d+e x) (a+b x+c x^2)^4 \, dx\)

Optimal. Leaf size=268 \[ a^4 d x+\frac {1}{2} a^3 x^2 (a e+4 b d)+\frac {2}{3} a^2 x^3 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac {1}{2} a x^4 \left (2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right )+\frac {1}{6} x^6 \left (6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+b^4 e+4 b^3 c d\right )+\frac {1}{5} x^5 \left (12 a^2 b c e+6 a^2 c^2 d+4 a b^3 e+12 a b^2 c d+b^4 d\right )+\frac {1}{4} c^2 x^8 \left (2 a c e+3 b^2 e+2 b c d\right )+\frac {2}{7} c x^7 \left (6 a b c e+2 a c^2 d+2 b^3 e+3 b^2 c d\right )+\frac {1}{9} c^3 x^9 (4 b e+c d)+\frac {1}{10} c^4 e x^{10} \]

[Out]

a^4*d*x+1/2*a^3*(a*e+4*b*d)*x^2+2/3*a^2*(2*a*b*e+2*a*c*d+3*b^2*d)*x^3+1/2*a*(2*a^2*c*e+3*a*b^2*e+6*a*b*c*d+2*b
^3*d)*x^4+1/5*(12*a^2*b*c*e+6*a^2*c^2*d+4*a*b^3*e+12*a*b^2*c*d+b^4*d)*x^5+1/6*(6*a^2*c^2*e+12*a*b^2*c*e+12*a*b
*c^2*d+b^4*e+4*b^3*c*d)*x^6+2/7*c*(6*a*b*c*e+2*a*c^2*d+2*b^3*e+3*b^2*c*d)*x^7+1/4*c^2*(2*a*c*e+3*b^2*e+2*b*c*d
)*x^8+1/9*c^3*(4*b*e+c*d)*x^9+1/10*c^4*e*x^10

________________________________________________________________________________________

Rubi [A]  time = 0.29, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {631} \[ \frac {1}{6} x^6 \left (6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+4 b^3 c d+b^4 e\right )+\frac {1}{5} x^5 \left (12 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+4 a b^3 e+b^4 d\right )+\frac {1}{2} a x^4 \left (2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right )+\frac {2}{3} a^2 x^3 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac {1}{2} a^3 x^2 (a e+4 b d)+a^4 d x+\frac {1}{4} c^2 x^8 \left (2 a c e+3 b^2 e+2 b c d\right )+\frac {2}{7} c x^7 \left (6 a b c e+2 a c^2 d+3 b^2 c d+2 b^3 e\right )+\frac {1}{9} c^3 x^9 (4 b e+c d)+\frac {1}{10} c^4 e x^{10} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(a + b*x + c*x^2)^4,x]

[Out]

a^4*d*x + (a^3*(4*b*d + a*e)*x^2)/2 + (2*a^2*(3*b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + (a*(2*b^3*d + 6*a*b*c*d +
3*a*b^2*e + 2*a^2*c*e)*x^4)/2 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*e + 12*a^2*b*c*e)*x^5)/5 + ((4*
b^3*c*d + 12*a*b*c^2*d + b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*x^6)/6 + (2*c*(3*b^2*c*d + 2*a*c^2*d + 2*b^3*e +
6*a*b*c*e)*x^7)/7 + (c^2*(2*b*c*d + 3*b^2*e + 2*a*c*e)*x^8)/4 + (c^3*(c*d + 4*b*e)*x^9)/9 + (c^4*e*x^10)/10

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

\begin {align*} \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx &=\int \left (a^4 d+a^3 (4 b d+a e) x+2 a^2 \left (3 b^2 d+2 a c d+2 a b e\right ) x^2+2 a \left (2 b^3 d+6 a b c d+3 a b^2 e+2 a^2 c e\right ) x^3+\left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 e+12 a^2 b c e\right ) x^4+\left (4 b^3 c d+12 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^5+2 c \left (3 b^2 c d+2 a c^2 d+2 b^3 e+6 a b c e\right ) x^6+2 c^2 \left (2 b c d+3 b^2 e+2 a c e\right ) x^7+c^3 (c d+4 b e) x^8+c^4 e x^9\right ) \, dx\\ &=a^4 d x+\frac {1}{2} a^3 (4 b d+a e) x^2+\frac {2}{3} a^2 \left (3 b^2 d+2 a c d+2 a b e\right ) x^3+\frac {1}{2} a \left (2 b^3 d+6 a b c d+3 a b^2 e+2 a^2 c e\right ) x^4+\frac {1}{5} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 e+12 a^2 b c e\right ) x^5+\frac {1}{6} \left (4 b^3 c d+12 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^6+\frac {2}{7} c \left (3 b^2 c d+2 a c^2 d+2 b^3 e+6 a b c e\right ) x^7+\frac {1}{4} c^2 \left (2 b c d+3 b^2 e+2 a c e\right ) x^8+\frac {1}{9} c^3 (c d+4 b e) x^9+\frac {1}{10} c^4 e x^{10}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 268, normalized size = 1.00 \[ a^4 d x+\frac {1}{2} a^3 x^2 (a e+4 b d)+\frac {2}{3} a^2 x^3 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac {1}{2} a x^4 \left (2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right )+\frac {1}{6} x^6 \left (6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+b^4 e+4 b^3 c d\right )+\frac {1}{5} x^5 \left (12 a^2 b c e+6 a^2 c^2 d+4 a b^3 e+12 a b^2 c d+b^4 d\right )+\frac {1}{4} c^2 x^8 \left (2 a c e+3 b^2 e+2 b c d\right )+\frac {2}{7} c x^7 \left (6 a b c e+2 a c^2 d+2 b^3 e+3 b^2 c d\right )+\frac {1}{9} c^3 x^9 (4 b e+c d)+\frac {1}{10} c^4 e x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^4*d*x + (a^3*(4*b*d + a*e)*x^2)/2 + (2*a^2*(3*b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + (a*(2*b^3*d + 6*a*b*c*d +
3*a*b^2*e + 2*a^2*c*e)*x^4)/2 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*e + 12*a^2*b*c*e)*x^5)/5 + ((4*
b^3*c*d + 12*a*b*c^2*d + b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*x^6)/6 + (2*c*(3*b^2*c*d + 2*a*c^2*d + 2*b^3*e +
6*a*b*c*e)*x^7)/7 + (c^2*(2*b*c*d + 3*b^2*e + 2*a*c*e)*x^8)/4 + (c^3*(c*d + 4*b*e)*x^9)/9 + (c^4*e*x^10)/10

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 307, normalized size = 1.15 \[ \frac {1}{10} x^{10} e c^{4} + \frac {1}{9} x^{9} d c^{4} + \frac {4}{9} x^{9} e c^{3} b + \frac {1}{2} x^{8} d c^{3} b + \frac {3}{4} x^{8} e c^{2} b^{2} + \frac {1}{2} x^{8} e c^{3} a + \frac {6}{7} x^{7} d c^{2} b^{2} + \frac {4}{7} x^{7} e c b^{3} + \frac {4}{7} x^{7} d c^{3} a + \frac {12}{7} x^{7} e c^{2} b a + \frac {2}{3} x^{6} d c b^{3} + \frac {1}{6} x^{6} e b^{4} + 2 x^{6} d c^{2} b a + 2 x^{6} e c b^{2} a + x^{6} e c^{2} a^{2} + \frac {1}{5} x^{5} d b^{4} + \frac {12}{5} x^{5} d c b^{2} a + \frac {4}{5} x^{5} e b^{3} a + \frac {6}{5} x^{5} d c^{2} a^{2} + \frac {12}{5} x^{5} e c b a^{2} + x^{4} d b^{3} a + 3 x^{4} d c b a^{2} + \frac {3}{2} x^{4} e b^{2} a^{2} + x^{4} e c a^{3} + 2 x^{3} d b^{2} a^{2} + \frac {4}{3} x^{3} d c a^{3} + \frac {4}{3} x^{3} e b a^{3} + 2 x^{2} d b a^{3} + \frac {1}{2} x^{2} e a^{4} + x d a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*e*c^4 + 1/9*x^9*d*c^4 + 4/9*x^9*e*c^3*b + 1/2*x^8*d*c^3*b + 3/4*x^8*e*c^2*b^2 + 1/2*x^8*e*c^3*a + 6/
7*x^7*d*c^2*b^2 + 4/7*x^7*e*c*b^3 + 4/7*x^7*d*c^3*a + 12/7*x^7*e*c^2*b*a + 2/3*x^6*d*c*b^3 + 1/6*x^6*e*b^4 + 2
*x^6*d*c^2*b*a + 2*x^6*e*c*b^2*a + x^6*e*c^2*a^2 + 1/5*x^5*d*b^4 + 12/5*x^5*d*c*b^2*a + 4/5*x^5*e*b^3*a + 6/5*
x^5*d*c^2*a^2 + 12/5*x^5*e*c*b*a^2 + x^4*d*b^3*a + 3*x^4*d*c*b*a^2 + 3/2*x^4*e*b^2*a^2 + x^4*e*c*a^3 + 2*x^3*d
*b^2*a^2 + 4/3*x^3*d*c*a^3 + 4/3*x^3*e*b*a^3 + 2*x^2*d*b*a^3 + 1/2*x^2*e*a^4 + x*d*a^4

________________________________________________________________________________________

giac [A]  time = 0.17, size = 322, normalized size = 1.20 \[ \frac {1}{10} \, c^{4} x^{10} e + \frac {1}{9} \, c^{4} d x^{9} + \frac {4}{9} \, b c^{3} x^{9} e + \frac {1}{2} \, b c^{3} d x^{8} + \frac {3}{4} \, b^{2} c^{2} x^{8} e + \frac {1}{2} \, a c^{3} x^{8} e + \frac {6}{7} \, b^{2} c^{2} d x^{7} + \frac {4}{7} \, a c^{3} d x^{7} + \frac {4}{7} \, b^{3} c x^{7} e + \frac {12}{7} \, a b c^{2} x^{7} e + \frac {2}{3} \, b^{3} c d x^{6} + 2 \, a b c^{2} d x^{6} + \frac {1}{6} \, b^{4} x^{6} e + 2 \, a b^{2} c x^{6} e + a^{2} c^{2} x^{6} e + \frac {1}{5} \, b^{4} d x^{5} + \frac {12}{5} \, a b^{2} c d x^{5} + \frac {6}{5} \, a^{2} c^{2} d x^{5} + \frac {4}{5} \, a b^{3} x^{5} e + \frac {12}{5} \, a^{2} b c x^{5} e + a b^{3} d x^{4} + 3 \, a^{2} b c d x^{4} + \frac {3}{2} \, a^{2} b^{2} x^{4} e + a^{3} c x^{4} e + 2 \, a^{2} b^{2} d x^{3} + \frac {4}{3} \, a^{3} c d x^{3} + \frac {4}{3} \, a^{3} b x^{3} e + 2 \, a^{3} b d x^{2} + \frac {1}{2} \, a^{4} x^{2} e + a^{4} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^4*x^10*e + 1/9*c^4*d*x^9 + 4/9*b*c^3*x^9*e + 1/2*b*c^3*d*x^8 + 3/4*b^2*c^2*x^8*e + 1/2*a*c^3*x^8*e + 6/
7*b^2*c^2*d*x^7 + 4/7*a*c^3*d*x^7 + 4/7*b^3*c*x^7*e + 12/7*a*b*c^2*x^7*e + 2/3*b^3*c*d*x^6 + 2*a*b*c^2*d*x^6 +
1/6*b^4*x^6*e + 2*a*b^2*c*x^6*e + a^2*c^2*x^6*e + 1/5*b^4*d*x^5 + 12/5*a*b^2*c*d*x^5 + 6/5*a^2*c^2*d*x^5 + 4/
5*a*b^3*x^5*e + 12/5*a^2*b*c*x^5*e + a*b^3*d*x^4 + 3*a^2*b*c*d*x^4 + 3/2*a^2*b^2*x^4*e + a^3*c*x^4*e + 2*a^2*b
^2*d*x^3 + 4/3*a^3*c*d*x^3 + 4/3*a^3*b*x^3*e + 2*a^3*b*d*x^2 + 1/2*a^4*x^2*e + a^4*d*x

________________________________________________________________________________________

maple [A]  time = 0.04, size = 343, normalized size = 1.28 \[ \frac {c^{4} e \,x^{10}}{10}+\frac {\left (4 e b \,c^{3}+d \,c^{4}\right ) x^{9}}{9}+\frac {\left (4 b \,c^{3} d +\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) e \right ) x^{8}}{8}+\frac {\left (\left (4 b^{2} c^{2}+2 \left (2 a c +b^{2}\right ) c^{2}\right ) d +\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) e \right ) x^{7}}{7}+a^{4} d x +\frac {\left (\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) d +\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) e \right ) x^{6}}{6}+\frac {\left (\left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right ) d +\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) e \right ) x^{5}}{5}+\frac {\left (\left (4 a^{2} b c +4 \left (2 a c +b^{2}\right ) a b \right ) d +\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) e \right ) x^{4}}{4}+\frac {\left (4 a^{3} b e +\left (4 a^{2} b^{2}+2 \left (2 a c +b^{2}\right ) a^{2}\right ) d \right ) x^{3}}{3}+\frac {\left (e \,a^{4}+4 d \,a^{3} b \right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^4*e*x^10+1/9*(4*b*c^3*e+c^4*d)*x^9+1/8*(4*c^3*d*b+e*(4*b^2*c^2+2*(2*a*c+b^2)*c^2))*x^8+1/7*(d*(4*b^2*c^
2+2*(2*a*c+b^2)*c^2)+e*(4*a*b*c^2+4*(2*a*c+b^2)*b*c))*x^7+1/6*(d*(4*a*b*c^2+4*(2*a*c+b^2)*b*c)+e*(2*a^2*c^2+8*
a*b^2*c+(2*a*c+b^2)^2))*x^6+1/5*(d*(2*a^2*c^2+8*a*b^2*c+(2*a*c+b^2)^2)+e*(4*a^2*b*c+4*(2*a*c+b^2)*a*b))*x^5+1/
4*(d*(4*a^2*b*c+4*(2*a*c+b^2)*a*b)+e*(4*a^2*b^2+2*(2*a*c+b^2)*a^2))*x^4+1/3*(d*(4*a^2*b^2+2*(2*a*c+b^2)*a^2)+4
*e*a^3*b)*x^3+1/2*(a^4*e+4*a^3*b*d)*x^2+a^4*d*x

________________________________________________________________________________________

maxima [A]  time = 1.12, size = 276, normalized size = 1.03 \[ \frac {1}{10} \, c^{4} e x^{10} + \frac {1}{9} \, {\left (c^{4} d + 4 \, b c^{3} e\right )} x^{9} + \frac {1}{4} \, {\left (2 \, b c^{3} d + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e\right )} x^{8} + \frac {2}{7} \, {\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e\right )} x^{6} + a^{4} d x + \frac {1}{5} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e\right )} x^{5} + \frac {1}{2} \, {\left (2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e\right )} x^{4} + \frac {2}{3} \, {\left (2 \, a^{3} b e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^4*e*x^10 + 1/9*(c^4*d + 4*b*c^3*e)*x^9 + 1/4*(2*b*c^3*d + (3*b^2*c^2 + 2*a*c^3)*e)*x^8 + 2/7*((3*b^2*c^
2 + 2*a*c^3)*d + 2*(b^3*c + 3*a*b*c^2)*e)*x^7 + 1/6*(4*(b^3*c + 3*a*b*c^2)*d + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*
e)*x^6 + a^4*d*x + 1/5*((b^4 + 12*a*b^2*c + 6*a^2*c^2)*d + 4*(a*b^3 + 3*a^2*b*c)*e)*x^5 + 1/2*(2*(a*b^3 + 3*a^
2*b*c)*d + (3*a^2*b^2 + 2*a^3*c)*e)*x^4 + 2/3*(2*a^3*b*e + (3*a^2*b^2 + 2*a^3*c)*d)*x^3 + 1/2*(4*a^3*b*d + a^4
*e)*x^2

________________________________________________________________________________________

mupad [B]  time = 0.12, size = 263, normalized size = 0.98 \[ x^2\,\left (\frac {e\,a^4}{2}+2\,b\,d\,a^3\right )+x^9\,\left (\frac {d\,c^4}{9}+\frac {4\,b\,e\,c^3}{9}\right )+x^3\,\left (\frac {4\,e\,a^3\,b}{3}+\frac {4\,c\,d\,a^3}{3}+2\,d\,a^2\,b^2\right )+x^8\,\left (\frac {3\,e\,b^2\,c^2}{4}+\frac {d\,b\,c^3}{2}+\frac {a\,e\,c^3}{2}\right )+x^5\,\left (\frac {12\,e\,a^2\,b\,c}{5}+\frac {6\,d\,a^2\,c^2}{5}+\frac {4\,e\,a\,b^3}{5}+\frac {12\,d\,a\,b^2\,c}{5}+\frac {d\,b^4}{5}\right )+x^6\,\left (e\,a^2\,c^2+2\,e\,a\,b^2\,c+2\,d\,a\,b\,c^2+\frac {e\,b^4}{6}+\frac {2\,d\,b^3\,c}{3}\right )+x^4\,\left (c\,e\,a^3+\frac {3\,e\,a^2\,b^2}{2}+3\,c\,d\,a^2\,b+d\,a\,b^3\right )+x^7\,\left (\frac {4\,e\,b^3\,c}{7}+\frac {6\,d\,b^2\,c^2}{7}+\frac {12\,a\,e\,b\,c^2}{7}+\frac {4\,a\,d\,c^3}{7}\right )+\frac {c^4\,e\,x^{10}}{10}+a^4\,d\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)*(a + b*x + c*x^2)^4,x)

[Out]

x^2*((a^4*e)/2 + 2*a^3*b*d) + x^9*((c^4*d)/9 + (4*b*c^3*e)/9) + x^3*(2*a^2*b^2*d + (4*a^3*b*e)/3 + (4*a^3*c*d)
/3) + x^8*((3*b^2*c^2*e)/4 + (a*c^3*e)/2 + (b*c^3*d)/2) + x^5*((b^4*d)/5 + (6*a^2*c^2*d)/5 + (4*a*b^3*e)/5 + (
12*a*b^2*c*d)/5 + (12*a^2*b*c*e)/5) + x^6*((b^4*e)/6 + a^2*c^2*e + (2*b^3*c*d)/3 + 2*a*b*c^2*d + 2*a*b^2*c*e)
+ x^4*((3*a^2*b^2*e)/2 + a*b^3*d + a^3*c*e + 3*a^2*b*c*d) + x^7*((6*b^2*c^2*d)/7 + (4*a*c^3*d)/7 + (4*b^3*c*e)
/7 + (12*a*b*c^2*e)/7) + (c^4*e*x^10)/10 + a^4*d*x

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 313, normalized size = 1.17 \[ a^{4} d x + \frac {c^{4} e x^{10}}{10} + x^{9} \left (\frac {4 b c^{3} e}{9} + \frac {c^{4} d}{9}\right ) + x^{8} \left (\frac {a c^{3} e}{2} + \frac {3 b^{2} c^{2} e}{4} + \frac {b c^{3} d}{2}\right ) + x^{7} \left (\frac {12 a b c^{2} e}{7} + \frac {4 a c^{3} d}{7} + \frac {4 b^{3} c e}{7} + \frac {6 b^{2} c^{2} d}{7}\right ) + x^{6} \left (a^{2} c^{2} e + 2 a b^{2} c e + 2 a b c^{2} d + \frac {b^{4} e}{6} + \frac {2 b^{3} c d}{3}\right ) + x^{5} \left (\frac {12 a^{2} b c e}{5} + \frac {6 a^{2} c^{2} d}{5} + \frac {4 a b^{3} e}{5} + \frac {12 a b^{2} c d}{5} + \frac {b^{4} d}{5}\right ) + x^{4} \left (a^{3} c e + \frac {3 a^{2} b^{2} e}{2} + 3 a^{2} b c d + a b^{3} d\right ) + x^{3} \left (\frac {4 a^{3} b e}{3} + \frac {4 a^{3} c d}{3} + 2 a^{2} b^{2} d\right ) + x^{2} \left (\frac {a^{4} e}{2} + 2 a^{3} b d\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*d*x + c**4*e*x**10/10 + x**9*(4*b*c**3*e/9 + c**4*d/9) + x**8*(a*c**3*e/2 + 3*b**2*c**2*e/4 + b*c**3*d/2)
+ x**7*(12*a*b*c**2*e/7 + 4*a*c**3*d/7 + 4*b**3*c*e/7 + 6*b**2*c**2*d/7) + x**6*(a**2*c**2*e + 2*a*b**2*c*e +
2*a*b*c**2*d + b**4*e/6 + 2*b**3*c*d/3) + x**5*(12*a**2*b*c*e/5 + 6*a**2*c**2*d/5 + 4*a*b**3*e/5 + 12*a*b**2*
c*d/5 + b**4*d/5) + x**4*(a**3*c*e + 3*a**2*b**2*e/2 + 3*a**2*b*c*d + a*b**3*d) + x**3*(4*a**3*b*e/3 + 4*a**3*
c*d/3 + 2*a**2*b**2*d) + x**2*(a**4*e/2 + 2*a**3*b*d)

________________________________________________________________________________________