\(\int (d+e x) (a+b x+c x^2)^4 \, dx\) [2149]

Optimal result

Integrand size = 18, antiderivative size = 268 \[ \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx=a^4 d x+\frac {1}{2} a^3 (4 b d+a e) x^2+\frac {2}{3} a^2 \left (3 b^2 d+2 a c d+2 a b e\right ) x^3+\frac {1}{2} a \left (2 b^3 d+6 a b c d+3 a b^2 e+2 a^2 c e\right ) x^4+\frac {1}{5} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 e+12 a^2 b c e\right ) x^5+\frac {1}{6} \left (4 b^3 c d+12 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^6+\frac {2}{7} c \left (3 b^2 c d+2 a c^2 d+2 b^3 e+6 a b c e\right ) x^7+\frac {1}{4} c^2 \left (2 b c d+3 b^2 e+2 a c e\right ) x^8+\frac {1}{9} c^3 (c d+4 b e) x^9+\frac {1}{10} c^4 e x^{10} \]

[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] (verified)

Time = 0.19 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {645} \[ \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx=a^4 d x+\frac {1}{2} a^3 x^2 (a e+4 b d)+\frac {2}{3} a^2 x^3 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac {1}{2} a x^4 \left (2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right )+\frac {1}{6} x^6 \left (6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+b^4 e+4 b^3 c d\right )+\frac {1}{5} x^5 \left (12 a^2 b c e+6 a^2 c^2 d+4 a b^3 e+12 a b^2 c d+b^4 d\right )+\frac {1}{4} c^2 x^8 \left (2 a c e+3 b^2 e+2 b c d\right )+\frac {2}{7} c x^7 \left (6 a b c e+2 a c^2 d+2 b^3 e+3 b^2 c d\right )+\frac {1}{9} c^3 x^9 (4 b e+c d)+\frac {1}{10} c^4 e x^{10} \]

[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 645

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*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx=a^4 d x+\frac {1}{2} a^3 (4 b d+a e) x^2+\frac {2}{3} a^2 \left (3 b^2 d+2 a c d+2 a b e\right ) x^3+\frac {1}{2} a \left (2 b^3 d+6 a b c d+3 a b^2 e+2 a^2 c e\right ) x^4+\frac {1}{5} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 e+12 a^2 b c e\right ) x^5+\frac {1}{6} \left (4 b^3 c d+12 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^6+\frac {2}{7} c \left (3 b^2 c d+2 a c^2 d+2 b^3 e+6 a b c e\right ) x^7+\frac {1}{4} c^2 \left (2 b c d+3 b^2 e+2 a c e\right ) x^8+\frac {1}{9} c^3 (c d+4 b e) x^9+\frac {1}{10} c^4 e x^{10} \]

[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

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {c^{4} e \,x^{10}}{10}+\left (\frac {4}{9} b \,c^{3} e +\frac {1}{9} c^{4} d \right ) x^{9}+\left (\frac {1}{2} a \,c^{3} e +\frac {3}{4} b^{2} c^{2} e +\frac {1}{2} d \,c^{3} b \right ) x^{8}+\left (\frac {12}{7} a b \,c^{2} e +\frac {4}{7} a \,c^{3} d +\frac {4}{7} b^{3} c e +\frac {6}{7} b^{2} c^{2} d \right ) x^{7}+\left (a^{2} c^{2} e +2 a \,b^{2} c e +2 a b \,c^{2} d +\frac {1}{6} b^{4} e +\frac {2}{3} b^{3} d c \right ) x^{6}+\left (\frac {12}{5} a^{2} b c e +\frac {6}{5} a^{2} c^{2} d +\frac {4}{5} e a \,b^{3}+\frac {12}{5} a \,b^{2} c d +\frac {1}{5} d \,b^{4}\right ) x^{5}+\left (a^{3} c e +\frac {3}{2} a^{2} e \,b^{2}+3 a^{2} b c d +a \,b^{3} d \right ) x^{4}+\left (\frac {4}{3} a^{3} b e +\frac {4}{3} a^{3} c d +2 a^{2} b^{2} d \right ) x^{3}+\left (\frac {1}{2} e \,a^{4}+2 d \,a^{3} b \right ) x^{2}+a^{4} d x\)	\(264\)
gosper	\(\frac {6}{7} x^{7} b^{2} c^{2} d +\frac {12}{5} x^{5} a^{2} b c e +\frac {12}{7} x^{7} a b \,c^{2} e +2 x^{6} a b \,c^{2} d +\frac {12}{5} a \,b^{2} c d \,x^{5}+a \,b^{3} d \,x^{4}+2 x^{6} a \,b^{2} c e +\frac {4}{5} a \,b^{3} e \,x^{5}+\frac {1}{5} b^{4} d \,x^{5}+\frac {3}{2} a^{2} b^{2} e \,x^{4}+3 x^{4} a^{2} b c d +\frac {1}{2} a \,c^{3} e \,x^{8}+a^{2} c^{2} e \,x^{6}+a^{3} c e \,x^{4}+a^{4} d x +\frac {1}{6} e \,b^{4} x^{6}+\frac {4}{9} x^{9} b \,c^{3} e +\frac {4}{7} x^{7} b^{3} c e +2 x^{3} a^{2} b^{2} d +\frac {1}{2} x^{8} d \,c^{3} b +\frac {3}{4} x^{8} b^{2} c^{2} e +\frac {4}{3} a^{3} c d \,x^{3}+\frac {2}{3} b^{3} c d \,x^{6}+2 a^{3} b d \,x^{2}+\frac {4}{3} a^{3} b e \,x^{3}+\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a^{4} e \,x^{2}+\frac {4}{7} a \,c^{3} d \,x^{7}+\frac {6}{5} a^{2} c^{2} d \,x^{5}\)	\(308\)
risch	\(\frac {6}{7} x^{7} b^{2} c^{2} d +\frac {12}{5} x^{5} a^{2} b c e +\frac {12}{7} x^{7} a b \,c^{2} e +2 x^{6} a b \,c^{2} d +\frac {12}{5} a \,b^{2} c d \,x^{5}+a \,b^{3} d \,x^{4}+2 x^{6} a \,b^{2} c e +\frac {4}{5} a \,b^{3} e \,x^{5}+\frac {1}{5} b^{4} d \,x^{5}+\frac {3}{2} a^{2} b^{2} e \,x^{4}+3 x^{4} a^{2} b c d +\frac {1}{2} a \,c^{3} e \,x^{8}+a^{2} c^{2} e \,x^{6}+a^{3} c e \,x^{4}+a^{4} d x +\frac {1}{6} e \,b^{4} x^{6}+\frac {4}{9} x^{9} b \,c^{3} e +\frac {4}{7} x^{7} b^{3} c e +2 x^{3} a^{2} b^{2} d +\frac {1}{2} x^{8} d \,c^{3} b +\frac {3}{4} x^{8} b^{2} c^{2} e +\frac {4}{3} a^{3} c d \,x^{3}+\frac {2}{3} b^{3} c d \,x^{6}+2 a^{3} b d \,x^{2}+\frac {4}{3} a^{3} b e \,x^{3}+\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a^{4} e \,x^{2}+\frac {4}{7} a \,c^{3} d \,x^{7}+\frac {6}{5} a^{2} c^{2} d \,x^{5}\)	\(308\)
parallelrisch	\(\frac {6}{7} x^{7} b^{2} c^{2} d +\frac {12}{5} x^{5} a^{2} b c e +\frac {12}{7} x^{7} a b \,c^{2} e +2 x^{6} a b \,c^{2} d +\frac {12}{5} a \,b^{2} c d \,x^{5}+a \,b^{3} d \,x^{4}+2 x^{6} a \,b^{2} c e +\frac {4}{5} a \,b^{3} e \,x^{5}+\frac {1}{5} b^{4} d \,x^{5}+\frac {3}{2} a^{2} b^{2} e \,x^{4}+3 x^{4} a^{2} b c d +\frac {1}{2} a \,c^{3} e \,x^{8}+a^{2} c^{2} e \,x^{6}+a^{3} c e \,x^{4}+a^{4} d x +\frac {1}{6} e \,b^{4} x^{6}+\frac {4}{9} x^{9} b \,c^{3} e +\frac {4}{7} x^{7} b^{3} c e +2 x^{3} a^{2} b^{2} d +\frac {1}{2} x^{8} d \,c^{3} b +\frac {3}{4} x^{8} b^{2} c^{2} e +\frac {4}{3} a^{3} c d \,x^{3}+\frac {2}{3} b^{3} c d \,x^{6}+2 a^{3} b d \,x^{2}+\frac {4}{3} a^{3} b e \,x^{3}+\frac {1}{10} c^{4} e \,x^{10}+\frac {1}{9} c^{4} d \,x^{9}+\frac {1}{2} a^{4} e \,x^{2}+\frac {4}{7} a \,c^{3} d \,x^{7}+\frac {6}{5} a^{2} c^{2} d \,x^{5}\)	\(308\)
default	\(\frac {c^{4} e \,x^{10}}{10}+\frac {\left (4 b \,c^{3} e +c^{4} d \right ) x^{9}}{9}+\frac {\left (4 d \,c^{3} b +e \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )\right ) x^{8}}{8}+\frac {\left (d \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )+e \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )\right ) x^{7}}{7}+\frac {\left (d \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )+e \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )\right ) x^{6}}{6}+\frac {\left (d \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )+e \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )\right ) x^{5}}{5}+\frac {\left (d \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )+e \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )\right ) x^{4}}{4}+\frac {\left (d \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )+4 a^{3} b e \right ) x^{3}}{3}+\frac {\left (e \,a^{4}+4 d \,a^{3} b \right ) x^{2}}{2}+a^{4} d x\)	\(343\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} e x^{10} + \frac {1}{9} \, {\left (c^{4} d + 4 \, b c^{3} e\right )} x^{9} + \frac {1}{4} \, {\left (2 \, b c^{3} d + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e\right )} x^{8} + \frac {2}{7} \, {\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e\right )} x^{6} + a^{4} d x + \frac {1}{5} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e\right )} x^{5} + \frac {1}{2} \, {\left (2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e\right )} x^{4} + \frac {2}{3} \, {\left (2 \, a^{3} b e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \]

[In]

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

[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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.17 \[ \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx=a^{4} d x + \frac {c^{4} e x^{10}}{10} + x^{9} \cdot \left (\frac {4 b c^{3} e}{9} + \frac {c^{4} d}{9}\right ) + x^{8} \left (\frac {a c^{3} e}{2} + \frac {3 b^{2} c^{2} e}{4} + \frac {b c^{3} d}{2}\right ) + x^{7} \cdot \left (\frac {12 a b c^{2} e}{7} + \frac {4 a c^{3} d}{7} + \frac {4 b^{3} c e}{7} + \frac {6 b^{2} c^{2} d}{7}\right ) + x^{6} \left (a^{2} c^{2} e + 2 a b^{2} c e + 2 a b c^{2} d + \frac {b^{4} e}{6} + \frac {2 b^{3} c d}{3}\right ) + x^{5} \cdot \left (\frac {12 a^{2} b c e}{5} + \frac {6 a^{2} c^{2} d}{5} + \frac {4 a b^{3} e}{5} + \frac {12 a b^{2} c d}{5} + \frac {b^{4} d}{5}\right ) + x^{4} \left (a^{3} c e + \frac {3 a^{2} b^{2} e}{2} + 3 a^{2} b c d + a b^{3} d\right ) + x^{3} \cdot \left (\frac {4 a^{3} b e}{3} + \frac {4 a^{3} c d}{3} + 2 a^{2} b^{2} d\right ) + x^{2} \left (\frac {a^{4} e}{2} + 2 a^{3} b d\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} e x^{10} + \frac {1}{9} \, {\left (c^{4} d + 4 \, b c^{3} e\right )} x^{9} + \frac {1}{4} \, {\left (2 \, b c^{3} d + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e\right )} x^{8} + \frac {2}{7} \, {\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e\right )} x^{6} + a^{4} d x + \frac {1}{5} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e\right )} x^{5} + \frac {1}{2} \, {\left (2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e\right )} x^{4} + \frac {2}{3} \, {\left (2 \, a^{3} b e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.15 \[ \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} e x^{10} + \frac {1}{9} \, c^{4} d x^{9} + \frac {4}{9} \, b c^{3} e x^{9} + \frac {1}{2} \, b c^{3} d x^{8} + \frac {3}{4} \, b^{2} c^{2} e x^{8} + \frac {1}{2} \, a c^{3} e x^{8} + \frac {6}{7} \, b^{2} c^{2} d x^{7} + \frac {4}{7} \, a c^{3} d x^{7} + \frac {4}{7} \, b^{3} c e x^{7} + \frac {12}{7} \, a b c^{2} e x^{7} + \frac {2}{3} \, b^{3} c d x^{6} + 2 \, a b c^{2} d x^{6} + \frac {1}{6} \, b^{4} e x^{6} + 2 \, a b^{2} c e x^{6} + a^{2} c^{2} e x^{6} + \frac {1}{5} \, b^{4} d x^{5} + \frac {12}{5} \, a b^{2} c d x^{5} + \frac {6}{5} \, a^{2} c^{2} d x^{5} + \frac {4}{5} \, a b^{3} e x^{5} + \frac {12}{5} \, a^{2} b c e x^{5} + a b^{3} d x^{4} + 3 \, a^{2} b c d x^{4} + \frac {3}{2} \, a^{2} b^{2} e x^{4} + a^{3} c e x^{4} + 2 \, a^{2} b^{2} d x^{3} + \frac {4}{3} \, a^{3} c d x^{3} + \frac {4}{3} \, a^{3} b e x^{3} + 2 \, a^{3} b d x^{2} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.98 \[ \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx=x^2\,\left (\frac {e\,a^4}{2}+2\,b\,d\,a^3\right )+x^9\,\left (\frac {d\,c^4}{9}+\frac {4\,b\,e\,c^3}{9}\right )+x^3\,\left (\frac {4\,e\,a^3\,b}{3}+\frac {4\,c\,d\,a^3}{3}+2\,d\,a^2\,b^2\right )+x^8\,\left (\frac {3\,e\,b^2\,c^2}{4}+\frac {d\,b\,c^3}{2}+\frac {a\,e\,c^3}{2}\right )+x^5\,\left (\frac {12\,e\,a^2\,b\,c}{5}+\frac {6\,d\,a^2\,c^2}{5}+\frac {4\,e\,a\,b^3}{5}+\frac {12\,d\,a\,b^2\,c}{5}+\frac {d\,b^4}{5}\right )+x^6\,\left (e\,a^2\,c^2+2\,e\,a\,b^2\,c+2\,d\,a\,b\,c^2+\frac {e\,b^4}{6}+\frac {2\,d\,b^3\,c}{3}\right )+x^4\,\left (c\,e\,a^3+\frac {3\,e\,a^2\,b^2}{2}+3\,c\,d\,a^2\,b+d\,a\,b^3\right )+x^7\,\left (\frac {4\,e\,b^3\,c}{7}+\frac {6\,d\,b^2\,c^2}{7}+\frac {12\,a\,e\,b\,c^2}{7}+\frac {4\,a\,d\,c^3}{7}\right )+\frac {c^4\,e\,x^{10}}{10}+a^4\,d\,x \]

[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