\(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx\) [94]

Optimal result

Integrand size = 33, antiderivative size = 111 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^5}{5 e^4}+\frac {c d \left (c d^2-a e^2\right )^2 (d+e x)^6}{2 e^4}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^7}{7 e^4}+\frac {c^3 d^3 (d+e x)^8}{8 e^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.90 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{280} x \left (56 a^3 e^3 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+28 a^2 c d e^2 x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+8 a c^2 d^2 e x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+c^3 d^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )\right ) \] Input:

Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
 

Output:

(x*(56*a^3*e^3*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^ 
4) + 28*a^2*c*d*e^2*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 
 + 5*e^4*x^4) + 8*a*c^2*d^2*e*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 
+ 70*d*e^3*x^3 + 15*e^4*x^4) + c^3*d^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2 
*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4)))/280
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1121, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3 \, dx\)

\(\Big \downarrow \) 1121

\(\displaystyle \int \left (-\frac {3 c^2 d^2 (d+e x)^6 \left (c d^2-a e^2\right )}{e^3}+\frac {3 c d (d+e x)^5 \left (c d^2-a e^2\right )^2}{e^3}+\frac {(d+e x)^4 \left (a e^2-c d^2\right )^3}{e^3}+\frac {c^3 d^3 (d+e x)^7}{e^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 c^2 d^2 (d+e x)^7 \left (c d^2-a e^2\right )}{7 e^4}+\frac {c d (d+e x)^6 \left (c d^2-a e^2\right )^2}{2 e^4}-\frac {(d+e x)^5 \left (c d^2-a e^2\right )^3}{5 e^4}+\frac {c^3 d^3 (d+e x)^8}{8 e^4}\)

Input:

Int[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
 

Output:

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

Defintions of rubi rules used

rule 1121

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(247\) vs. \(2(103)=206\).

Time = 1.34 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.23

method	result	size
norman	\(\frac {d^{3} c^{3} e^{4} x^{8}}{8}+\left (\frac {3}{7} a \,e^{5} c^{2} d^{2}+\frac {4}{7} d^{4} c^{3} e^{3}\right ) x^{7}+\left (\frac {1}{2} a^{2} e^{6} c d +2 a \,e^{4} c^{2} d^{3}+d^{5} c^{3} e^{2}\right ) x^{6}+\left (\frac {1}{5} a^{3} e^{7}+\frac {12}{5} a^{2} e^{5} c \,d^{2}+\frac {18}{5} a \,e^{3} c^{2} d^{4}+\frac {4}{5} d^{6} c^{3} e \right ) x^{5}+\left (e^{6} a^{3} d +\frac {9}{2} a^{2} c \,d^{3} e^{4}+3 a \,c^{2} d^{5} e^{2}+\frac {1}{4} c^{3} d^{7}\right ) x^{4}+\left (2 e^{5} a^{3} d^{2}+4 a^{2} e^{3} c \,d^{4}+a e \,c^{2} d^{6}\right ) x^{3}+\left (2 e^{4} a^{3} d^{3}+\frac {3}{2} a^{2} e^{2} c \,d^{5}\right ) x^{2}+e^{3} a^{3} d^{4} x\)	\(248\)
risch	\(\frac {1}{8} d^{3} c^{3} e^{4} x^{8}+\frac {3}{7} x^{7} a \,e^{5} c^{2} d^{2}+\frac {4}{7} x^{7} d^{4} c^{3} e^{3}+\frac {1}{2} x^{6} a^{2} e^{6} c d +2 x^{6} a \,e^{4} c^{2} d^{3}+x^{6} d^{5} c^{3} e^{2}+\frac {1}{5} x^{5} a^{3} e^{7}+\frac {12}{5} x^{5} a^{2} e^{5} c \,d^{2}+\frac {18}{5} x^{5} a \,e^{3} c^{2} d^{4}+\frac {4}{5} x^{5} d^{6} c^{3} e +x^{4} e^{6} a^{3} d +\frac {9}{2} x^{4} a^{2} c \,d^{3} e^{4}+3 x^{4} a \,c^{2} d^{5} e^{2}+\frac {1}{4} x^{4} c^{3} d^{7}+2 a^{3} d^{2} e^{5} x^{3}+4 a^{2} c \,d^{4} e^{3} x^{3}+a \,c^{2} d^{6} e \,x^{3}+2 x^{2} e^{4} a^{3} d^{3}+\frac {3}{2} x^{2} a^{2} e^{2} c \,d^{5}+e^{3} a^{3} d^{4} x\)	\(272\)
parallelrisch	\(\frac {1}{8} d^{3} c^{3} e^{4} x^{8}+\frac {3}{7} x^{7} a \,e^{5} c^{2} d^{2}+\frac {4}{7} x^{7} d^{4} c^{3} e^{3}+\frac {1}{2} x^{6} a^{2} e^{6} c d +2 x^{6} a \,e^{4} c^{2} d^{3}+x^{6} d^{5} c^{3} e^{2}+\frac {1}{5} x^{5} a^{3} e^{7}+\frac {12}{5} x^{5} a^{2} e^{5} c \,d^{2}+\frac {18}{5} x^{5} a \,e^{3} c^{2} d^{4}+\frac {4}{5} x^{5} d^{6} c^{3} e +x^{4} e^{6} a^{3} d +\frac {9}{2} x^{4} a^{2} c \,d^{3} e^{4}+3 x^{4} a \,c^{2} d^{5} e^{2}+\frac {1}{4} x^{4} c^{3} d^{7}+2 a^{3} d^{2} e^{5} x^{3}+4 a^{2} c \,d^{4} e^{3} x^{3}+a \,c^{2} d^{6} e \,x^{3}+2 x^{2} e^{4} a^{3} d^{3}+\frac {3}{2} x^{2} a^{2} e^{2} c \,d^{5}+e^{3} a^{3} d^{4} x\)	\(272\)
gosper	\(\frac {x \left (35 d^{3} c^{3} e^{4} x^{7}+120 x^{6} a \,e^{5} c^{2} d^{2}+160 x^{6} d^{4} c^{3} e^{3}+140 x^{5} a^{2} e^{6} c d +560 x^{5} a \,e^{4} c^{2} d^{3}+280 x^{5} d^{5} c^{3} e^{2}+56 x^{4} a^{3} e^{7}+672 x^{4} a^{2} e^{5} c \,d^{2}+1008 x^{4} a \,e^{3} c^{2} d^{4}+224 x^{4} d^{6} c^{3} e +280 x^{3} e^{6} a^{3} d +1260 x^{3} a^{2} c \,d^{3} e^{4}+840 x^{3} a \,c^{2} d^{5} e^{2}+70 x^{3} c^{3} d^{7}+560 a^{3} d^{2} e^{5} x^{2}+1120 a^{2} c \,d^{4} e^{3} x^{2}+280 a \,c^{2} d^{6} e \,x^{2}+560 x \,e^{4} a^{3} d^{3}+420 x \,a^{2} e^{2} c \,d^{5}+280 e^{3} a^{3} d^{4}\right )}{280}\)	\(274\)
orering	\(\frac {x \left (35 d^{3} c^{3} e^{4} x^{7}+120 x^{6} a \,e^{5} c^{2} d^{2}+160 x^{6} d^{4} c^{3} e^{3}+140 x^{5} a^{2} e^{6} c d +560 x^{5} a \,e^{4} c^{2} d^{3}+280 x^{5} d^{5} c^{3} e^{2}+56 x^{4} a^{3} e^{7}+672 x^{4} a^{2} e^{5} c \,d^{2}+1008 x^{4} a \,e^{3} c^{2} d^{4}+224 x^{4} d^{6} c^{3} e +280 x^{3} e^{6} a^{3} d +1260 x^{3} a^{2} c \,d^{3} e^{4}+840 x^{3} a \,c^{2} d^{5} e^{2}+70 x^{3} c^{3} d^{7}+560 a^{3} d^{2} e^{5} x^{2}+1120 a^{2} c \,d^{4} e^{3} x^{2}+280 a \,c^{2} d^{6} e \,x^{2}+560 x \,e^{4} a^{3} d^{3}+420 x \,a^{2} e^{2} c \,d^{5}+280 e^{3} a^{3} d^{4}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{3}}{280 \left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}\)	\(318\)
default	\(\frac {d^{3} c^{3} e^{4} x^{8}}{8}+\frac {\left (d^{4} c^{3} e^{3}+3 e^{3} \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d^{2}\right ) x^{7}}{7}+\frac {\left (3 d^{3} \left (a \,e^{2}+c \,d^{2}\right ) e^{2} c^{2}+e \left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} d e c +d e c \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (d \left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} d e c +d e c \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right )+e \left (4 a \,d^{2} e^{2} c \left (a \,e^{2}+c \,d^{2}\right )+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right )\right ) x^{5}}{5}+\frac {\left (d \left (4 a \,d^{2} e^{2} c \left (a \,e^{2}+c \,d^{2}\right )+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right )+e \left (a d e \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e +d^{3} e^{3} c \,a^{2}\right )\right ) x^{4}}{4}+\frac {\left (d \left (a d e \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e +d^{3} e^{3} c \,a^{2}\right )+3 e^{3} a^{2} d^{2} \left (a \,e^{2}+c \,d^{2}\right )\right ) x^{3}}{3}+\frac {\left (3 d^{3} a^{2} e^{2} \left (a \,e^{2}+c \,d^{2}\right )+e^{4} a^{3} d^{3}\right ) x^{2}}{2}+e^{3} a^{3} d^{4} x\)	\(531\)

Input:

int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x,method=_RETURNVERBOSE)
 

Output:

1/8*d^3*c^3*e^4*x^8+(3/7*a*e^5*c^2*d^2+4/7*d^4*c^3*e^3)*x^7+(1/2*a^2*e^6*c 
*d+2*a*e^4*c^2*d^3+d^5*c^3*e^2)*x^6+(1/5*a^3*e^7+12/5*a^2*e^5*c*d^2+18/5*a 
*e^3*c^2*d^4+4/5*d^6*c^3*e)*x^5+(e^6*a^3*d+9/2*a^2*c*d^3*e^4+3*a*c^2*d^5*e 
^2+1/4*c^3*d^7)*x^4+(2*a^3*d^2*e^5+4*a^2*c*d^4*e^3+a*c^2*d^6*e)*x^3+(2*e^4 
*a^3*d^3+3/2*a^2*e^2*c*d^5)*x^2+e^3*a^3*d^4*x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (103) = 206\).

Time = 0.07 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.26 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} d^{3} e^{4} x^{8} + a^{3} d^{4} e^{3} x + \frac {1}{7} \, {\left (4 \, c^{3} d^{4} e^{3} + 3 \, a c^{2} d^{2} e^{5}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, c^{3} d^{5} e^{2} + 4 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, c^{3} d^{6} e + 18 \, a c^{2} d^{4} e^{3} + 12 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{7} + 12 \, a c^{2} d^{5} e^{2} + 18 \, a^{2} c d^{3} e^{4} + 4 \, a^{3} d e^{6}\right )} x^{4} + {\left (a c^{2} d^{6} e + 4 \, a^{2} c d^{4} e^{3} + 2 \, a^{3} d^{2} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{5} e^{2} + 4 \, a^{3} d^{3} e^{4}\right )} x^{2} \] Input:

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas 
")
 

Output:

1/8*c^3*d^3*e^4*x^8 + a^3*d^4*e^3*x + 1/7*(4*c^3*d^4*e^3 + 3*a*c^2*d^2*e^5 
)*x^7 + 1/2*(2*c^3*d^5*e^2 + 4*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^6 + 1/5*(4*c 
^3*d^6*e + 18*a*c^2*d^4*e^3 + 12*a^2*c*d^2*e^5 + a^3*e^7)*x^5 + 1/4*(c^3*d 
^7 + 12*a*c^2*d^5*e^2 + 18*a^2*c*d^3*e^4 + 4*a^3*d*e^6)*x^4 + (a*c^2*d^6*e 
 + 4*a^2*c*d^4*e^3 + 2*a^3*d^2*e^5)*x^3 + 1/2*(3*a^2*c*d^5*e^2 + 4*a^3*d^3 
*e^4)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (99) = 198\).

Time = 0.04 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.43 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=a^{3} d^{4} e^{3} x + \frac {c^{3} d^{3} e^{4} x^{8}}{8} + x^{7} \cdot \left (\frac {3 a c^{2} d^{2} e^{5}}{7} + \frac {4 c^{3} d^{4} e^{3}}{7}\right ) + x^{6} \left (\frac {a^{2} c d e^{6}}{2} + 2 a c^{2} d^{3} e^{4} + c^{3} d^{5} e^{2}\right ) + x^{5} \left (\frac {a^{3} e^{7}}{5} + \frac {12 a^{2} c d^{2} e^{5}}{5} + \frac {18 a c^{2} d^{4} e^{3}}{5} + \frac {4 c^{3} d^{6} e}{5}\right ) + x^{4} \left (a^{3} d e^{6} + \frac {9 a^{2} c d^{3} e^{4}}{2} + 3 a c^{2} d^{5} e^{2} + \frac {c^{3} d^{7}}{4}\right ) + x^{3} \cdot \left (2 a^{3} d^{2} e^{5} + 4 a^{2} c d^{4} e^{3} + a c^{2} d^{6} e\right ) + x^{2} \cdot \left (2 a^{3} d^{3} e^{4} + \frac {3 a^{2} c d^{5} e^{2}}{2}\right ) \] Input:

integrate((e*x+d)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
 

Output:

a**3*d**4*e**3*x + c**3*d**3*e**4*x**8/8 + x**7*(3*a*c**2*d**2*e**5/7 + 4* 
c**3*d**4*e**3/7) + x**6*(a**2*c*d*e**6/2 + 2*a*c**2*d**3*e**4 + c**3*d**5 
*e**2) + x**5*(a**3*e**7/5 + 12*a**2*c*d**2*e**5/5 + 18*a*c**2*d**4*e**3/5 
 + 4*c**3*d**6*e/5) + x**4*(a**3*d*e**6 + 9*a**2*c*d**3*e**4/2 + 3*a*c**2* 
d**5*e**2 + c**3*d**7/4) + x**3*(2*a**3*d**2*e**5 + 4*a**2*c*d**4*e**3 + a 
*c**2*d**6*e) + x**2*(2*a**3*d**3*e**4 + 3*a**2*c*d**5*e**2/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (103) = 206\).

Time = 0.03 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.26 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} d^{3} e^{4} x^{8} + a^{3} d^{4} e^{3} x + \frac {1}{7} \, {\left (4 \, c^{3} d^{4} e^{3} + 3 \, a c^{2} d^{2} e^{5}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, c^{3} d^{5} e^{2} + 4 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, c^{3} d^{6} e + 18 \, a c^{2} d^{4} e^{3} + 12 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{7} + 12 \, a c^{2} d^{5} e^{2} + 18 \, a^{2} c d^{3} e^{4} + 4 \, a^{3} d e^{6}\right )} x^{4} + {\left (a c^{2} d^{6} e + 4 \, a^{2} c d^{4} e^{3} + 2 \, a^{3} d^{2} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{5} e^{2} + 4 \, a^{3} d^{3} e^{4}\right )} x^{2} \] Input:

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima 
")
 

Output:

1/8*c^3*d^3*e^4*x^8 + a^3*d^4*e^3*x + 1/7*(4*c^3*d^4*e^3 + 3*a*c^2*d^2*e^5 
)*x^7 + 1/2*(2*c^3*d^5*e^2 + 4*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^6 + 1/5*(4*c 
^3*d^6*e + 18*a*c^2*d^4*e^3 + 12*a^2*c*d^2*e^5 + a^3*e^7)*x^5 + 1/4*(c^3*d 
^7 + 12*a*c^2*d^5*e^2 + 18*a^2*c*d^3*e^4 + 4*a^3*d*e^6)*x^4 + (a*c^2*d^6*e 
 + 4*a^2*c*d^4*e^3 + 2*a^3*d^2*e^5)*x^3 + 1/2*(3*a^2*c*d^5*e^2 + 4*a^3*d^3 
*e^4)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (103) = 206\).

Time = 0.14 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.44 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} d^{3} e^{4} x^{8} + \frac {4}{7} \, c^{3} d^{4} e^{3} x^{7} + \frac {3}{7} \, a c^{2} d^{2} e^{5} x^{7} + c^{3} d^{5} e^{2} x^{6} + 2 \, a c^{2} d^{3} e^{4} x^{6} + \frac {1}{2} \, a^{2} c d e^{6} x^{6} + \frac {4}{5} \, c^{3} d^{6} e x^{5} + \frac {18}{5} \, a c^{2} d^{4} e^{3} x^{5} + \frac {12}{5} \, a^{2} c d^{2} e^{5} x^{5} + \frac {1}{5} \, a^{3} e^{7} x^{5} + \frac {1}{4} \, c^{3} d^{7} x^{4} + 3 \, a c^{2} d^{5} e^{2} x^{4} + \frac {9}{2} \, a^{2} c d^{3} e^{4} x^{4} + a^{3} d e^{6} x^{4} + a c^{2} d^{6} e x^{3} + 4 \, a^{2} c d^{4} e^{3} x^{3} + 2 \, a^{3} d^{2} e^{5} x^{3} + \frac {3}{2} \, a^{2} c d^{5} e^{2} x^{2} + 2 \, a^{3} d^{3} e^{4} x^{2} + a^{3} d^{4} e^{3} x \] Input:

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")
 

Output:

1/8*c^3*d^3*e^4*x^8 + 4/7*c^3*d^4*e^3*x^7 + 3/7*a*c^2*d^2*e^5*x^7 + c^3*d^ 
5*e^2*x^6 + 2*a*c^2*d^3*e^4*x^6 + 1/2*a^2*c*d*e^6*x^6 + 4/5*c^3*d^6*e*x^5 
+ 18/5*a*c^2*d^4*e^3*x^5 + 12/5*a^2*c*d^2*e^5*x^5 + 1/5*a^3*e^7*x^5 + 1/4* 
c^3*d^7*x^4 + 3*a*c^2*d^5*e^2*x^4 + 9/2*a^2*c*d^3*e^4*x^4 + a^3*d*e^6*x^4 
+ a*c^2*d^6*e*x^3 + 4*a^2*c*d^4*e^3*x^3 + 2*a^3*d^2*e^5*x^3 + 3/2*a^2*c*d^ 
5*e^2*x^2 + 2*a^3*d^3*e^4*x^2 + a^3*d^4*e^3*x
 

Mupad [B] (verification not implemented)

Time = 5.60 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.18 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=x^4\,\left (a^3\,d\,e^6+\frac {9\,a^2\,c\,d^3\,e^4}{2}+3\,a\,c^2\,d^5\,e^2+\frac {c^3\,d^7}{4}\right )+x^5\,\left (\frac {a^3\,e^7}{5}+\frac {12\,a^2\,c\,d^2\,e^5}{5}+\frac {18\,a\,c^2\,d^4\,e^3}{5}+\frac {4\,c^3\,d^6\,e}{5}\right )+a^3\,d^4\,e^3\,x+\frac {c^3\,d^3\,e^4\,x^8}{8}+a\,d^2\,e\,x^3\,\left (2\,a^2\,e^4+4\,a\,c\,d^2\,e^2+c^2\,d^4\right )+\frac {c\,d\,e^2\,x^6\,\left (a^2\,e^4+4\,a\,c\,d^2\,e^2+2\,c^2\,d^4\right )}{2}+\frac {a^2\,d^3\,e^2\,x^2\,\left (3\,c\,d^2+4\,a\,e^2\right )}{2}+\frac {c^2\,d^2\,e^3\,x^7\,\left (4\,c\,d^2+3\,a\,e^2\right )}{7} \] Input:

int((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)
 

Output:

x^4*((c^3*d^7)/4 + a^3*d*e^6 + 3*a*c^2*d^5*e^2 + (9*a^2*c*d^3*e^4)/2) + x^ 
5*((a^3*e^7)/5 + (4*c^3*d^6*e)/5 + (18*a*c^2*d^4*e^3)/5 + (12*a^2*c*d^2*e^ 
5)/5) + a^3*d^4*e^3*x + (c^3*d^3*e^4*x^8)/8 + a*d^2*e*x^3*(2*a^2*e^4 + c^2 
*d^4 + 4*a*c*d^2*e^2) + (c*d*e^2*x^6*(a^2*e^4 + 2*c^2*d^4 + 4*a*c*d^2*e^2) 
)/2 + (a^2*d^3*e^2*x^2*(4*a*e^2 + 3*c*d^2))/2 + (c^2*d^2*e^3*x^7*(3*a*e^2 
+ 4*c*d^2))/7
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.46 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {x \left (35 c^{3} d^{3} e^{4} x^{7}+120 a \,c^{2} d^{2} e^{5} x^{6}+160 c^{3} d^{4} e^{3} x^{6}+140 a^{2} c d \,e^{6} x^{5}+560 a \,c^{2} d^{3} e^{4} x^{5}+280 c^{3} d^{5} e^{2} x^{5}+56 a^{3} e^{7} x^{4}+672 a^{2} c \,d^{2} e^{5} x^{4}+1008 a \,c^{2} d^{4} e^{3} x^{4}+224 c^{3} d^{6} e \,x^{4}+280 a^{3} d \,e^{6} x^{3}+1260 a^{2} c \,d^{3} e^{4} x^{3}+840 a \,c^{2} d^{5} e^{2} x^{3}+70 c^{3} d^{7} x^{3}+560 a^{3} d^{2} e^{5} x^{2}+1120 a^{2} c \,d^{4} e^{3} x^{2}+280 a \,c^{2} d^{6} e \,x^{2}+560 a^{3} d^{3} e^{4} x +420 a^{2} c \,d^{5} e^{2} x +280 a^{3} d^{4} e^{3}\right )}{280} \] Input:

int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
 

Output:

(x*(280*a**3*d**4*e**3 + 560*a**3*d**3*e**4*x + 560*a**3*d**2*e**5*x**2 + 
280*a**3*d*e**6*x**3 + 56*a**3*e**7*x**4 + 420*a**2*c*d**5*e**2*x + 1120*a 
**2*c*d**4*e**3*x**2 + 1260*a**2*c*d**3*e**4*x**3 + 672*a**2*c*d**2*e**5*x 
**4 + 140*a**2*c*d*e**6*x**5 + 280*a*c**2*d**6*e*x**2 + 840*a*c**2*d**5*e* 
*2*x**3 + 1008*a*c**2*d**4*e**3*x**4 + 560*a*c**2*d**3*e**4*x**5 + 120*a*c 
**2*d**2*e**5*x**6 + 70*c**3*d**7*x**3 + 224*c**3*d**6*e*x**4 + 280*c**3*d 
**5*e**2*x**5 + 160*c**3*d**4*e**3*x**6 + 35*c**3*d**3*e**4*x**7))/280