\(\int (d+e x)^3 (a+b (d+e x)^2+c (d+e x)^4)^3 \, dx\) [609]

Optimal result

Integrand size = 30, antiderivative size = 138 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx=\frac {a^3 (d+e x)^4}{4 e}+\frac {a^2 b (d+e x)^6}{2 e}+\frac {3 a \left (b^2+a c\right ) (d+e x)^8}{8 e}+\frac {b \left (b^2+6 a c\right ) (d+e x)^{10}}{10 e}+\frac {c \left (b^2+a c\right ) (d+e x)^{12}}{4 e}+\frac {3 b c^2 (d+e x)^{14}}{14 e}+\frac {c^3 (d+e x)^{16}}{16 e} \]

[Out]

1/4*a^3*(e*x+d)^4/e+1/2*a^2*b*(e*x+d)^6/e+3/8*a*(a*c+b^2)*(e*x+d)^8/e+1/10*b*(6*a*c+b^2)*(e*x+d)^10/e+1/4*c*(a
*c+b^2)*(e*x+d)^12/e+3/14*b*c^2*(e*x+d)^14/e+1/16*c^3*(e*x+d)^16/e

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1156, 1128, 645} \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx=\frac {a^3 (d+e x)^4}{4 e}+\frac {a^2 b (d+e x)^6}{2 e}+\frac {c \left (a c+b^2\right ) (d+e x)^{12}}{4 e}+\frac {b \left (6 a c+b^2\right ) (d+e x)^{10}}{10 e}+\frac {3 a \left (a c+b^2\right ) (d+e x)^8}{8 e}+\frac {3 b c^2 (d+e x)^{14}}{14 e}+\frac {c^3 (d+e x)^{16}}{16 e} \]

[In]

Int[(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]

[Out]

(a^3*(d + e*x)^4)/(4*e) + (a^2*b*(d + e*x)^6)/(2*e) + (3*a*(b^2 + a*c)*(d + e*x)^8)/(8*e) + (b*(b^2 + 6*a*c)*(
d + e*x)^10)/(10*e) + (c*(b^2 + a*c)*(d + e*x)^12)/(4*e) + (3*b*c^2*(d + e*x)^14)/(14*e) + (c^3*(d + e*x)^16)/
(16*e)

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])

Rule 1128

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^3 \left (a+b x^2+c x^4\right )^3 \, dx,x,d+e x\right )}{e} \\ & = \frac {\text {Subst}\left (\int x \left (a+b x+c x^2\right )^3 \, dx,x,(d+e x)^2\right )}{2 e} \\ & = \frac {\text {Subst}\left (\int \left (a^3 x+3 a^2 b x^2+3 a \left (b^2+a c\right ) x^3+b \left (b^2+6 a c\right ) x^4+3 c \left (b^2+a c\right ) x^5+3 b c^2 x^6+c^3 x^7\right ) \, dx,x,(d+e x)^2\right )}{2 e} \\ & = \frac {a^3 (d+e x)^4}{4 e}+\frac {a^2 b (d+e x)^6}{2 e}+\frac {3 a \left (b^2+a c\right ) (d+e x)^8}{8 e}+\frac {b \left (b^2+6 a c\right ) (d+e x)^{10}}{10 e}+\frac {c \left (b^2+a c\right ) (d+e x)^{12}}{4 e}+\frac {3 b c^2 (d+e x)^{14}}{14 e}+\frac {c^3 (d+e x)^{16}}{16 e} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(797\) vs. \(2(138)=276\).

Time = 0.20 (sec) , antiderivative size = 797, normalized size of antiderivative = 5.78 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx=d^3 \left (a+b d^2+c d^4\right )^3 x+\frac {3}{2} d^2 \left (a+b d^2+c d^4\right )^2 \left (a+3 b d^2+5 c d^4\right ) e x^2+d \left (a^3+10 a^2 b d^2+21 a b^2 d^4+21 a^2 c d^4+12 b^3 d^6+72 a b c d^6+55 b^2 c d^8+55 a c^2 d^8+78 b c^2 d^{10}+35 c^3 d^{12}\right ) e^2 x^3+\frac {1}{4} \left (a^3+30 a^2 b d^2+105 a b^2 d^4+105 a^2 c d^4+84 b^3 d^6+504 a b c d^6+495 b^2 c d^8+495 a c^2 d^8+858 b c^2 d^{10}+455 c^3 d^{12}\right ) e^3 x^4+\frac {3}{5} d \left (5 a^2 b+35 a b^2 d^2+35 a^2 c d^2+42 b^3 d^4+252 a b c d^4+330 b^2 c d^6+330 a c^2 d^6+715 b c^2 d^8+455 c^3 d^{10}\right ) e^4 x^5+\frac {1}{2} \left (a^2 b+21 a b^2 d^2+21 a^2 c d^2+42 b^3 d^4+252 a b c d^4+462 b^2 c d^6+462 a c^2 d^6+1287 b c^2 d^8+1001 c^3 d^{10}\right ) e^5 x^6+\frac {1}{7} d \left (21 a b^2+21 a^2 c+84 b^3 d^2+504 a b c d^2+1386 b^2 c d^4+1386 a c^2 d^4+5148 b c^2 d^6+5005 c^3 d^8\right ) e^6 x^7+\frac {3}{8} \left (a b^2+a^2 c+12 b^3 d^2+72 a b c d^2+330 b^2 c d^4+330 a c^2 d^4+1716 b c^2 d^6+2145 c^3 d^8\right ) e^7 x^8+d \left (b^3+6 a b c+55 b^2 c d^2+55 a c^2 d^2+429 b c^2 d^4+715 c^3 d^6\right ) e^8 x^9+\frac {1}{10} \left (b^3+6 a b c+165 b^2 c d^2+165 a c^2 d^2+2145 b c^2 d^4+5005 c^3 d^6\right ) e^9 x^{10}+3 c d \left (b^2+a c+26 b c d^2+91 c^2 d^4\right ) e^{10} x^{11}+\frac {1}{4} c \left (b^2+a c+78 b c d^2+455 c^2 d^4\right ) e^{11} x^{12}+c^2 d \left (3 b+35 c d^2\right ) e^{12} x^{13}+\frac {3}{14} c^2 \left (b+35 c d^2\right ) e^{13} x^{14}+c^3 d e^{14} x^{15}+\frac {1}{16} c^3 e^{15} x^{16} \]

[In]

Integrate[(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]

[Out]

d^3*(a + b*d^2 + c*d^4)^3*x + (3*d^2*(a + b*d^2 + c*d^4)^2*(a + 3*b*d^2 + 5*c*d^4)*e*x^2)/2 + d*(a^3 + 10*a^2*
b*d^2 + 21*a*b^2*d^4 + 21*a^2*c*d^4 + 12*b^3*d^6 + 72*a*b*c*d^6 + 55*b^2*c*d^8 + 55*a*c^2*d^8 + 78*b*c^2*d^10
+ 35*c^3*d^12)*e^2*x^3 + ((a^3 + 30*a^2*b*d^2 + 105*a*b^2*d^4 + 105*a^2*c*d^4 + 84*b^3*d^6 + 504*a*b*c*d^6 + 4
95*b^2*c*d^8 + 495*a*c^2*d^8 + 858*b*c^2*d^10 + 455*c^3*d^12)*e^3*x^4)/4 + (3*d*(5*a^2*b + 35*a*b^2*d^2 + 35*a
^2*c*d^2 + 42*b^3*d^4 + 252*a*b*c*d^4 + 330*b^2*c*d^6 + 330*a*c^2*d^6 + 715*b*c^2*d^8 + 455*c^3*d^10)*e^4*x^5)
/5 + ((a^2*b + 21*a*b^2*d^2 + 21*a^2*c*d^2 + 42*b^3*d^4 + 252*a*b*c*d^4 + 462*b^2*c*d^6 + 462*a*c^2*d^6 + 1287
*b*c^2*d^8 + 1001*c^3*d^10)*e^5*x^6)/2 + (d*(21*a*b^2 + 21*a^2*c + 84*b^3*d^2 + 504*a*b*c*d^2 + 1386*b^2*c*d^4
+ 1386*a*c^2*d^4 + 5148*b*c^2*d^6 + 5005*c^3*d^8)*e^6*x^7)/7 + (3*(a*b^2 + a^2*c + 12*b^3*d^2 + 72*a*b*c*d^2
+ 330*b^2*c*d^4 + 330*a*c^2*d^4 + 1716*b*c^2*d^6 + 2145*c^3*d^8)*e^7*x^8)/8 + d*(b^3 + 6*a*b*c + 55*b^2*c*d^2
+ 55*a*c^2*d^2 + 429*b*c^2*d^4 + 715*c^3*d^6)*e^8*x^9 + ((b^3 + 6*a*b*c + 165*b^2*c*d^2 + 165*a*c^2*d^2 + 2145
*b*c^2*d^4 + 5005*c^3*d^6)*e^9*x^10)/10 + 3*c*d*(b^2 + a*c + 26*b*c*d^2 + 91*c^2*d^4)*e^10*x^11 + (c*(b^2 + a*
c + 78*b*c*d^2 + 455*c^2*d^4)*e^11*x^12)/4 + c^2*d*(3*b + 35*c*d^2)*e^12*x^13 + (3*c^2*(b + 35*c*d^2)*e^13*x^1
4)/14 + c^3*d*e^14*x^15 + (c^3*e^15*x^16)/16

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1129\) vs. \(2(124)=248\).

Time = 0.64 (sec) , antiderivative size = 1130, normalized size of antiderivative = 8.19

method	result	size
norman	\(\text {Expression too large to display}\)	\(1130\)
gosper	\(\text {Expression too large to display}\)	\(1315\)
risch	\(\text {Expression too large to display}\)	\(1336\)
parallelrisch	\(\text {Expression too large to display}\)	\(1336\)
default	\(\text {Expression too large to display}\)	\(7550\)

[In]

int((e*x+d)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x,method=_RETURNVERBOSE)

[Out]

(c^3*d^15+3*b*c^2*d^13+3*a*c^2*d^11+3*b^2*c*d^11+6*a*b*c*d^9+b^3*d^9+3*a^2*c*d^7+3*a*b^2*d^7+3*a^2*b*d^5+a^3*d
^3)*x+(15/2*c^3*d^14*e+39/2*b*c^2*d^12*e+33/2*a*c^2*d^10*e+33/2*b^2*c*d^10*e+27*a*b*c*d^8*e+9/2*b^3*d^8*e+21/2
*a^2*c*d^6*e+21/2*a*b^2*d^6*e+15/2*a^2*b*d^4*e+3/2*a^3*d^2*e)*x^2+(35*c^3*d^13*e^2+78*b*c^2*d^11*e^2+55*a*c^2*
d^9*e^2+55*b^2*c*d^9*e^2+72*a*b*c*d^7*e^2+12*b^3*d^7*e^2+21*a^2*c*d^5*e^2+21*a*b^2*d^5*e^2+10*a^2*b*d^3*e^2+a^
3*d*e^2)*x^3+(455/4*c^3*d^12*e^3+429/2*b*c^2*d^10*e^3+495/4*a*c^2*d^8*e^3+495/4*b^2*c*d^8*e^3+126*a*b*c*d^6*e^
3+21*b^3*d^6*e^3+105/4*a^2*c*d^4*e^3+105/4*a*b^2*d^4*e^3+15/2*a^2*b*d^2*e^3+1/4*a^3*e^3)*x^4+(273*c^3*d^11*e^4
+429*b*c^2*d^9*e^4+198*a*c^2*d^7*e^4+198*b^2*c*d^7*e^4+756/5*a*b*c*d^5*e^4+126/5*b^3*d^5*e^4+21*a^2*c*d^3*e^4+
21*a*b^2*d^3*e^4+3*a^2*b*d*e^4)*x^5+(1001/2*c^3*d^10*e^5+1287/2*b*c^2*d^8*e^5+231*a*c^2*d^6*e^5+231*b^2*c*d^6*
e^5+126*a*b*c*d^4*e^5+21*b^3*d^4*e^5+21/2*a^2*c*d^2*e^5+21/2*a*b^2*d^2*e^5+1/2*a^2*b*e^5)*x^6+(715*c^3*d^9*e^6
+5148/7*b*c^2*d^7*e^6+198*a*c^2*d^5*e^6+198*b^2*c*d^5*e^6+72*a*b*c*d^3*e^6+12*b^3*d^3*e^6+3*a^2*c*d*e^6+3*a*b^
2*d*e^6)*x^7+(6435/8*c^3*d^8*e^7+1287/2*b*c^2*d^6*e^7+495/4*a*c^2*d^4*e^7+495/4*b^2*c*d^4*e^7+27*a*b*c*d^2*e^7
+9/2*b^3*d^2*e^7+3/8*a^2*c*e^7+3/8*a*b^2*e^7)*x^8+(715*c^3*d^7*e^8+429*b*c^2*d^5*e^8+55*a*c^2*d^3*e^8+55*b^2*c
*d^3*e^8+6*a*b*c*d*e^8+b^3*d*e^8)*x^9+(1001/2*c^3*d^6*e^9+429/2*b*c^2*d^4*e^9+33/2*a*c^2*d^2*e^9+33/2*b^2*c*d^
2*e^9+3/5*a*b*c*e^9+1/10*b^3*e^9)*x^10+(273*c^3*d^5*e^10+78*b*c^2*d^3*e^10+3*a*c^2*d*e^10+3*b^2*c*d*e^10)*x^11
+(455/4*d^4*c^3*e^11+39/2*b*c^2*d^2*e^11+1/4*a*c^2*e^11+1/4*b^2*c*e^11)*x^12+(35*c^3*d^3*e^12+3*b*c^2*d*e^12)*
x^13+(15/2*d^2*e^13*c^3+3/14*b*c^2*e^13)*x^14+d*e^14*c^3*x^15+1/16*e^15*c^3*x^16

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (124) = 248\).

Time = 0.26 (sec) , antiderivative size = 872, normalized size of antiderivative = 6.32 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx=\frac {1}{16} \, c^{3} e^{15} x^{16} + c^{3} d e^{14} x^{15} + \frac {3}{14} \, {\left (35 \, c^{3} d^{2} + b c^{2}\right )} e^{13} x^{14} + {\left (35 \, c^{3} d^{3} + 3 \, b c^{2} d\right )} e^{12} x^{13} + \frac {1}{4} \, {\left (455 \, c^{3} d^{4} + 78 \, b c^{2} d^{2} + b^{2} c + a c^{2}\right )} e^{11} x^{12} + 3 \, {\left (91 \, c^{3} d^{5} + 26 \, b c^{2} d^{3} + {\left (b^{2} c + a c^{2}\right )} d\right )} e^{10} x^{11} + \frac {1}{10} \, {\left (5005 \, c^{3} d^{6} + 2145 \, b c^{2} d^{4} + b^{3} + 6 \, a b c + 165 \, {\left (b^{2} c + a c^{2}\right )} d^{2}\right )} e^{9} x^{10} + {\left (715 \, c^{3} d^{7} + 429 \, b c^{2} d^{5} + 55 \, {\left (b^{2} c + a c^{2}\right )} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d\right )} e^{8} x^{9} + \frac {3}{8} \, {\left (2145 \, c^{3} d^{8} + 1716 \, b c^{2} d^{6} + 330 \, {\left (b^{2} c + a c^{2}\right )} d^{4} + a b^{2} + a^{2} c + 12 \, {\left (b^{3} + 6 \, a b c\right )} d^{2}\right )} e^{7} x^{8} + \frac {1}{7} \, {\left (5005 \, c^{3} d^{9} + 5148 \, b c^{2} d^{7} + 1386 \, {\left (b^{2} c + a c^{2}\right )} d^{5} + 84 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} + 21 \, {\left (a b^{2} + a^{2} c\right )} d\right )} e^{6} x^{7} + \frac {1}{2} \, {\left (1001 \, c^{3} d^{10} + 1287 \, b c^{2} d^{8} + 462 \, {\left (b^{2} c + a c^{2}\right )} d^{6} + 42 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} + a^{2} b + 21 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{5} x^{6} + \frac {3}{5} \, {\left (455 \, c^{3} d^{11} + 715 \, b c^{2} d^{9} + 330 \, {\left (b^{2} c + a c^{2}\right )} d^{7} + 42 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} + 5 \, a^{2} b d + 35 \, {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} e^{4} x^{5} + \frac {1}{4} \, {\left (455 \, c^{3} d^{12} + 858 \, b c^{2} d^{10} + 495 \, {\left (b^{2} c + a c^{2}\right )} d^{8} + 84 \, {\left (b^{3} + 6 \, a b c\right )} d^{6} + 30 \, a^{2} b d^{2} + 105 \, {\left (a b^{2} + a^{2} c\right )} d^{4} + a^{3}\right )} e^{3} x^{4} + {\left (35 \, c^{3} d^{13} + 78 \, b c^{2} d^{11} + 55 \, {\left (b^{2} c + a c^{2}\right )} d^{9} + 12 \, {\left (b^{3} + 6 \, a b c\right )} d^{7} + 10 \, a^{2} b d^{3} + 21 \, {\left (a b^{2} + a^{2} c\right )} d^{5} + a^{3} d\right )} e^{2} x^{3} + \frac {3}{2} \, {\left (5 \, c^{3} d^{14} + 13 \, b c^{2} d^{12} + 11 \, {\left (b^{2} c + a c^{2}\right )} d^{10} + 3 \, {\left (b^{3} + 6 \, a b c\right )} d^{8} + 5 \, a^{2} b d^{4} + 7 \, {\left (a b^{2} + a^{2} c\right )} d^{6} + a^{3} d^{2}\right )} e x^{2} + {\left (c^{3} d^{15} + 3 \, b c^{2} d^{13} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{11} + {\left (b^{3} + 6 \, a b c\right )} d^{9} + 3 \, a^{2} b d^{5} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{7} + a^{3} d^{3}\right )} x \]

[In]

integrate((e*x+d)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="fricas")

[Out]

1/16*c^3*e^15*x^16 + c^3*d*e^14*x^15 + 3/14*(35*c^3*d^2 + b*c^2)*e^13*x^14 + (35*c^3*d^3 + 3*b*c^2*d)*e^12*x^1
3 + 1/4*(455*c^3*d^4 + 78*b*c^2*d^2 + b^2*c + a*c^2)*e^11*x^12 + 3*(91*c^3*d^5 + 26*b*c^2*d^3 + (b^2*c + a*c^2
)*d)*e^10*x^11 + 1/10*(5005*c^3*d^6 + 2145*b*c^2*d^4 + b^3 + 6*a*b*c + 165*(b^2*c + a*c^2)*d^2)*e^9*x^10 + (71
5*c^3*d^7 + 429*b*c^2*d^5 + 55*(b^2*c + a*c^2)*d^3 + (b^3 + 6*a*b*c)*d)*e^8*x^9 + 3/8*(2145*c^3*d^8 + 1716*b*c
^2*d^6 + 330*(b^2*c + a*c^2)*d^4 + a*b^2 + a^2*c + 12*(b^3 + 6*a*b*c)*d^2)*e^7*x^8 + 1/7*(5005*c^3*d^9 + 5148*
b*c^2*d^7 + 1386*(b^2*c + a*c^2)*d^5 + 84*(b^3 + 6*a*b*c)*d^3 + 21*(a*b^2 + a^2*c)*d)*e^6*x^7 + 1/2*(1001*c^3*
d^10 + 1287*b*c^2*d^8 + 462*(b^2*c + a*c^2)*d^6 + 42*(b^3 + 6*a*b*c)*d^4 + a^2*b + 21*(a*b^2 + a^2*c)*d^2)*e^5
*x^6 + 3/5*(455*c^3*d^11 + 715*b*c^2*d^9 + 330*(b^2*c + a*c^2)*d^7 + 42*(b^3 + 6*a*b*c)*d^5 + 5*a^2*b*d + 35*(
a*b^2 + a^2*c)*d^3)*e^4*x^5 + 1/4*(455*c^3*d^12 + 858*b*c^2*d^10 + 495*(b^2*c + a*c^2)*d^8 + 84*(b^3 + 6*a*b*c
)*d^6 + 30*a^2*b*d^2 + 105*(a*b^2 + a^2*c)*d^4 + a^3)*e^3*x^4 + (35*c^3*d^13 + 78*b*c^2*d^11 + 55*(b^2*c + a*c
^2)*d^9 + 12*(b^3 + 6*a*b*c)*d^7 + 10*a^2*b*d^3 + 21*(a*b^2 + a^2*c)*d^5 + a^3*d)*e^2*x^3 + 3/2*(5*c^3*d^14 +
13*b*c^2*d^12 + 11*(b^2*c + a*c^2)*d^10 + 3*(b^3 + 6*a*b*c)*d^8 + 5*a^2*b*d^4 + 7*(a*b^2 + a^2*c)*d^6 + a^3*d^
2)*e*x^2 + (c^3*d^15 + 3*b*c^2*d^13 + 3*(b^2*c + a*c^2)*d^11 + (b^3 + 6*a*b*c)*d^9 + 3*a^2*b*d^5 + 3*(a*b^2 +
a^2*c)*d^7 + a^3*d^3)*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1314 vs. \(2 (117) = 234\).

Time = 0.13 (sec) , antiderivative size = 1314, normalized size of antiderivative = 9.52 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**3*(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)

[Out]

c**3*d*e**14*x**15 + c**3*e**15*x**16/16 + x**14*(3*b*c**2*e**13/14 + 15*c**3*d**2*e**13/2) + x**13*(3*b*c**2*
d*e**12 + 35*c**3*d**3*e**12) + x**12*(a*c**2*e**11/4 + b**2*c*e**11/4 + 39*b*c**2*d**2*e**11/2 + 455*c**3*d**
4*e**11/4) + x**11*(3*a*c**2*d*e**10 + 3*b**2*c*d*e**10 + 78*b*c**2*d**3*e**10 + 273*c**3*d**5*e**10) + x**10*
(3*a*b*c*e**9/5 + 33*a*c**2*d**2*e**9/2 + b**3*e**9/10 + 33*b**2*c*d**2*e**9/2 + 429*b*c**2*d**4*e**9/2 + 1001
*c**3*d**6*e**9/2) + x**9*(6*a*b*c*d*e**8 + 55*a*c**2*d**3*e**8 + b**3*d*e**8 + 55*b**2*c*d**3*e**8 + 429*b*c*
*2*d**5*e**8 + 715*c**3*d**7*e**8) + x**8*(3*a**2*c*e**7/8 + 3*a*b**2*e**7/8 + 27*a*b*c*d**2*e**7 + 495*a*c**2
*d**4*e**7/4 + 9*b**3*d**2*e**7/2 + 495*b**2*c*d**4*e**7/4 + 1287*b*c**2*d**6*e**7/2 + 6435*c**3*d**8*e**7/8)
+ x**7*(3*a**2*c*d*e**6 + 3*a*b**2*d*e**6 + 72*a*b*c*d**3*e**6 + 198*a*c**2*d**5*e**6 + 12*b**3*d**3*e**6 + 19
8*b**2*c*d**5*e**6 + 5148*b*c**2*d**7*e**6/7 + 715*c**3*d**9*e**6) + x**6*(a**2*b*e**5/2 + 21*a**2*c*d**2*e**5
/2 + 21*a*b**2*d**2*e**5/2 + 126*a*b*c*d**4*e**5 + 231*a*c**2*d**6*e**5 + 21*b**3*d**4*e**5 + 231*b**2*c*d**6*
e**5 + 1287*b*c**2*d**8*e**5/2 + 1001*c**3*d**10*e**5/2) + x**5*(3*a**2*b*d*e**4 + 21*a**2*c*d**3*e**4 + 21*a*
b**2*d**3*e**4 + 756*a*b*c*d**5*e**4/5 + 198*a*c**2*d**7*e**4 + 126*b**3*d**5*e**4/5 + 198*b**2*c*d**7*e**4 +
429*b*c**2*d**9*e**4 + 273*c**3*d**11*e**4) + x**4*(a**3*e**3/4 + 15*a**2*b*d**2*e**3/2 + 105*a**2*c*d**4*e**3
/4 + 105*a*b**2*d**4*e**3/4 + 126*a*b*c*d**6*e**3 + 495*a*c**2*d**8*e**3/4 + 21*b**3*d**6*e**3 + 495*b**2*c*d*
*8*e**3/4 + 429*b*c**2*d**10*e**3/2 + 455*c**3*d**12*e**3/4) + x**3*(a**3*d*e**2 + 10*a**2*b*d**3*e**2 + 21*a*
*2*c*d**5*e**2 + 21*a*b**2*d**5*e**2 + 72*a*b*c*d**7*e**2 + 55*a*c**2*d**9*e**2 + 12*b**3*d**7*e**2 + 55*b**2*
c*d**9*e**2 + 78*b*c**2*d**11*e**2 + 35*c**3*d**13*e**2) + x**2*(3*a**3*d**2*e/2 + 15*a**2*b*d**4*e/2 + 21*a**
2*c*d**6*e/2 + 21*a*b**2*d**6*e/2 + 27*a*b*c*d**8*e + 33*a*c**2*d**10*e/2 + 9*b**3*d**8*e/2 + 33*b**2*c*d**10*
e/2 + 39*b*c**2*d**12*e/2 + 15*c**3*d**14*e/2) + x*(a**3*d**3 + 3*a**2*b*d**5 + 3*a**2*c*d**7 + 3*a*b**2*d**7
+ 6*a*b*c*d**9 + 3*a*c**2*d**11 + b**3*d**9 + 3*b**2*c*d**11 + 3*b*c**2*d**13 + c**3*d**15)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (124) = 248\).

Time = 0.22 (sec) , antiderivative size = 872, normalized size of antiderivative = 6.32 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx=\frac {1}{16} \, c^{3} e^{15} x^{16} + c^{3} d e^{14} x^{15} + \frac {3}{14} \, {\left (35 \, c^{3} d^{2} + b c^{2}\right )} e^{13} x^{14} + {\left (35 \, c^{3} d^{3} + 3 \, b c^{2} d\right )} e^{12} x^{13} + \frac {1}{4} \, {\left (455 \, c^{3} d^{4} + 78 \, b c^{2} d^{2} + b^{2} c + a c^{2}\right )} e^{11} x^{12} + 3 \, {\left (91 \, c^{3} d^{5} + 26 \, b c^{2} d^{3} + {\left (b^{2} c + a c^{2}\right )} d\right )} e^{10} x^{11} + \frac {1}{10} \, {\left (5005 \, c^{3} d^{6} + 2145 \, b c^{2} d^{4} + b^{3} + 6 \, a b c + 165 \, {\left (b^{2} c + a c^{2}\right )} d^{2}\right )} e^{9} x^{10} + {\left (715 \, c^{3} d^{7} + 429 \, b c^{2} d^{5} + 55 \, {\left (b^{2} c + a c^{2}\right )} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d\right )} e^{8} x^{9} + \frac {3}{8} \, {\left (2145 \, c^{3} d^{8} + 1716 \, b c^{2} d^{6} + 330 \, {\left (b^{2} c + a c^{2}\right )} d^{4} + a b^{2} + a^{2} c + 12 \, {\left (b^{3} + 6 \, a b c\right )} d^{2}\right )} e^{7} x^{8} + \frac {1}{7} \, {\left (5005 \, c^{3} d^{9} + 5148 \, b c^{2} d^{7} + 1386 \, {\left (b^{2} c + a c^{2}\right )} d^{5} + 84 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} + 21 \, {\left (a b^{2} + a^{2} c\right )} d\right )} e^{6} x^{7} + \frac {1}{2} \, {\left (1001 \, c^{3} d^{10} + 1287 \, b c^{2} d^{8} + 462 \, {\left (b^{2} c + a c^{2}\right )} d^{6} + 42 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} + a^{2} b + 21 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{5} x^{6} + \frac {3}{5} \, {\left (455 \, c^{3} d^{11} + 715 \, b c^{2} d^{9} + 330 \, {\left (b^{2} c + a c^{2}\right )} d^{7} + 42 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} + 5 \, a^{2} b d + 35 \, {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} e^{4} x^{5} + \frac {1}{4} \, {\left (455 \, c^{3} d^{12} + 858 \, b c^{2} d^{10} + 495 \, {\left (b^{2} c + a c^{2}\right )} d^{8} + 84 \, {\left (b^{3} + 6 \, a b c\right )} d^{6} + 30 \, a^{2} b d^{2} + 105 \, {\left (a b^{2} + a^{2} c\right )} d^{4} + a^{3}\right )} e^{3} x^{4} + {\left (35 \, c^{3} d^{13} + 78 \, b c^{2} d^{11} + 55 \, {\left (b^{2} c + a c^{2}\right )} d^{9} + 12 \, {\left (b^{3} + 6 \, a b c\right )} d^{7} + 10 \, a^{2} b d^{3} + 21 \, {\left (a b^{2} + a^{2} c\right )} d^{5} + a^{3} d\right )} e^{2} x^{3} + \frac {3}{2} \, {\left (5 \, c^{3} d^{14} + 13 \, b c^{2} d^{12} + 11 \, {\left (b^{2} c + a c^{2}\right )} d^{10} + 3 \, {\left (b^{3} + 6 \, a b c\right )} d^{8} + 5 \, a^{2} b d^{4} + 7 \, {\left (a b^{2} + a^{2} c\right )} d^{6} + a^{3} d^{2}\right )} e x^{2} + {\left (c^{3} d^{15} + 3 \, b c^{2} d^{13} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{11} + {\left (b^{3} + 6 \, a b c\right )} d^{9} + 3 \, a^{2} b d^{5} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{7} + a^{3} d^{3}\right )} x \]

[In]

integrate((e*x+d)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="maxima")

[Out]

1/16*c^3*e^15*x^16 + c^3*d*e^14*x^15 + 3/14*(35*c^3*d^2 + b*c^2)*e^13*x^14 + (35*c^3*d^3 + 3*b*c^2*d)*e^12*x^1
3 + 1/4*(455*c^3*d^4 + 78*b*c^2*d^2 + b^2*c + a*c^2)*e^11*x^12 + 3*(91*c^3*d^5 + 26*b*c^2*d^3 + (b^2*c + a*c^2
)*d)*e^10*x^11 + 1/10*(5005*c^3*d^6 + 2145*b*c^2*d^4 + b^3 + 6*a*b*c + 165*(b^2*c + a*c^2)*d^2)*e^9*x^10 + (71
5*c^3*d^7 + 429*b*c^2*d^5 + 55*(b^2*c + a*c^2)*d^3 + (b^3 + 6*a*b*c)*d)*e^8*x^9 + 3/8*(2145*c^3*d^8 + 1716*b*c
^2*d^6 + 330*(b^2*c + a*c^2)*d^4 + a*b^2 + a^2*c + 12*(b^3 + 6*a*b*c)*d^2)*e^7*x^8 + 1/7*(5005*c^3*d^9 + 5148*
b*c^2*d^7 + 1386*(b^2*c + a*c^2)*d^5 + 84*(b^3 + 6*a*b*c)*d^3 + 21*(a*b^2 + a^2*c)*d)*e^6*x^7 + 1/2*(1001*c^3*
d^10 + 1287*b*c^2*d^8 + 462*(b^2*c + a*c^2)*d^6 + 42*(b^3 + 6*a*b*c)*d^4 + a^2*b + 21*(a*b^2 + a^2*c)*d^2)*e^5
*x^6 + 3/5*(455*c^3*d^11 + 715*b*c^2*d^9 + 330*(b^2*c + a*c^2)*d^7 + 42*(b^3 + 6*a*b*c)*d^5 + 5*a^2*b*d + 35*(
a*b^2 + a^2*c)*d^3)*e^4*x^5 + 1/4*(455*c^3*d^12 + 858*b*c^2*d^10 + 495*(b^2*c + a*c^2)*d^8 + 84*(b^3 + 6*a*b*c
)*d^6 + 30*a^2*b*d^2 + 105*(a*b^2 + a^2*c)*d^4 + a^3)*e^3*x^4 + (35*c^3*d^13 + 78*b*c^2*d^11 + 55*(b^2*c + a*c
^2)*d^9 + 12*(b^3 + 6*a*b*c)*d^7 + 10*a^2*b*d^3 + 21*(a*b^2 + a^2*c)*d^5 + a^3*d)*e^2*x^3 + 3/2*(5*c^3*d^14 +
13*b*c^2*d^12 + 11*(b^2*c + a*c^2)*d^10 + 3*(b^3 + 6*a*b*c)*d^8 + 5*a^2*b*d^4 + 7*(a*b^2 + a^2*c)*d^6 + a^3*d^
2)*e*x^2 + (c^3*d^15 + 3*b*c^2*d^13 + 3*(b^2*c + a*c^2)*d^11 + (b^3 + 6*a*b*c)*d^9 + 3*a^2*b*d^5 + 3*(a*b^2 +
a^2*c)*d^7 + a^3*d^3)*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1079 vs. \(2 (124) = 248\).

Time = 0.31 (sec) , antiderivative size = 1079, normalized size of antiderivative = 7.82 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="giac")

[Out]

1/2*(e*x^2 + 2*d*x)*c^3*d^14 + 7/4*(e*x^2 + 2*d*x)^2*c^3*d^12*e + 7/2*(e*x^2 + 2*d*x)^3*c^3*d^10*e^2 + 35/8*(e
*x^2 + 2*d*x)^4*c^3*d^8*e^3 + 7/2*(e*x^2 + 2*d*x)^5*c^3*d^6*e^4 + 7/4*(e*x^2 + 2*d*x)^6*c^3*d^4*e^5 + 1/2*(e*x
^2 + 2*d*x)^7*c^3*d^2*e^6 + 1/16*(e*x^2 + 2*d*x)^8*c^3*e^7 + 3/2*(e*x^2 + 2*d*x)*b*c^2*d^12 + 9/2*(e*x^2 + 2*d
*x)^2*b*c^2*d^10*e + 15/2*(e*x^2 + 2*d*x)^3*b*c^2*d^8*e^2 + 15/2*(e*x^2 + 2*d*x)^4*b*c^2*d^6*e^3 + 9/2*(e*x^2
+ 2*d*x)^5*b*c^2*d^4*e^4 + 3/2*(e*x^2 + 2*d*x)^6*b*c^2*d^2*e^5 + 3/14*(e*x^2 + 2*d*x)^7*b*c^2*e^6 + 3/2*(e*x^2
+ 2*d*x)*b^2*c*d^10 + 3/2*(e*x^2 + 2*d*x)*a*c^2*d^10 + 15/4*(e*x^2 + 2*d*x)^2*b^2*c*d^8*e + 15/4*(e*x^2 + 2*d
*x)^2*a*c^2*d^8*e + 5*(e*x^2 + 2*d*x)^3*b^2*c*d^6*e^2 + 5*(e*x^2 + 2*d*x)^3*a*c^2*d^6*e^2 + 15/4*(e*x^2 + 2*d*
x)^4*b^2*c*d^4*e^3 + 15/4*(e*x^2 + 2*d*x)^4*a*c^2*d^4*e^3 + 3/2*(e*x^2 + 2*d*x)^5*b^2*c*d^2*e^4 + 3/2*(e*x^2 +
2*d*x)^5*a*c^2*d^2*e^4 + 1/4*(e*x^2 + 2*d*x)^6*b^2*c*e^5 + 1/4*(e*x^2 + 2*d*x)^6*a*c^2*e^5 + 1/2*(e*x^2 + 2*d
*x)*b^3*d^8 + 3*(e*x^2 + 2*d*x)*a*b*c*d^8 + (e*x^2 + 2*d*x)^2*b^3*d^6*e + 6*(e*x^2 + 2*d*x)^2*a*b*c*d^6*e + (e
*x^2 + 2*d*x)^3*b^3*d^4*e^2 + 6*(e*x^2 + 2*d*x)^3*a*b*c*d^4*e^2 + 1/2*(e*x^2 + 2*d*x)^4*b^3*d^2*e^3 + 3*(e*x^2
+ 2*d*x)^4*a*b*c*d^2*e^3 + 1/10*(e*x^2 + 2*d*x)^5*b^3*e^4 + 3/5*(e*x^2 + 2*d*x)^5*a*b*c*e^4 + 3/2*(e*x^2 + 2*
d*x)*a*b^2*d^6 + 3/2*(e*x^2 + 2*d*x)*a^2*c*d^6 + 9/4*(e*x^2 + 2*d*x)^2*a*b^2*d^4*e + 9/4*(e*x^2 + 2*d*x)^2*a^2
*c*d^4*e + 3/2*(e*x^2 + 2*d*x)^3*a*b^2*d^2*e^2 + 3/2*(e*x^2 + 2*d*x)^3*a^2*c*d^2*e^2 + 3/8*(e*x^2 + 2*d*x)^4*a
*b^2*e^3 + 3/8*(e*x^2 + 2*d*x)^4*a^2*c*e^3 + 3/2*(e*x^2 + 2*d*x)*a^2*b*d^4 + 3/2*(e*x^2 + 2*d*x)^2*a^2*b*d^2*e
+ 1/2*(e*x^2 + 2*d*x)^3*a^2*b*e^2 + 1/2*(e*x^2 + 2*d*x)*a^3*d^2 + 1/4*(e*x^2 + 2*d*x)^2*a^3*e

Mupad [B] (verification not implemented)

Time = 9.02 (sec) , antiderivative size = 777, normalized size of antiderivative = 5.63 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx=\frac {3\,e^7\,x^8\,\left (a^2\,c+a\,b^2+72\,a\,b\,c\,d^2+330\,a\,c^2\,d^4+12\,b^3\,d^2+330\,b^2\,c\,d^4+1716\,b\,c^2\,d^6+2145\,c^3\,d^8\right )}{8}+\frac {e^5\,x^6\,\left (a^2\,b+21\,a^2\,c\,d^2+21\,a\,b^2\,d^2+252\,a\,b\,c\,d^4+462\,a\,c^2\,d^6+42\,b^3\,d^4+462\,b^2\,c\,d^6+1287\,b\,c^2\,d^8+1001\,c^3\,d^{10}\right )}{2}+\frac {e^9\,x^{10}\,\left (b^3+165\,b^2\,c\,d^2+2145\,b\,c^2\,d^4+6\,a\,b\,c+5005\,c^3\,d^6+165\,a\,c^2\,d^2\right )}{10}+\frac {c^3\,e^{15}\,x^{16}}{16}+d^3\,x\,{\left (c\,d^4+b\,d^2+a\right )}^3+\frac {e^3\,x^4\,\left (a^3+30\,a^2\,b\,d^2+105\,a^2\,c\,d^4+105\,a\,b^2\,d^4+504\,a\,b\,c\,d^6+495\,a\,c^2\,d^8+84\,b^3\,d^6+495\,b^2\,c\,d^8+858\,b\,c^2\,d^{10}+455\,c^3\,d^{12}\right )}{4}+\frac {3\,c^2\,e^{13}\,x^{14}\,\left (35\,c\,d^2+b\right )}{14}+c^3\,d\,e^{14}\,x^{15}+d\,e^2\,x^3\,\left (a^3+10\,a^2\,b\,d^2+21\,a^2\,c\,d^4+21\,a\,b^2\,d^4+72\,a\,b\,c\,d^6+55\,a\,c^2\,d^8+12\,b^3\,d^6+55\,b^2\,c\,d^8+78\,b\,c^2\,d^{10}+35\,c^3\,d^{12}\right )+\frac {c\,e^{11}\,x^{12}\,\left (b^2+78\,b\,c\,d^2+455\,c^2\,d^4+a\,c\right )}{4}+\frac {d\,e^6\,x^7\,\left (21\,a^2\,c+21\,a\,b^2+504\,a\,b\,c\,d^2+1386\,a\,c^2\,d^4+84\,b^3\,d^2+1386\,b^2\,c\,d^4+5148\,b\,c^2\,d^6+5005\,c^3\,d^8\right )}{7}+\frac {3\,d\,e^4\,x^5\,\left (5\,a^2\,b+35\,a^2\,c\,d^2+35\,a\,b^2\,d^2+252\,a\,b\,c\,d^4+330\,a\,c^2\,d^6+42\,b^3\,d^4+330\,b^2\,c\,d^6+715\,b\,c^2\,d^8+455\,c^3\,d^{10}\right )}{5}+d\,e^8\,x^9\,\left (b^3+55\,b^2\,c\,d^2+429\,b\,c^2\,d^4+6\,a\,b\,c+715\,c^3\,d^6+55\,a\,c^2\,d^2\right )+\frac {3\,d^2\,e\,x^2\,{\left (c\,d^4+b\,d^2+a\right )}^2\,\left (5\,c\,d^4+3\,b\,d^2+a\right )}{2}+c^2\,d\,e^{12}\,x^{13}\,\left (35\,c\,d^2+3\,b\right )+3\,c\,d\,e^{10}\,x^{11}\,\left (b^2+26\,b\,c\,d^2+91\,c^2\,d^4+a\,c\right ) \]

[In]

int((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x)

[Out]

(3*e^7*x^8*(a*b^2 + a^2*c + 12*b^3*d^2 + 2145*c^3*d^8 + 330*a*c^2*d^4 + 330*b^2*c*d^4 + 1716*b*c^2*d^6 + 72*a*
b*c*d^2))/8 + (e^5*x^6*(a^2*b + 42*b^3*d^4 + 1001*c^3*d^10 + 21*a*b^2*d^2 + 21*a^2*c*d^2 + 462*a*c^2*d^6 + 462
*b^2*c*d^6 + 1287*b*c^2*d^8 + 252*a*b*c*d^4))/2 + (e^9*x^10*(b^3 + 5005*c^3*d^6 + 165*a*c^2*d^2 + 165*b^2*c*d^
2 + 2145*b*c^2*d^4 + 6*a*b*c))/10 + (c^3*e^15*x^16)/16 + d^3*x*(a + b*d^2 + c*d^4)^3 + (e^3*x^4*(a^3 + 84*b^3*
d^6 + 455*c^3*d^12 + 30*a^2*b*d^2 + 105*a*b^2*d^4 + 105*a^2*c*d^4 + 495*a*c^2*d^8 + 495*b^2*c*d^8 + 858*b*c^2*
d^10 + 504*a*b*c*d^6))/4 + (3*c^2*e^13*x^14*(b + 35*c*d^2))/14 + c^3*d*e^14*x^15 + d*e^2*x^3*(a^3 + 12*b^3*d^6
+ 35*c^3*d^12 + 10*a^2*b*d^2 + 21*a*b^2*d^4 + 21*a^2*c*d^4 + 55*a*c^2*d^8 + 55*b^2*c*d^8 + 78*b*c^2*d^10 + 72
*a*b*c*d^6) + (c*e^11*x^12*(a*c + b^2 + 455*c^2*d^4 + 78*b*c*d^2))/4 + (d*e^6*x^7*(21*a*b^2 + 21*a^2*c + 84*b^
3*d^2 + 5005*c^3*d^8 + 1386*a*c^2*d^4 + 1386*b^2*c*d^4 + 5148*b*c^2*d^6 + 504*a*b*c*d^2))/7 + (3*d*e^4*x^5*(5*
a^2*b + 42*b^3*d^4 + 455*c^3*d^10 + 35*a*b^2*d^2 + 35*a^2*c*d^2 + 330*a*c^2*d^6 + 330*b^2*c*d^6 + 715*b*c^2*d^
8 + 252*a*b*c*d^4))/5 + d*e^8*x^9*(b^3 + 715*c^3*d^6 + 55*a*c^2*d^2 + 55*b^2*c*d^2 + 429*b*c^2*d^4 + 6*a*b*c)
+ (3*d^2*e*x^2*(a + b*d^2 + c*d^4)^2*(a + 3*b*d^2 + 5*c*d^4))/2 + c^2*d*e^12*x^13*(3*b + 35*c*d^2) + 3*c*d*e^1
0*x^11*(a*c + b^2 + 91*c^2*d^4 + 26*b*c*d^2)