\(\int (d+e x)^3 (a+b (d+e x)^2+c (d+e x)^4)^2 \, dx\) [608]

Optimal result

Integrand size = 30, antiderivative size = 89 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx=\frac {a^2 (d+e x)^4}{4 e}+\frac {a b (d+e x)^6}{3 e}+\frac {\left (b^2+2 a c\right ) (d+e x)^8}{8 e}+\frac {b c (d+e x)^{10}}{5 e}+\frac {c^2 (d+e x)^{12}}{12 e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 89, 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} \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx=\frac {a^2 (d+e x)^4}{4 e}+\frac {\left (2 a c+b^2\right ) (d+e x)^8}{8 e}+\frac {a b (d+e x)^6}{3 e}+\frac {b c (d+e x)^{10}}{5 e}+\frac {c^2 (d+e x)^{12}}{12 e} \]

[In]

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

[Out]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(401\) vs. \(2(89)=178\).

Time = 0.08 (sec) , antiderivative size = 401, normalized size of antiderivative = 4.51 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx=d^3 \left (a+b d^2+c d^4\right )^2 x+\frac {1}{2} d^2 \left (3 a^2+10 a b d^2+7 b^2 d^4+14 a c d^4+18 b c d^6+11 c^2 d^8\right ) e x^2+\frac {1}{3} d \left (3 a^2+20 a b d^2+21 b^2 d^4+42 a c d^4+72 b c d^6+55 c^2 d^8\right ) e^2 x^3+\frac {1}{4} \left (a^2+20 a b d^2+35 b^2 d^4+70 a c d^4+168 b c d^6+165 c^2 d^8\right ) e^3 x^4+\frac {1}{5} d \left (10 a b+35 b^2 d^2+70 a c d^2+252 b c d^4+330 c^2 d^6\right ) e^4 x^5+\frac {1}{6} \left (2 a b+21 b^2 d^2+42 a c d^2+252 b c d^4+462 c^2 d^6\right ) e^5 x^6+d \left (b^2+2 a c+24 b c d^2+66 c^2 d^4\right ) e^6 x^7+\frac {1}{8} \left (b^2+2 a c+72 b c d^2+330 c^2 d^4\right ) e^7 x^8+\frac {1}{3} c d \left (6 b+55 c d^2\right ) e^8 x^9+\frac {1}{10} c \left (2 b+55 c d^2\right ) e^9 x^{10}+c^2 d e^{10} x^{11}+\frac {1}{12} c^2 e^{11} x^{12} \]

[In]

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

[Out]

d^3*(a + b*d^2 + c*d^4)^2*x + (d^2*(3*a^2 + 10*a*b*d^2 + 7*b^2*d^4 + 14*a*c*d^4 + 18*b*c*d^6 + 11*c^2*d^8)*e*x
^2)/2 + (d*(3*a^2 + 20*a*b*d^2 + 21*b^2*d^4 + 42*a*c*d^4 + 72*b*c*d^6 + 55*c^2*d^8)*e^2*x^3)/3 + ((a^2 + 20*a*
b*d^2 + 35*b^2*d^4 + 70*a*c*d^4 + 168*b*c*d^6 + 165*c^2*d^8)*e^3*x^4)/4 + (d*(10*a*b + 35*b^2*d^2 + 70*a*c*d^2
+ 252*b*c*d^4 + 330*c^2*d^6)*e^4*x^5)/5 + ((2*a*b + 21*b^2*d^2 + 42*a*c*d^2 + 252*b*c*d^4 + 462*c^2*d^6)*e^5*
x^6)/6 + d*(b^2 + 2*a*c + 24*b*c*d^2 + 66*c^2*d^4)*e^6*x^7 + ((b^2 + 2*a*c + 72*b*c*d^2 + 330*c^2*d^4)*e^7*x^8
)/8 + (c*d*(6*b + 55*c*d^2)*e^8*x^9)/3 + (c*(2*b + 55*c*d^2)*e^9*x^10)/10 + c^2*d*e^10*x^11 + (c^2*e^11*x^12)/
12

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(493\) vs. \(2(79)=158\).

Time = 0.63 (sec) , antiderivative size = 494, normalized size of antiderivative = 5.55

method	result	size
norman	\(\frac {e^{11} c^{2} x^{12}}{12}+d \,e^{10} c^{2} x^{11}+\left (\frac {11}{2} d^{2} e^{9} c^{2}+\frac {1}{5} b c \,e^{9}\right ) x^{10}+\left (\frac {55}{3} d^{3} c^{2} e^{8}+2 b c d \,e^{8}\right ) x^{9}+\left (\frac {165}{4} c^{2} d^{4} e^{7}+9 b c \,d^{2} e^{7}+\frac {1}{4} a c \,e^{7}+\frac {1}{8} b^{2} e^{7}\right ) x^{8}+\left (66 c^{2} d^{5} e^{6}+24 b c \,d^{3} e^{6}+2 a c d \,e^{6}+b^{2} d \,e^{6}\right ) x^{7}+\left (77 c^{2} d^{6} e^{5}+42 b c \,d^{4} e^{5}+7 a c \,d^{2} e^{5}+\frac {7}{2} b^{2} d^{2} e^{5}+\frac {1}{3} a b \,e^{5}\right ) x^{6}+\left (66 c^{2} d^{7} e^{4}+\frac {252}{5} b c \,d^{5} e^{4}+14 a c \,d^{3} e^{4}+7 b^{2} d^{3} e^{4}+2 a b d \,e^{4}\right ) x^{5}+\left (\frac {165}{4} c^{2} d^{8} e^{3}+42 b c \,d^{6} e^{3}+\frac {35}{2} a c \,d^{4} e^{3}+\frac {35}{4} b^{2} d^{4} e^{3}+5 e^{3} a b \,d^{2}+\frac {1}{4} e^{3} a^{2}\right ) x^{4}+\left (\frac {55}{3} c^{2} d^{9} e^{2}+24 b c \,d^{7} e^{2}+14 a c \,d^{5} e^{2}+7 b^{2} d^{5} e^{2}+\frac {20}{3} a b \,d^{3} e^{2}+d \,e^{2} a^{2}\right ) x^{3}+\left (\frac {11}{2} c^{2} d^{10} e +9 b c \,d^{8} e +7 a c \,d^{6} e +\frac {7}{2} b^{2} d^{6} e +5 a b \,d^{4} e +\frac {3}{2} a^{2} d^{2} e \right ) x^{2}+\left (c^{2} d^{11}+2 b c \,d^{9}+2 a c \,d^{7}+b^{2} d^{7}+2 a b \,d^{5}+a^{2} d^{3}\right ) x\)	\(494\)
gosper	\(\frac {x \left (10 e^{11} c^{2} x^{11}+120 d \,e^{10} c^{2} x^{10}+660 x^{9} d^{2} e^{9} c^{2}+2200 x^{8} d^{3} c^{2} e^{8}+24 x^{9} b c \,e^{9}+4950 x^{7} c^{2} d^{4} e^{7}+240 x^{8} b c d \,e^{8}+7920 c^{2} d^{5} e^{6} x^{6}+1080 x^{7} b c \,d^{2} e^{7}+9240 x^{5} c^{2} d^{6} e^{5}+2880 b c \,d^{3} e^{6} x^{6}+7920 x^{4} c^{2} d^{7} e^{4}+30 x^{7} a c \,e^{7}+15 x^{7} b^{2} e^{7}+5040 x^{5} b c \,d^{4} e^{5}+4950 x^{3} c^{2} d^{8} e^{3}+240 a c d \,e^{6} x^{6}+120 b^{2} d \,e^{6} x^{6}+6048 x^{4} b c \,d^{5} e^{4}+2200 x^{2} c^{2} d^{9} e^{2}+840 x^{5} a c \,d^{2} e^{5}+420 x^{5} b^{2} d^{2} e^{5}+5040 x^{3} b c \,d^{6} e^{3}+660 x \,c^{2} d^{10} e +1680 x^{4} a c \,d^{3} e^{4}+840 x^{4} b^{2} d^{3} e^{4}+2880 x^{2} b c \,d^{7} e^{2}+120 c^{2} d^{11}+40 x^{5} a b \,e^{5}+2100 x^{3} a c \,d^{4} e^{3}+1050 x^{3} b^{2} d^{4} e^{3}+1080 x b c \,d^{8} e +240 x^{4} a b d \,e^{4}+1680 x^{2} a c \,d^{5} e^{2}+840 x^{2} b^{2} d^{5} e^{2}+240 b c \,d^{9}+600 x^{3} e^{3} a b \,d^{2}+840 x a c \,d^{6} e +420 x \,b^{2} d^{6} e +800 x^{2} a b \,d^{3} e^{2}+240 a c \,d^{7}+120 b^{2} d^{7}+30 x^{3} e^{3} a^{2}+600 x a b \,d^{4} e +120 x^{2} d \,e^{2} a^{2}+240 a b \,d^{5}+180 x \,a^{2} d^{2} e +120 a^{2} d^{3}\right )}{120}\)	\(563\)
risch	\(24 b c \,d^{3} e^{6} x^{7}+2 x^{9} b c d \,e^{8}+9 x^{8} b c \,d^{2} e^{7}+42 x^{6} b c \,d^{4} e^{5}+7 x^{6} a c \,d^{2} e^{5}+\frac {11}{2} x^{10} d^{2} e^{9} c^{2}+\frac {1}{5} x^{10} b c \,e^{9}+\frac {55}{3} x^{9} d^{3} c^{2} e^{8}+\frac {165}{4} x^{8} c^{2} d^{4} e^{7}+\frac {1}{4} x^{8} a c \,e^{7}+77 x^{6} c^{2} d^{6} e^{5}+\frac {7}{2} x^{6} b^{2} d^{2} e^{5}+\frac {1}{3} x^{6} a b \,e^{5}+66 x^{5} c^{2} d^{7} e^{4}+7 x^{5} b^{2} d^{3} e^{4}+\frac {165}{4} x^{4} c^{2} d^{8} e^{3}+\frac {35}{4} x^{4} b^{2} d^{4} e^{3}+\frac {55}{3} x^{3} c^{2} d^{9} e^{2}+7 x^{3} b^{2} d^{5} e^{2}+x^{3} d \,e^{2} a^{2}+\frac {11}{2} x^{2} c^{2} d^{10} e +\frac {7}{2} x^{2} b^{2} d^{6} e +\frac {3}{2} x^{2} a^{2} d^{2} e +66 c^{2} d^{5} e^{6} x^{7}+b^{2} d \,e^{6} x^{7}+2 b c \,d^{9} x +2 a c \,d^{7} x +2 a b \,d^{5} x +c^{2} d^{11} x +b^{2} d^{7} x +d \,e^{10} c^{2} x^{11}+\frac {1}{8} x^{8} b^{2} e^{7}+\frac {1}{4} x^{4} e^{3} a^{2}+\frac {252}{5} x^{5} b c \,d^{5} e^{4}+14 x^{5} a c \,d^{3} e^{4}+2 x^{5} a b d \,e^{4}+42 x^{4} b c \,d^{6} e^{3}+\frac {35}{2} x^{4} a c \,d^{4} e^{3}+5 x^{4} e^{3} a b \,d^{2}+24 x^{3} b c \,d^{7} e^{2}+14 x^{3} a c \,d^{5} e^{2}+\frac {20}{3} x^{3} a b \,d^{3} e^{2}+9 x^{2} b c \,d^{8} e +7 x^{2} a c \,d^{6} e +5 x^{2} a b \,d^{4} e +2 a c d \,e^{6} x^{7}+a^{2} d^{3} x +\frac {1}{12} e^{11} c^{2} x^{12}\)	\(572\)
parallelrisch	\(24 b c \,d^{3} e^{6} x^{7}+2 x^{9} b c d \,e^{8}+9 x^{8} b c \,d^{2} e^{7}+42 x^{6} b c \,d^{4} e^{5}+7 x^{6} a c \,d^{2} e^{5}+\frac {11}{2} x^{10} d^{2} e^{9} c^{2}+\frac {1}{5} x^{10} b c \,e^{9}+\frac {55}{3} x^{9} d^{3} c^{2} e^{8}+\frac {165}{4} x^{8} c^{2} d^{4} e^{7}+\frac {1}{4} x^{8} a c \,e^{7}+77 x^{6} c^{2} d^{6} e^{5}+\frac {7}{2} x^{6} b^{2} d^{2} e^{5}+\frac {1}{3} x^{6} a b \,e^{5}+66 x^{5} c^{2} d^{7} e^{4}+7 x^{5} b^{2} d^{3} e^{4}+\frac {165}{4} x^{4} c^{2} d^{8} e^{3}+\frac {35}{4} x^{4} b^{2} d^{4} e^{3}+\frac {55}{3} x^{3} c^{2} d^{9} e^{2}+7 x^{3} b^{2} d^{5} e^{2}+x^{3} d \,e^{2} a^{2}+\frac {11}{2} x^{2} c^{2} d^{10} e +\frac {7}{2} x^{2} b^{2} d^{6} e +\frac {3}{2} x^{2} a^{2} d^{2} e +66 c^{2} d^{5} e^{6} x^{7}+b^{2} d \,e^{6} x^{7}+2 b c \,d^{9} x +2 a c \,d^{7} x +2 a b \,d^{5} x +c^{2} d^{11} x +b^{2} d^{7} x +d \,e^{10} c^{2} x^{11}+\frac {1}{8} x^{8} b^{2} e^{7}+\frac {1}{4} x^{4} e^{3} a^{2}+\frac {252}{5} x^{5} b c \,d^{5} e^{4}+14 x^{5} a c \,d^{3} e^{4}+2 x^{5} a b d \,e^{4}+42 x^{4} b c \,d^{6} e^{3}+\frac {35}{2} x^{4} a c \,d^{4} e^{3}+5 x^{4} e^{3} a b \,d^{2}+24 x^{3} b c \,d^{7} e^{2}+14 x^{3} a c \,d^{5} e^{2}+\frac {20}{3} x^{3} a b \,d^{3} e^{2}+9 x^{2} b c \,d^{8} e +7 x^{2} a c \,d^{6} e +5 x^{2} a b \,d^{4} e +2 a c d \,e^{6} x^{7}+a^{2} d^{3} x +\frac {1}{12} e^{11} c^{2} x^{12}\)	\(572\)
default	\(\text {Expression too large to display}\)	\(1314\)

[In]

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

[Out]

1/12*e^11*c^2*x^12+d*e^10*c^2*x^11+(11/2*d^2*e^9*c^2+1/5*b*c*e^9)*x^10+(55/3*d^3*c^2*e^8+2*b*c*d*e^8)*x^9+(165
/4*c^2*d^4*e^7+9*b*c*d^2*e^7+1/4*a*c*e^7+1/8*b^2*e^7)*x^8+(66*c^2*d^5*e^6+24*b*c*d^3*e^6+2*a*c*d*e^6+b^2*d*e^6
)*x^7+(77*c^2*d^6*e^5+42*b*c*d^4*e^5+7*a*c*d^2*e^5+7/2*b^2*d^2*e^5+1/3*a*b*e^5)*x^6+(66*c^2*d^7*e^4+252/5*b*c*
d^5*e^4+14*a*c*d^3*e^4+7*b^2*d^3*e^4+2*a*b*d*e^4)*x^5+(165/4*c^2*d^8*e^3+42*b*c*d^6*e^3+35/2*a*c*d^4*e^3+35/4*
b^2*d^4*e^3+5*e^3*a*b*d^2+1/4*e^3*a^2)*x^4+(55/3*c^2*d^9*e^2+24*b*c*d^7*e^2+14*a*c*d^5*e^2+7*b^2*d^5*e^2+20/3*
a*b*d^3*e^2+d*e^2*a^2)*x^3+(11/2*c^2*d^10*e+9*b*c*d^8*e+7*a*c*d^6*e+7/2*b^2*d^6*e+5*a*b*d^4*e+3/2*a^2*d^2*e)*x
^2+(c^2*d^11+2*b*c*d^9+2*a*c*d^7+b^2*d^7+2*a*b*d^5+a^2*d^3)*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (79) = 158\).

Time = 0.25 (sec) , antiderivative size = 403, normalized size of antiderivative = 4.53 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx=\frac {1}{12} \, c^{2} e^{11} x^{12} + c^{2} d e^{10} x^{11} + \frac {1}{10} \, {\left (55 \, c^{2} d^{2} + 2 \, b c\right )} e^{9} x^{10} + \frac {1}{3} \, {\left (55 \, c^{2} d^{3} + 6 \, b c d\right )} e^{8} x^{9} + \frac {1}{8} \, {\left (330 \, c^{2} d^{4} + 72 \, b c d^{2} + b^{2} + 2 \, a c\right )} e^{7} x^{8} + {\left (66 \, c^{2} d^{5} + 24 \, b c d^{3} + {\left (b^{2} + 2 \, a c\right )} d\right )} e^{6} x^{7} + \frac {1}{6} \, {\left (462 \, c^{2} d^{6} + 252 \, b c d^{4} + 21 \, {\left (b^{2} + 2 \, a c\right )} d^{2} + 2 \, a b\right )} e^{5} x^{6} + \frac {1}{5} \, {\left (330 \, c^{2} d^{7} + 252 \, b c d^{5} + 35 \, {\left (b^{2} + 2 \, a c\right )} d^{3} + 10 \, a b d\right )} e^{4} x^{5} + \frac {1}{4} \, {\left (165 \, c^{2} d^{8} + 168 \, b c d^{6} + 35 \, {\left (b^{2} + 2 \, a c\right )} d^{4} + 20 \, a b d^{2} + a^{2}\right )} e^{3} x^{4} + \frac {1}{3} \, {\left (55 \, c^{2} d^{9} + 72 \, b c d^{7} + 21 \, {\left (b^{2} + 2 \, a c\right )} d^{5} + 20 \, a b d^{3} + 3 \, a^{2} d\right )} e^{2} x^{3} + \frac {1}{2} \, {\left (11 \, c^{2} d^{10} + 18 \, b c d^{8} + 7 \, {\left (b^{2} + 2 \, a c\right )} d^{6} + 10 \, a b d^{4} + 3 \, a^{2} d^{2}\right )} e x^{2} + {\left (c^{2} d^{11} + 2 \, b c d^{9} + {\left (b^{2} + 2 \, a c\right )} d^{7} + 2 \, a b d^{5} + a^{2} d^{3}\right )} x \]

[In]

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

[Out]

1/12*c^2*e^11*x^12 + c^2*d*e^10*x^11 + 1/10*(55*c^2*d^2 + 2*b*c)*e^9*x^10 + 1/3*(55*c^2*d^3 + 6*b*c*d)*e^8*x^9
+ 1/8*(330*c^2*d^4 + 72*b*c*d^2 + b^2 + 2*a*c)*e^7*x^8 + (66*c^2*d^5 + 24*b*c*d^3 + (b^2 + 2*a*c)*d)*e^6*x^7
+ 1/6*(462*c^2*d^6 + 252*b*c*d^4 + 21*(b^2 + 2*a*c)*d^2 + 2*a*b)*e^5*x^6 + 1/5*(330*c^2*d^7 + 252*b*c*d^5 + 35
*(b^2 + 2*a*c)*d^3 + 10*a*b*d)*e^4*x^5 + 1/4*(165*c^2*d^8 + 168*b*c*d^6 + 35*(b^2 + 2*a*c)*d^4 + 20*a*b*d^2 +
a^2)*e^3*x^4 + 1/3*(55*c^2*d^9 + 72*b*c*d^7 + 21*(b^2 + 2*a*c)*d^5 + 20*a*b*d^3 + 3*a^2*d)*e^2*x^3 + 1/2*(11*c
^2*d^10 + 18*b*c*d^8 + 7*(b^2 + 2*a*c)*d^6 + 10*a*b*d^4 + 3*a^2*d^2)*e*x^2 + (c^2*d^11 + 2*b*c*d^9 + (b^2 + 2*
a*c)*d^7 + 2*a*b*d^5 + a^2*d^3)*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (71) = 142\).

Time = 0.07 (sec) , antiderivative size = 559, normalized size of antiderivative = 6.28 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx=c^{2} d e^{10} x^{11} + \frac {c^{2} e^{11} x^{12}}{12} + x^{10} \left (\frac {b c e^{9}}{5} + \frac {11 c^{2} d^{2} e^{9}}{2}\right ) + x^{9} \cdot \left (2 b c d e^{8} + \frac {55 c^{2} d^{3} e^{8}}{3}\right ) + x^{8} \left (\frac {a c e^{7}}{4} + \frac {b^{2} e^{7}}{8} + 9 b c d^{2} e^{7} + \frac {165 c^{2} d^{4} e^{7}}{4}\right ) + x^{7} \cdot \left (2 a c d e^{6} + b^{2} d e^{6} + 24 b c d^{3} e^{6} + 66 c^{2} d^{5} e^{6}\right ) + x^{6} \left (\frac {a b e^{5}}{3} + 7 a c d^{2} e^{5} + \frac {7 b^{2} d^{2} e^{5}}{2} + 42 b c d^{4} e^{5} + 77 c^{2} d^{6} e^{5}\right ) + x^{5} \cdot \left (2 a b d e^{4} + 14 a c d^{3} e^{4} + 7 b^{2} d^{3} e^{4} + \frac {252 b c d^{5} e^{4}}{5} + 66 c^{2} d^{7} e^{4}\right ) + x^{4} \left (\frac {a^{2} e^{3}}{4} + 5 a b d^{2} e^{3} + \frac {35 a c d^{4} e^{3}}{2} + \frac {35 b^{2} d^{4} e^{3}}{4} + 42 b c d^{6} e^{3} + \frac {165 c^{2} d^{8} e^{3}}{4}\right ) + x^{3} \left (a^{2} d e^{2} + \frac {20 a b d^{3} e^{2}}{3} + 14 a c d^{5} e^{2} + 7 b^{2} d^{5} e^{2} + 24 b c d^{7} e^{2} + \frac {55 c^{2} d^{9} e^{2}}{3}\right ) + x^{2} \cdot \left (\frac {3 a^{2} d^{2} e}{2} + 5 a b d^{4} e + 7 a c d^{6} e + \frac {7 b^{2} d^{6} e}{2} + 9 b c d^{8} e + \frac {11 c^{2} d^{10} e}{2}\right ) + x \left (a^{2} d^{3} + 2 a b d^{5} + 2 a c d^{7} + b^{2} d^{7} + 2 b c d^{9} + c^{2} d^{11}\right ) \]

[In]

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

[Out]

c**2*d*e**10*x**11 + c**2*e**11*x**12/12 + x**10*(b*c*e**9/5 + 11*c**2*d**2*e**9/2) + x**9*(2*b*c*d*e**8 + 55*
c**2*d**3*e**8/3) + x**8*(a*c*e**7/4 + b**2*e**7/8 + 9*b*c*d**2*e**7 + 165*c**2*d**4*e**7/4) + x**7*(2*a*c*d*e
**6 + b**2*d*e**6 + 24*b*c*d**3*e**6 + 66*c**2*d**5*e**6) + x**6*(a*b*e**5/3 + 7*a*c*d**2*e**5 + 7*b**2*d**2*e
**5/2 + 42*b*c*d**4*e**5 + 77*c**2*d**6*e**5) + x**5*(2*a*b*d*e**4 + 14*a*c*d**3*e**4 + 7*b**2*d**3*e**4 + 252
*b*c*d**5*e**4/5 + 66*c**2*d**7*e**4) + x**4*(a**2*e**3/4 + 5*a*b*d**2*e**3 + 35*a*c*d**4*e**3/2 + 35*b**2*d**
4*e**3/4 + 42*b*c*d**6*e**3 + 165*c**2*d**8*e**3/4) + x**3*(a**2*d*e**2 + 20*a*b*d**3*e**2/3 + 14*a*c*d**5*e**
2 + 7*b**2*d**5*e**2 + 24*b*c*d**7*e**2 + 55*c**2*d**9*e**2/3) + x**2*(3*a**2*d**2*e/2 + 5*a*b*d**4*e + 7*a*c*
d**6*e + 7*b**2*d**6*e/2 + 9*b*c*d**8*e + 11*c**2*d**10*e/2) + x*(a**2*d**3 + 2*a*b*d**5 + 2*a*c*d**7 + b**2*d
**7 + 2*b*c*d**9 + c**2*d**11)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (79) = 158\).

Time = 0.21 (sec) , antiderivative size = 403, normalized size of antiderivative = 4.53 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx=\frac {1}{12} \, c^{2} e^{11} x^{12} + c^{2} d e^{10} x^{11} + \frac {1}{10} \, {\left (55 \, c^{2} d^{2} + 2 \, b c\right )} e^{9} x^{10} + \frac {1}{3} \, {\left (55 \, c^{2} d^{3} + 6 \, b c d\right )} e^{8} x^{9} + \frac {1}{8} \, {\left (330 \, c^{2} d^{4} + 72 \, b c d^{2} + b^{2} + 2 \, a c\right )} e^{7} x^{8} + {\left (66 \, c^{2} d^{5} + 24 \, b c d^{3} + {\left (b^{2} + 2 \, a c\right )} d\right )} e^{6} x^{7} + \frac {1}{6} \, {\left (462 \, c^{2} d^{6} + 252 \, b c d^{4} + 21 \, {\left (b^{2} + 2 \, a c\right )} d^{2} + 2 \, a b\right )} e^{5} x^{6} + \frac {1}{5} \, {\left (330 \, c^{2} d^{7} + 252 \, b c d^{5} + 35 \, {\left (b^{2} + 2 \, a c\right )} d^{3} + 10 \, a b d\right )} e^{4} x^{5} + \frac {1}{4} \, {\left (165 \, c^{2} d^{8} + 168 \, b c d^{6} + 35 \, {\left (b^{2} + 2 \, a c\right )} d^{4} + 20 \, a b d^{2} + a^{2}\right )} e^{3} x^{4} + \frac {1}{3} \, {\left (55 \, c^{2} d^{9} + 72 \, b c d^{7} + 21 \, {\left (b^{2} + 2 \, a c\right )} d^{5} + 20 \, a b d^{3} + 3 \, a^{2} d\right )} e^{2} x^{3} + \frac {1}{2} \, {\left (11 \, c^{2} d^{10} + 18 \, b c d^{8} + 7 \, {\left (b^{2} + 2 \, a c\right )} d^{6} + 10 \, a b d^{4} + 3 \, a^{2} d^{2}\right )} e x^{2} + {\left (c^{2} d^{11} + 2 \, b c d^{9} + {\left (b^{2} + 2 \, a c\right )} d^{7} + 2 \, a b d^{5} + a^{2} d^{3}\right )} x \]

[In]

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

[Out]

1/12*c^2*e^11*x^12 + c^2*d*e^10*x^11 + 1/10*(55*c^2*d^2 + 2*b*c)*e^9*x^10 + 1/3*(55*c^2*d^3 + 6*b*c*d)*e^8*x^9
+ 1/8*(330*c^2*d^4 + 72*b*c*d^2 + b^2 + 2*a*c)*e^7*x^8 + (66*c^2*d^5 + 24*b*c*d^3 + (b^2 + 2*a*c)*d)*e^6*x^7
+ 1/6*(462*c^2*d^6 + 252*b*c*d^4 + 21*(b^2 + 2*a*c)*d^2 + 2*a*b)*e^5*x^6 + 1/5*(330*c^2*d^7 + 252*b*c*d^5 + 35
*(b^2 + 2*a*c)*d^3 + 10*a*b*d)*e^4*x^5 + 1/4*(165*c^2*d^8 + 168*b*c*d^6 + 35*(b^2 + 2*a*c)*d^4 + 20*a*b*d^2 +
a^2)*e^3*x^4 + 1/3*(55*c^2*d^9 + 72*b*c*d^7 + 21*(b^2 + 2*a*c)*d^5 + 20*a*b*d^3 + 3*a^2*d)*e^2*x^3 + 1/2*(11*c
^2*d^10 + 18*b*c*d^8 + 7*(b^2 + 2*a*c)*d^6 + 10*a*b*d^4 + 3*a^2*d^2)*e*x^2 + (c^2*d^11 + 2*b*c*d^9 + (b^2 + 2*
a*c)*d^7 + 2*a*b*d^5 + a^2*d^3)*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (79) = 158\).

Time = 0.30 (sec) , antiderivative size = 475, normalized size of antiderivative = 5.34 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx=\frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} c^{2} d^{10} + \frac {5}{4} \, {\left (e x^{2} + 2 \, d x\right )}^{2} c^{2} d^{8} e + \frac {5}{3} \, {\left (e x^{2} + 2 \, d x\right )}^{3} c^{2} d^{6} e^{2} + \frac {5}{4} \, {\left (e x^{2} + 2 \, d x\right )}^{4} c^{2} d^{4} e^{3} + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )}^{5} c^{2} d^{2} e^{4} + \frac {1}{12} \, {\left (e x^{2} + 2 \, d x\right )}^{6} c^{2} e^{5} + {\left (e x^{2} + 2 \, d x\right )} b c d^{8} + 2 \, {\left (e x^{2} + 2 \, d x\right )}^{2} b c d^{6} e + 2 \, {\left (e x^{2} + 2 \, d x\right )}^{3} b c d^{4} e^{2} + {\left (e x^{2} + 2 \, d x\right )}^{4} b c d^{2} e^{3} + \frac {1}{5} \, {\left (e x^{2} + 2 \, d x\right )}^{5} b c e^{4} + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} b^{2} d^{6} + {\left (e x^{2} + 2 \, d x\right )} a c d^{6} + \frac {3}{4} \, {\left (e x^{2} + 2 \, d x\right )}^{2} b^{2} d^{4} e + \frac {3}{2} \, {\left (e x^{2} + 2 \, d x\right )}^{2} a c d^{4} e + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )}^{3} b^{2} d^{2} e^{2} + {\left (e x^{2} + 2 \, d x\right )}^{3} a c d^{2} e^{2} + \frac {1}{8} \, {\left (e x^{2} + 2 \, d x\right )}^{4} b^{2} e^{3} + \frac {1}{4} \, {\left (e x^{2} + 2 \, d x\right )}^{4} a c e^{3} + {\left (e x^{2} + 2 \, d x\right )} a b d^{4} + {\left (e x^{2} + 2 \, d x\right )}^{2} a b d^{2} e + \frac {1}{3} \, {\left (e x^{2} + 2 \, d x\right )}^{3} a b e^{2} + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} a^{2} d^{2} + \frac {1}{4} \, {\left (e x^{2} + 2 \, d x\right )}^{2} a^{2} e \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.88 (sec) , antiderivative size = 383, normalized size of antiderivative = 4.30 \[ \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx=\frac {e^7\,x^8\,\left (b^2+72\,b\,c\,d^2+330\,c^2\,d^4+2\,a\,c\right )}{8}+\frac {e^5\,x^6\,\left (21\,b^2\,d^2+252\,b\,c\,d^4+2\,a\,b+462\,c^2\,d^6+42\,a\,c\,d^2\right )}{6}+\frac {e^3\,x^4\,\left (a^2+20\,a\,b\,d^2+70\,a\,c\,d^4+35\,b^2\,d^4+168\,b\,c\,d^6+165\,c^2\,d^8\right )}{4}+\frac {c^2\,e^{11}\,x^{12}}{12}+d^3\,x\,{\left (c\,d^4+b\,d^2+a\right )}^2+\frac {c\,e^9\,x^{10}\,\left (55\,c\,d^2+2\,b\right )}{10}+c^2\,d\,e^{10}\,x^{11}+\frac {d^2\,e\,x^2\,\left (3\,a^2+10\,a\,b\,d^2+14\,a\,c\,d^4+7\,b^2\,d^4+18\,b\,c\,d^6+11\,c^2\,d^8\right )}{2}+\frac {d\,e^2\,x^3\,\left (3\,a^2+20\,a\,b\,d^2+42\,a\,c\,d^4+21\,b^2\,d^4+72\,b\,c\,d^6+55\,c^2\,d^8\right )}{3}+d\,e^6\,x^7\,\left (b^2+24\,b\,c\,d^2+66\,c^2\,d^4+2\,a\,c\right )+\frac {d\,e^4\,x^5\,\left (35\,b^2\,d^2+252\,b\,c\,d^4+10\,a\,b+330\,c^2\,d^6+70\,a\,c\,d^2\right )}{5}+\frac {c\,d\,e^8\,x^9\,\left (55\,c\,d^2+6\,b\right )}{3} \]

[In]

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

[Out]

(e^7*x^8*(2*a*c + b^2 + 330*c^2*d^4 + 72*b*c*d^2))/8 + (e^5*x^6*(2*a*b + 21*b^2*d^2 + 462*c^2*d^6 + 42*a*c*d^2
+ 252*b*c*d^4))/6 + (e^3*x^4*(a^2 + 35*b^2*d^4 + 165*c^2*d^8 + 20*a*b*d^2 + 70*a*c*d^4 + 168*b*c*d^6))/4 + (c
^2*e^11*x^12)/12 + d^3*x*(a + b*d^2 + c*d^4)^2 + (c*e^9*x^10*(2*b + 55*c*d^2))/10 + c^2*d*e^10*x^11 + (d^2*e*x
^2*(3*a^2 + 7*b^2*d^4 + 11*c^2*d^8 + 10*a*b*d^2 + 14*a*c*d^4 + 18*b*c*d^6))/2 + (d*e^2*x^3*(3*a^2 + 21*b^2*d^4
+ 55*c^2*d^8 + 20*a*b*d^2 + 42*a*c*d^4 + 72*b*c*d^6))/3 + d*e^6*x^7*(2*a*c + b^2 + 66*c^2*d^4 + 24*b*c*d^2) +
(d*e^4*x^5*(10*a*b + 35*b^2*d^2 + 330*c^2*d^6 + 70*a*c*d^2 + 252*b*c*d^4))/5 + (c*d*e^8*x^9*(6*b + 55*c*d^2))
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