3.612 \(\int (d f+e f x)^3 (a+b (d+e x)^2+c (d+e x)^4)^3 \, dx\)

Optimal. Leaf size=159 \[ \frac{a^2 b f^3 (d+e x)^6}{2 e}+\frac{a^3 f^3 (d+e x)^4}{4 e}+\frac{c f^3 \left (a c+b^2\right ) (d+e x)^{12}}{4 e}+\frac{b f^3 \left (6 a c+b^2\right ) (d+e x)^{10}}{10 e}+\frac{3 a f^3 \left (a c+b^2\right ) (d+e x)^8}{8 e}+\frac{3 b c^2 f^3 (d+e x)^{14}}{14 e}+\frac{c^3 f^3 (d+e x)^{16}}{16 e} \]

[Out]

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

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Rubi [A]  time = 0.315301, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1142, 1114, 631} \[ \frac{a^2 b f^3 (d+e x)^6}{2 e}+\frac{a^3 f^3 (d+e x)^4}{4 e}+\frac{c f^3 \left (a c+b^2\right ) (d+e x)^{12}}{4 e}+\frac{b f^3 \left (6 a c+b^2\right ) (d+e x)^{10}}{10 e}+\frac{3 a f^3 \left (a c+b^2\right ) (d+e x)^8}{8 e}+\frac{3 b c^2 f^3 (d+e x)^{14}}{14 e}+\frac{c^3 f^3 (d+e x)^{16}}{16 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1142

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]

Rule 1114

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 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 f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx &=\frac{f^3 \operatorname{Subst}\left (\int x^3 \left (a+b x^2+c x^4\right )^3 \, dx,x,d+e x\right )}{e}\\ &=\frac{f^3 \operatorname{Subst}\left (\int x \left (a+b x+c x^2\right )^3 \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac{f^3 \operatorname{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 f^3 (d+e x)^4}{4 e}+\frac{a^2 b f^3 (d+e x)^6}{2 e}+\frac{3 a \left (b^2+a c\right ) f^3 (d+e x)^8}{8 e}+\frac{b \left (b^2+6 a c\right ) f^3 (d+e x)^{10}}{10 e}+\frac{c \left (b^2+a c\right ) f^3 (d+e x)^{12}}{4 e}+\frac{3 b c^2 f^3 (d+e x)^{14}}{14 e}+\frac{c^3 f^3 (d+e x)^{16}}{16 e}\\ \end{align*}

Mathematica [B]  time = 0.0408879, size = 801, normalized size = 5.04 \[ f^3 \left (\frac{1}{16} c^3 e^{15} x^{16}+c^3 d e^{14} x^{15}+\frac{3}{14} c^2 \left (35 c d^2+b\right ) e^{13} x^{14}+c^2 d \left (35 c d^2+3 b\right ) e^{12} x^{13}+\frac{1}{4} c \left (455 c^2 d^4+78 b c d^2+b^2+a c\right ) e^{11} x^{12}+3 c d \left (91 c^2 d^4+26 b c d^2+b^2+a c\right ) e^{10} x^{11}+\frac{1}{10} \left (5005 c^3 d^6+2145 b c^2 d^4+165 a c^2 d^2+165 b^2 c d^2+b^3+6 a b c\right ) e^9 x^{10}+d \left (715 c^3 d^6+429 b c^2 d^4+55 a c^2 d^2+55 b^2 c d^2+b^3+6 a b c\right ) e^8 x^9+\frac{3}{8} \left (2145 c^3 d^8+1716 b c^2 d^6+330 a c^2 d^4+330 b^2 c d^4+12 b^3 d^2+72 a b c d^2+a b^2+a^2 c\right ) e^7 x^8+\frac{1}{7} d \left (5005 c^3 d^8+5148 b c^2 d^6+1386 a c^2 d^4+1386 b^2 c d^4+84 b^3 d^2+504 a b c d^2+21 a b^2+21 a^2 c\right ) e^6 x^7+\frac{1}{2} \left (1001 c^3 d^{10}+1287 b c^2 d^8+462 a c^2 d^6+462 b^2 c d^6+42 b^3 d^4+252 a b c d^4+21 a b^2 d^2+21 a^2 c d^2+a^2 b\right ) e^5 x^6+\frac{3}{5} d \left (455 c^3 d^{10}+715 b c^2 d^8+330 a c^2 d^6+330 b^2 c d^6+42 b^3 d^4+252 a b c d^4+35 a b^2 d^2+35 a^2 c d^2+5 a^2 b\right ) e^4 x^5+\frac{1}{4} \left (455 c^3 d^{12}+858 b c^2 d^{10}+495 a c^2 d^8+495 b^2 c d^8+84 b^3 d^6+504 a b c d^6+105 a b^2 d^4+105 a^2 c d^4+30 a^2 b d^2+a^3\right ) e^3 x^4+d \left (35 c^3 d^{12}+78 b c^2 d^{10}+55 a c^2 d^8+55 b^2 c d^8+12 b^3 d^6+72 a b c d^6+21 a b^2 d^4+21 a^2 c d^4+10 a^2 b d^2+a^3\right ) e^2 x^3+\frac{3}{2} d^2 \left (c d^4+b d^2+a\right )^2 \left (5 c d^4+3 b d^2+a\right ) e x^2+d^3 \left (c d^4+b d^2+a\right )^3 x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

f^3*(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 + 495*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^1
3*x^14)/14 + c^3*d*e^14*x^15 + (c^3*e^15*x^16)/16)

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Maple [B]  time = 0.001, size = 7697, normalized size = 48.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [B]  time = 1.00452, size = 1242, normalized size = 7.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*c^3*e^15*f^3*x^16 + c^3*d*e^14*f^3*x^15 + 3/14*(35*c^3*d^2 + b*c^2)*e^13*f^3*x^14 + (35*c^3*d^3 + 3*b*c^2
*d)*e^12*f^3*x^13 + 1/4*(455*c^3*d^4 + 78*b*c^2*d^2 + b^2*c + a*c^2)*e^11*f^3*x^12 + 3*(91*c^3*d^5 + 26*b*c^2*
d^3 + (b^2*c + a*c^2)*d)*e^10*f^3*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*f^3*x^10 + (715*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*f^3*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
*f^3*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*f^3*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*f^3*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*f^3*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*f^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*f^3*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*f^3*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)*f^3*x

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Fricas [B]  time = 1.56755, size = 3555, normalized size = 22.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*x^16*f^3*e^15*c^3 + x^15*f^3*e^14*d*c^3 + 15/2*x^14*f^3*e^13*d^2*c^3 + 35*x^13*f^3*e^12*d^3*c^3 + 455/4*x
^12*f^3*e^11*d^4*c^3 + 3/14*x^14*f^3*e^13*c^2*b + 273*x^11*f^3*e^10*d^5*c^3 + 3*x^13*f^3*e^12*d*c^2*b + 1001/2
*x^10*f^3*e^9*d^6*c^3 + 39/2*x^12*f^3*e^11*d^2*c^2*b + 715*x^9*f^3*e^8*d^7*c^3 + 78*x^11*f^3*e^10*d^3*c^2*b +
6435/8*x^8*f^3*e^7*d^8*c^3 + 429/2*x^10*f^3*e^9*d^4*c^2*b + 1/4*x^12*f^3*e^11*c*b^2 + 1/4*x^12*f^3*e^11*c^2*a
+ 715*x^7*f^3*e^6*d^9*c^3 + 429*x^9*f^3*e^8*d^5*c^2*b + 3*x^11*f^3*e^10*d*c*b^2 + 3*x^11*f^3*e^10*d*c^2*a + 10
01/2*x^6*f^3*e^5*d^10*c^3 + 1287/2*x^8*f^3*e^7*d^6*c^2*b + 33/2*x^10*f^3*e^9*d^2*c*b^2 + 33/2*x^10*f^3*e^9*d^2
*c^2*a + 273*x^5*f^3*e^4*d^11*c^3 + 5148/7*x^7*f^3*e^6*d^7*c^2*b + 55*x^9*f^3*e^8*d^3*c*b^2 + 55*x^9*f^3*e^8*d
^3*c^2*a + 455/4*x^4*f^3*e^3*d^12*c^3 + 1287/2*x^6*f^3*e^5*d^8*c^2*b + 495/4*x^8*f^3*e^7*d^4*c*b^2 + 1/10*x^10
*f^3*e^9*b^3 + 495/4*x^8*f^3*e^7*d^4*c^2*a + 3/5*x^10*f^3*e^9*c*b*a + 35*x^3*f^3*e^2*d^13*c^3 + 429*x^5*f^3*e^
4*d^9*c^2*b + 198*x^7*f^3*e^6*d^5*c*b^2 + x^9*f^3*e^8*d*b^3 + 198*x^7*f^3*e^6*d^5*c^2*a + 6*x^9*f^3*e^8*d*c*b*
a + 15/2*x^2*f^3*e*d^14*c^3 + 429/2*x^4*f^3*e^3*d^10*c^2*b + 231*x^6*f^3*e^5*d^6*c*b^2 + 9/2*x^8*f^3*e^7*d^2*b
^3 + 231*x^6*f^3*e^5*d^6*c^2*a + 27*x^8*f^3*e^7*d^2*c*b*a + x*f^3*d^15*c^3 + 78*x^3*f^3*e^2*d^11*c^2*b + 198*x
^5*f^3*e^4*d^7*c*b^2 + 12*x^7*f^3*e^6*d^3*b^3 + 198*x^5*f^3*e^4*d^7*c^2*a + 72*x^7*f^3*e^6*d^3*c*b*a + 39/2*x^
2*f^3*e*d^12*c^2*b + 495/4*x^4*f^3*e^3*d^8*c*b^2 + 21*x^6*f^3*e^5*d^4*b^3 + 495/4*x^4*f^3*e^3*d^8*c^2*a + 126*
x^6*f^3*e^5*d^4*c*b*a + 3/8*x^8*f^3*e^7*b^2*a + 3/8*x^8*f^3*e^7*c*a^2 + 3*x*f^3*d^13*c^2*b + 55*x^3*f^3*e^2*d^
9*c*b^2 + 126/5*x^5*f^3*e^4*d^5*b^3 + 55*x^3*f^3*e^2*d^9*c^2*a + 756/5*x^5*f^3*e^4*d^5*c*b*a + 3*x^7*f^3*e^6*d
*b^2*a + 3*x^7*f^3*e^6*d*c*a^2 + 33/2*x^2*f^3*e*d^10*c*b^2 + 21*x^4*f^3*e^3*d^6*b^3 + 33/2*x^2*f^3*e*d^10*c^2*
a + 126*x^4*f^3*e^3*d^6*c*b*a + 21/2*x^6*f^3*e^5*d^2*b^2*a + 21/2*x^6*f^3*e^5*d^2*c*a^2 + 3*x*f^3*d^11*c*b^2 +
12*x^3*f^3*e^2*d^7*b^3 + 3*x*f^3*d^11*c^2*a + 72*x^3*f^3*e^2*d^7*c*b*a + 21*x^5*f^3*e^4*d^3*b^2*a + 21*x^5*f^
3*e^4*d^3*c*a^2 + 9/2*x^2*f^3*e*d^8*b^3 + 27*x^2*f^3*e*d^8*c*b*a + 105/4*x^4*f^3*e^3*d^4*b^2*a + 105/4*x^4*f^3
*e^3*d^4*c*a^2 + 1/2*x^6*f^3*e^5*b*a^2 + x*f^3*d^9*b^3 + 6*x*f^3*d^9*c*b*a + 21*x^3*f^3*e^2*d^5*b^2*a + 21*x^3
*f^3*e^2*d^5*c*a^2 + 3*x^5*f^3*e^4*d*b*a^2 + 21/2*x^2*f^3*e*d^6*b^2*a + 21/2*x^2*f^3*e*d^6*c*a^2 + 15/2*x^4*f^
3*e^3*d^2*b*a^2 + 3*x*f^3*d^7*b^2*a + 3*x*f^3*d^7*c*a^2 + 10*x^3*f^3*e^2*d^3*b*a^2 + 15/2*x^2*f^3*e*d^4*b*a^2
+ 1/4*x^4*f^3*e^3*a^3 + 3*x*f^3*d^5*b*a^2 + x^3*f^3*e^2*d*a^3 + 3/2*x^2*f^3*e*d^2*a^3 + x*f^3*d^3*a^3

________________________________________________________________________________________

Sympy [B]  time = 0.317605, size = 1654, normalized size = 10.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.15179, size = 2113, normalized size = 13.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*c^3*f^3*x^16*e^15 + c^3*d*f^3*x^15*e^14 + 15/2*c^3*d^2*f^3*x^14*e^13 + 35*c^3*d^3*f^3*x^13*e^12 + 455/4*c
^3*d^4*f^3*x^12*e^11 + 273*c^3*d^5*f^3*x^11*e^10 + 1001/2*c^3*d^6*f^3*x^10*e^9 + 715*c^3*d^7*f^3*x^9*e^8 + 643
5/8*c^3*d^8*f^3*x^8*e^7 + 715*c^3*d^9*f^3*x^7*e^6 + 1001/2*c^3*d^10*f^3*x^6*e^5 + 273*c^3*d^11*f^3*x^5*e^4 + 4
55/4*c^3*d^12*f^3*x^4*e^3 + 35*c^3*d^13*f^3*x^3*e^2 + 15/2*c^3*d^14*f^3*x^2*e + c^3*d^15*f^3*x + 3/14*b*c^2*f^
3*x^14*e^13 + 3*b*c^2*d*f^3*x^13*e^12 + 39/2*b*c^2*d^2*f^3*x^12*e^11 + 78*b*c^2*d^3*f^3*x^11*e^10 + 429/2*b*c^
2*d^4*f^3*x^10*e^9 + 429*b*c^2*d^5*f^3*x^9*e^8 + 1287/2*b*c^2*d^6*f^3*x^8*e^7 + 5148/7*b*c^2*d^7*f^3*x^7*e^6 +
1287/2*b*c^2*d^8*f^3*x^6*e^5 + 429*b*c^2*d^9*f^3*x^5*e^4 + 429/2*b*c^2*d^10*f^3*x^4*e^3 + 78*b*c^2*d^11*f^3*x
^3*e^2 + 39/2*b*c^2*d^12*f^3*x^2*e + 3*b*c^2*d^13*f^3*x + 1/4*b^2*c*f^3*x^12*e^11 + 1/4*a*c^2*f^3*x^12*e^11 +
3*b^2*c*d*f^3*x^11*e^10 + 3*a*c^2*d*f^3*x^11*e^10 + 33/2*b^2*c*d^2*f^3*x^10*e^9 + 33/2*a*c^2*d^2*f^3*x^10*e^9
+ 55*b^2*c*d^3*f^3*x^9*e^8 + 55*a*c^2*d^3*f^3*x^9*e^8 + 495/4*b^2*c*d^4*f^3*x^8*e^7 + 495/4*a*c^2*d^4*f^3*x^8*
e^7 + 198*b^2*c*d^5*f^3*x^7*e^6 + 198*a*c^2*d^5*f^3*x^7*e^6 + 231*b^2*c*d^6*f^3*x^6*e^5 + 231*a*c^2*d^6*f^3*x^
6*e^5 + 198*b^2*c*d^7*f^3*x^5*e^4 + 198*a*c^2*d^7*f^3*x^5*e^4 + 495/4*b^2*c*d^8*f^3*x^4*e^3 + 495/4*a*c^2*d^8*
f^3*x^4*e^3 + 55*b^2*c*d^9*f^3*x^3*e^2 + 55*a*c^2*d^9*f^3*x^3*e^2 + 33/2*b^2*c*d^10*f^3*x^2*e + 33/2*a*c^2*d^1
0*f^3*x^2*e + 3*b^2*c*d^11*f^3*x + 3*a*c^2*d^11*f^3*x + 1/10*b^3*f^3*x^10*e^9 + 3/5*a*b*c*f^3*x^10*e^9 + b^3*d
*f^3*x^9*e^8 + 6*a*b*c*d*f^3*x^9*e^8 + 9/2*b^3*d^2*f^3*x^8*e^7 + 27*a*b*c*d^2*f^3*x^8*e^7 + 12*b^3*d^3*f^3*x^7
*e^6 + 72*a*b*c*d^3*f^3*x^7*e^6 + 21*b^3*d^4*f^3*x^6*e^5 + 126*a*b*c*d^4*f^3*x^6*e^5 + 126/5*b^3*d^5*f^3*x^5*e
^4 + 756/5*a*b*c*d^5*f^3*x^5*e^4 + 21*b^3*d^6*f^3*x^4*e^3 + 126*a*b*c*d^6*f^3*x^4*e^3 + 12*b^3*d^7*f^3*x^3*e^2
+ 72*a*b*c*d^7*f^3*x^3*e^2 + 9/2*b^3*d^8*f^3*x^2*e + 27*a*b*c*d^8*f^3*x^2*e + b^3*d^9*f^3*x + 6*a*b*c*d^9*f^3
*x + 3/8*a*b^2*f^3*x^8*e^7 + 3/8*a^2*c*f^3*x^8*e^7 + 3*a*b^2*d*f^3*x^7*e^6 + 3*a^2*c*d*f^3*x^7*e^6 + 21/2*a*b^
2*d^2*f^3*x^6*e^5 + 21/2*a^2*c*d^2*f^3*x^6*e^5 + 21*a*b^2*d^3*f^3*x^5*e^4 + 21*a^2*c*d^3*f^3*x^5*e^4 + 105/4*a
*b^2*d^4*f^3*x^4*e^3 + 105/4*a^2*c*d^4*f^3*x^4*e^3 + 21*a*b^2*d^5*f^3*x^3*e^2 + 21*a^2*c*d^5*f^3*x^3*e^2 + 21/
2*a*b^2*d^6*f^3*x^2*e + 21/2*a^2*c*d^6*f^3*x^2*e + 3*a*b^2*d^7*f^3*x + 3*a^2*c*d^7*f^3*x + 1/2*a^2*b*f^3*x^6*e
^5 + 3*a^2*b*d*f^3*x^5*e^4 + 15/2*a^2*b*d^2*f^3*x^4*e^3 + 10*a^2*b*d^3*f^3*x^3*e^2 + 15/2*a^2*b*d^4*f^3*x^2*e
+ 3*a^2*b*d^5*f^3*x + 1/4*a^3*f^3*x^4*e^3 + a^3*d*f^3*x^3*e^2 + 3/2*a^3*d^2*f^3*x^2*e + a^3*d^3*f^3*x