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

[Out]

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

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Rubi [A]  time = 0.32, antiderivative size = 159, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [B]  time = 0.04, 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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fricas [B]  time = 0.76, size = 1635, normalized size = 10.28 \[ \text {result too large to display} \]

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

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giac [B]  time = 0.51, size = 1360, normalized size = 8.55 \[ \text {result too large to display} \]

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]

1/2*(f*x^2*e + 2*d*f*x)*c^3*d^14*f^2 + 3/2*(f*x^2*e + 2*d*f*x)*b*c^2*d^12*f^2 + 3/2*(f*x^2*e + 2*d*f*x)*b^2*c*
d^10*f^2 + 3/2*(f*x^2*e + 2*d*f*x)*a*c^2*d^10*f^2 + 1/2*(f*x^2*e + 2*d*f*x)*b^3*d^8*f^2 + 3*(f*x^2*e + 2*d*f*x
)*a*b*c*d^8*f^2 + 3/2*(f*x^2*e + 2*d*f*x)*a*b^2*d^6*f^2 + 3/2*(f*x^2*e + 2*d*f*x)*a^2*c*d^6*f^2 + 3/2*(f*x^2*e
+ 2*d*f*x)*a^2*b*d^4*f^2 + 1/2*(f*x^2*e + 2*d*f*x)*a^3*d^2*f^2 + 1/560*(980*(f*x^2*e + 2*d*f*x)^2*c^3*d^12*f^
6*e + 1960*(f*x^2*e + 2*d*f*x)^3*c^3*d^10*f^5*e^2 + 2520*(f*x^2*e + 2*d*f*x)^2*b*c^2*d^10*f^6*e + 2450*(f*x^2*
e + 2*d*f*x)^4*c^3*d^8*f^4*e^3 + 4200*(f*x^2*e + 2*d*f*x)^3*b*c^2*d^8*f^5*e^2 + 2100*(f*x^2*e + 2*d*f*x)^2*b^2
*c*d^8*f^6*e + 2100*(f*x^2*e + 2*d*f*x)^2*a*c^2*d^8*f^6*e + 1960*(f*x^2*e + 2*d*f*x)^5*c^3*d^6*f^3*e^4 + 4200*
(f*x^2*e + 2*d*f*x)^4*b*c^2*d^6*f^4*e^3 + 2800*(f*x^2*e + 2*d*f*x)^3*b^2*c*d^6*f^5*e^2 + 2800*(f*x^2*e + 2*d*f
*x)^3*a*c^2*d^6*f^5*e^2 + 560*(f*x^2*e + 2*d*f*x)^2*b^3*d^6*f^6*e + 3360*(f*x^2*e + 2*d*f*x)^2*a*b*c*d^6*f^6*e
+ 980*(f*x^2*e + 2*d*f*x)^6*c^3*d^4*f^2*e^5 + 2520*(f*x^2*e + 2*d*f*x)^5*b*c^2*d^4*f^3*e^4 + 2100*(f*x^2*e +
2*d*f*x)^4*b^2*c*d^4*f^4*e^3 + 2100*(f*x^2*e + 2*d*f*x)^4*a*c^2*d^4*f^4*e^3 + 560*(f*x^2*e + 2*d*f*x)^3*b^3*d^
4*f^5*e^2 + 3360*(f*x^2*e + 2*d*f*x)^3*a*b*c*d^4*f^5*e^2 + 1260*(f*x^2*e + 2*d*f*x)^2*a*b^2*d^4*f^6*e + 1260*(
f*x^2*e + 2*d*f*x)^2*a^2*c*d^4*f^6*e + 280*(f*x^2*e + 2*d*f*x)^7*c^3*d^2*f*e^6 + 840*(f*x^2*e + 2*d*f*x)^6*b*c
^2*d^2*f^2*e^5 + 840*(f*x^2*e + 2*d*f*x)^5*b^2*c*d^2*f^3*e^4 + 840*(f*x^2*e + 2*d*f*x)^5*a*c^2*d^2*f^3*e^4 + 2
80*(f*x^2*e + 2*d*f*x)^4*b^3*d^2*f^4*e^3 + 1680*(f*x^2*e + 2*d*f*x)^4*a*b*c*d^2*f^4*e^3 + 840*(f*x^2*e + 2*d*f
*x)^3*a*b^2*d^2*f^5*e^2 + 840*(f*x^2*e + 2*d*f*x)^3*a^2*c*d^2*f^5*e^2 + 840*(f*x^2*e + 2*d*f*x)^2*a^2*b*d^2*f^
6*e + 35*(f*x^2*e + 2*d*f*x)^8*c^3*e^7 + 120*(f*x^2*e + 2*d*f*x)^7*b*c^2*f*e^6 + 140*(f*x^2*e + 2*d*f*x)^6*b^2
*c*f^2*e^5 + 140*(f*x^2*e + 2*d*f*x)^6*a*c^2*f^2*e^5 + 56*(f*x^2*e + 2*d*f*x)^5*b^3*f^3*e^4 + 336*(f*x^2*e + 2
*d*f*x)^5*a*b*c*f^3*e^4 + 210*(f*x^2*e + 2*d*f*x)^4*a*b^2*f^4*e^3 + 210*(f*x^2*e + 2*d*f*x)^4*a^2*c*f^4*e^3 +
280*(f*x^2*e + 2*d*f*x)^3*a^2*b*f^5*e^2 + 140*(f*x^2*e + 2*d*f*x)^2*a^3*f^6*e)/f^5

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maple [B]  time = 0.00, size = 7697, normalized size = 48.41 \[ \text {output too large to display} \]

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 = 1.16, size = 920, normalized size = 5.79 \[ \frac {1}{16} \, c^{3} e^{15} f^{3} x^{16} + c^{3} d e^{14} f^{3} x^{15} + \frac {3}{14} \, {\left (35 \, c^{3} d^{2} + b c^{2}\right )} e^{13} f^{3} x^{14} + {\left (35 \, c^{3} d^{3} + 3 \, b c^{2} d\right )} e^{12} f^{3} 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} f^{3} 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} f^{3} 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} f^{3} 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} f^{3} 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} f^{3} 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} f^{3} 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} f^{3} 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} f^{3} 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} f^{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} f^{3} 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 f^{3} 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 )} f^{3} x \]

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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mupad [B]  time = 1.65, size = 825, normalized size = 5.19 \[ \frac {3\,e^7\,f^3\,x^8\,\left (a^2\,c+a\,b^2+72\,a\,b\,c\,d^2+330\,a\,c^2\,d^4+12\,b^3\,d^2+330\,b^2\,c\,d^4+1716\,b\,c^2\,d^6+2145\,c^3\,d^8\right )}{8}+\frac {e^5\,f^3\,x^6\,\left (a^2\,b+21\,a^2\,c\,d^2+21\,a\,b^2\,d^2+252\,a\,b\,c\,d^4+462\,a\,c^2\,d^6+42\,b^3\,d^4+462\,b^2\,c\,d^6+1287\,b\,c^2\,d^8+1001\,c^3\,d^{10}\right )}{2}+\frac {e^9\,f^3\,x^{10}\,\left (b^3+165\,b^2\,c\,d^2+2145\,b\,c^2\,d^4+6\,a\,b\,c+5005\,c^3\,d^6+165\,a\,c^2\,d^2\right )}{10}+\frac {c^3\,e^{15}\,f^3\,x^{16}}{16}+d^3\,f^3\,x\,{\left (c\,d^4+b\,d^2+a\right )}^3+\frac {e^3\,f^3\,x^4\,\left (a^3+30\,a^2\,b\,d^2+105\,a^2\,c\,d^4+105\,a\,b^2\,d^4+504\,a\,b\,c\,d^6+495\,a\,c^2\,d^8+84\,b^3\,d^6+495\,b^2\,c\,d^8+858\,b\,c^2\,d^{10}+455\,c^3\,d^{12}\right )}{4}+\frac {c\,e^{11}\,f^3\,x^{12}\,\left (b^2+78\,b\,c\,d^2+455\,c^2\,d^4+a\,c\right )}{4}+\frac {d\,e^6\,f^3\,x^7\,\left (21\,a^2\,c+21\,a\,b^2+504\,a\,b\,c\,d^2+1386\,a\,c^2\,d^4+84\,b^3\,d^2+1386\,b^2\,c\,d^4+5148\,b\,c^2\,d^6+5005\,c^3\,d^8\right )}{7}+\frac {3\,d\,e^4\,f^3\,x^5\,\left (5\,a^2\,b+35\,a^2\,c\,d^2+35\,a\,b^2\,d^2+252\,a\,b\,c\,d^4+330\,a\,c^2\,d^6+42\,b^3\,d^4+330\,b^2\,c\,d^6+715\,b\,c^2\,d^8+455\,c^3\,d^{10}\right )}{5}+d\,e^8\,f^3\,x^9\,\left (b^3+55\,b^2\,c\,d^2+429\,b\,c^2\,d^4+6\,a\,b\,c+715\,c^3\,d^6+55\,a\,c^2\,d^2\right )+\frac {3\,c^2\,e^{13}\,f^3\,x^{14}\,\left (35\,c\,d^2+b\right )}{14}+c^3\,d\,e^{14}\,f^3\,x^{15}+d\,e^2\,f^3\,x^3\,\left (a^3+10\,a^2\,b\,d^2+21\,a^2\,c\,d^4+21\,a\,b^2\,d^4+72\,a\,b\,c\,d^6+55\,a\,c^2\,d^8+12\,b^3\,d^6+55\,b^2\,c\,d^8+78\,b\,c^2\,d^{10}+35\,c^3\,d^{12}\right )+\frac {3\,d^2\,e\,f^3\,x^2\,{\left (c\,d^4+b\,d^2+a\right )}^2\,\left (5\,c\,d^4+3\,b\,d^2+a\right )}{2}+c^2\,d\,e^{12}\,f^3\,x^{13}\,\left (35\,c\,d^2+3\,b\right )+3\,c\,d\,e^{10}\,f^3\,x^{11}\,\left (b^2+26\,b\,c\,d^2+91\,c^2\,d^4+a\,c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*e^7*f^3*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 + 7
2*a*b*c*d^2))/8 + (e^5*f^3*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*f^3*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*f^3*x^16)/16 + d^3*f^3*x*(a + b*d^2 + c*d^4)^3 + (e^
3*f^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 + 49
5*b^2*c*d^8 + 858*b*c^2*d^10 + 504*a*b*c*d^6))/4 + (c*e^11*f^3*x^12*(a*c + b^2 + 455*c^2*d^4 + 78*b*c*d^2))/4
+ (d*e^6*f^3*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*f^3*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*f^3*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*c^2*e^13*f^3*x^14*(b + 35*c*d^2))/14 + c^3*d*e^14*f^
3*x^15 + d*e^2*f^3*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) + (3*d^2*e*f^3*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*f^3*x^13*(3*b + 35*c*d^2) + 3*c*d*e^10*f^3*x^11*(a*c + b^2 + 91*c^2*d^4 + 26*b*c*d^2)

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sympy [B]  time = 0.39, size = 1654, normalized size = 10.40 \[ \text {result too large to display} \]

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)

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