\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^{12}} \, dx\) [2162]

Optimal result

Integrand size = 20, antiderivative size = 440 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^4}{11 e^9 (d+e x)^{11}}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{5 e^9 (d+e x)^{10}}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{9 e^9 (d+e x)^9}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{2 e^9 (d+e x)^8}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{7 e^9 (d+e x)^7}+\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (d+e x)^6}-\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{5 e^9 (d+e x)^5}+\frac {c^3 (2 c d-b e)}{e^9 (d+e x)^4}-\frac {c^4}{3 e^9 (d+e x)^3} \]

[Out]

-1/11*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)^11+2/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^10-2/9*(a*e^2-b*
d*e+c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))/e^9/(e*x+d)^9+1/2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c
^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))/e^9/(e*x+d)^8+1/7*(-70*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(-3*a*e+5*b*d)+20*c^3*d^
2*e*(-3*a*e+7*b*d)-6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))/e^9/(e*x+d)^7+2/3*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*
e^2-c*e*(-3*a*e+7*b*d))/e^9/(e*x+d)^6-2/5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))/e^9/(e*x+d)^5+c^3*(-b*
e+2*c*d)/e^9/(e*x+d)^4-1/3*c^4/e^9/(e*x+d)^3

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx=-\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{7 e^9 (d+e x)^7}-\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac {2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^6}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9 (d+e x)^8}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^9 (d+e x)^9}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9 (d+e x)^{10}}-\frac {\left (a e^2-b d e+c d^2\right )^4}{11 e^9 (d+e x)^{11}}+\frac {c^3 (2 c d-b e)}{e^9 (d+e x)^4}-\frac {c^4}{3 e^9 (d+e x)^3} \]

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^12,x]

[Out]

-1/11*(c*d^2 - b*d*e + a*e^2)^4/(e^9*(d + e*x)^11) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(5*e^9*(d + e
*x)^10) - (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(9*e^9*(d + e*x)^9) + (
(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(2*e^9*(d + e*x)^8) - (70*c
^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b
*d*e + a^2*e^2))/(7*e^9*(d + e*x)^7) + (2*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(3*e^9*
(d + e*x)^6) - (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(5*e^9*(d + e*x)^5) + (c^3*(2*c*d - b*e)
)/(e^9*(d + e*x)^4) - c^4/(3*e^9*(d + e*x)^3)

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^{12}}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^{11}}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^8 (d+e x)^{10}}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right )}{e^8 (d+e x)^9}+\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{e^8 (d+e x)^8}+\frac {4 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right )}{e^8 (d+e x)^7}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^8 (d+e x)^6}-\frac {4 c^3 (2 c d-b e)}{e^8 (d+e x)^5}+\frac {c^4}{e^8 (d+e x)^4}\right ) \, dx \\ & = -\frac {\left (c d^2-b d e+a e^2\right )^4}{11 e^9 (d+e x)^{11}}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{5 e^9 (d+e x)^{10}}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{9 e^9 (d+e x)^9}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{2 e^9 (d+e x)^8}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{7 e^9 (d+e x)^7}+\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (d+e x)^6}-\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{5 e^9 (d+e x)^5}+\frac {c^3 (2 c d-b e)}{e^9 (d+e x)^4}-\frac {c^4}{3 e^9 (d+e x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx=-\frac {14 c^4 \left (d^8+11 d^7 e x+55 d^6 e^2 x^2+165 d^5 e^3 x^3+330 d^4 e^4 x^4+462 d^3 e^5 x^5+462 d^2 e^6 x^6+330 d e^7 x^7+165 e^8 x^8\right )+3 e^4 \left (210 a^4 e^4+84 a^3 b e^3 (d+11 e x)+28 a^2 b^2 e^2 \left (d^2+11 d e x+55 e^2 x^2\right )+7 a b^3 e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+b^4 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+c e^3 \left (56 a^3 e^3 \left (d^2+11 d e x+55 e^2 x^2\right )+63 a^2 b e^2 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+36 a b^2 e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+10 b^3 \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )+6 c^2 e^2 \left (3 a^2 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+3 b^2 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )+3 c^3 e \left (4 a e \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )+7 b \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )\right )}{6930 e^9 (d+e x)^{11}} \]

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^12,x]

[Out]

-1/6930*(14*c^4*(d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2 + 165*d^5*e^3*x^3 + 330*d^4*e^4*x^4 + 462*d^3*e^5*x^5 + 462
*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165*e^8*x^8) + 3*e^4*(210*a^4*e^4 + 84*a^3*b*e^3*(d + 11*e*x) + 28*a^2*b^2*e^2*
(d^2 + 11*d*e*x + 55*e^2*x^2) + 7*a*b^3*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + b^4*(d^4 + 11*d^3*
e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4)) + c*e^3*(56*a^3*e^3*(d^2 + 11*d*e*x + 55*e^2*x^2) + 63*a^
2*b*e^2*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 36*a*b^2*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*
d*e^3*x^3 + 330*e^4*x^4) + 10*b^3*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e
^5*x^5)) + 6*c^2*e^2*(3*a^2*e^2*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + 5*a*b*e*(d
^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) + 3*b^2*(d^6 + 11*d^5*e*x +
55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6)) + 3*c^3*e*(4*a*e*(d^6 + 11*
d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6) + 7*b*(d^7 + 11*d^
6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7)))/
(e^9*(d + e*x)^11)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(886\) vs. \(2(425)=850\).

Time = 7.77 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.02

method	result	size
risch	\(\frac {-\frac {c^{4} x^{8}}{3 e}-\frac {c^{3} \left (3 b e +2 c d \right ) x^{7}}{3 e^{2}}-\frac {c^{2} \left (12 a c \,e^{2}+18 b^{2} e^{2}+21 b c d e +14 c^{2} d^{2}\right ) x^{6}}{15 e^{3}}-\frac {c \left (30 a b c \,e^{3}+12 c^{2} a d \,e^{2}+10 b^{3} e^{3}+18 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e +14 c^{3} d^{3}\right ) x^{5}}{15 e^{4}}-\frac {\left (18 c^{2} a^{2} e^{4}+36 a \,b^{2} c \,e^{4}+30 a b \,c^{2} d \,e^{3}+12 c^{3} a \,d^{2} e^{2}+3 b^{4} e^{4}+10 b^{3} c d \,e^{3}+18 b^{2} c^{2} d^{2} e^{2}+21 b \,c^{3} d^{3} e +14 c^{4} d^{4}\right ) x^{4}}{21 e^{5}}-\frac {\left (63 a^{2} b c \,e^{5}+18 d \,e^{4} a^{2} c^{2}+21 a \,b^{3} e^{5}+36 a \,b^{2} c d \,e^{4}+30 a b \,c^{2} d^{2} e^{3}+12 d^{3} e^{2} c^{3} a +3 b^{4} d \,e^{4}+10 b^{3} c \,d^{2} e^{3}+18 b^{2} c^{2} d^{3} e^{2}+21 b \,c^{3} d^{4} e +14 c^{4} d^{5}\right ) x^{3}}{42 e^{6}}-\frac {\left (56 e^{6} c \,a^{3}+84 a^{2} b^{2} e^{6}+63 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}+21 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}+30 a b \,c^{2} d^{3} e^{3}+12 d^{4} e^{2} c^{3} a +3 b^{4} d^{2} e^{4}+10 b^{3} c \,d^{3} e^{3}+18 b^{2} c^{2} d^{4} e^{2}+21 b \,c^{3} d^{5} e +14 d^{6} c^{4}\right ) x^{2}}{126 e^{7}}-\frac {\left (252 a^{3} b \,e^{7}+56 d \,e^{6} c \,a^{3}+84 a^{2} b^{2} d \,e^{6}+63 a^{2} b c \,d^{2} e^{5}+18 d^{3} e^{4} a^{2} c^{2}+21 a \,b^{3} d^{2} e^{5}+36 a \,b^{2} c \,d^{3} e^{4}+30 a b \,c^{2} d^{4} e^{3}+12 d^{5} e^{2} c^{3} a +3 b^{4} d^{3} e^{4}+10 b^{3} c \,d^{4} e^{3}+18 b^{2} c^{2} d^{5} e^{2}+21 b \,c^{3} d^{6} e +14 d^{7} c^{4}\right ) x}{630 e^{8}}-\frac {630 a^{4} e^{8}+252 a^{3} b d \,e^{7}+56 a^{3} c \,d^{2} e^{6}+84 a^{2} b^{2} d^{2} e^{6}+63 a^{2} b c \,d^{3} e^{5}+18 a^{2} c^{2} d^{4} e^{4}+21 a \,b^{3} d^{3} e^{5}+36 a \,b^{2} c \,d^{4} e^{4}+30 a b \,c^{2} d^{5} e^{3}+12 a \,c^{3} d^{6} e^{2}+3 b^{4} d^{4} e^{4}+10 b^{3} c \,d^{5} e^{3}+18 b^{2} c^{2} d^{6} e^{2}+21 b \,c^{3} d^{7} e +14 c^{4} d^{8}}{6930 e^{9}}}{\left (e x +d \right )^{11}}\)	\(887\)
default	\(-\frac {4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}-36 a^{2} b c d \,e^{5}+36 d^{2} e^{4} a^{2} c^{2}-12 a \,b^{3} d \,e^{5}+72 a \,b^{2} c \,d^{2} e^{4}-120 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} c^{3} a +6 b^{4} d^{2} e^{4}-40 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-84 b \,c^{3} d^{5} e +28 d^{6} c^{4}}{9 e^{9} \left (e x +d \right )^{9}}-\frac {4 a^{3} b \,e^{7}-8 d \,e^{6} c \,a^{3}-12 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}-24 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}-48 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}-24 d^{5} e^{2} c^{3} a -4 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}-36 b^{2} c^{2} d^{5} e^{2}+28 b \,c^{3} d^{6} e -8 d^{7} c^{4}}{10 e^{9} \left (e x +d \right )^{10}}-\frac {2 c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right )}{5 e^{9} \left (e x +d \right )^{5}}-\frac {12 a^{2} b c \,e^{5}-24 d \,e^{4} a^{2} c^{2}+4 a \,b^{3} e^{5}-48 a \,b^{2} c d \,e^{4}+120 a b \,c^{2} d^{2} e^{3}-80 d^{3} e^{2} c^{3} a -4 b^{4} d \,e^{4}+40 b^{3} c \,d^{2} e^{3}-120 b^{2} c^{2} d^{3} e^{2}+140 b \,c^{3} d^{4} e -56 c^{4} d^{5}}{8 e^{9} \left (e x +d \right )^{8}}-\frac {c^{4}}{3 e^{9} \left (e x +d \right )^{3}}-\frac {6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}-20 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}-140 b \,c^{3} d^{3} e +70 c^{4} d^{4}}{7 e^{9} \left (e x +d \right )^{7}}-\frac {c^{3} \left (b e -2 c d \right )}{e^{9} \left (e x +d \right )^{4}}-\frac {a^{4} e^{8}-4 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-12 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}-12 a b \,c^{2} d^{5} e^{3}+4 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}-4 b^{3} c \,d^{5} e^{3}+6 b^{2} c^{2} d^{6} e^{2}-4 b \,c^{3} d^{7} e +c^{4} d^{8}}{11 e^{9} \left (e x +d \right )^{11}}-\frac {2 c \left (3 a b c \,e^{3}-6 c^{2} a d \,e^{2}+b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right )}{3 e^{9} \left (e x +d \right )^{6}}\)	\(914\)
norman	\(\frac {-\frac {c^{4} x^{8}}{3 e}-\frac {\left (3 e^{3} c^{3} b +2 d \,e^{2} c^{4}\right ) x^{7}}{3 e^{4}}-\frac {\left (12 a \,c^{3} e^{4}+18 b^{2} c^{2} e^{4}+21 d \,e^{3} c^{3} b +14 d^{2} e^{2} c^{4}\right ) x^{6}}{15 e^{5}}-\frac {\left (30 b \,c^{2} a \,e^{5}+12 d \,e^{4} c^{3} a +10 b^{3} c \,e^{5}+18 b^{2} c^{2} d \,e^{4}+21 b \,c^{3} d^{2} e^{3}+14 d^{3} e^{2} c^{4}\right ) x^{5}}{15 e^{6}}-\frac {\left (18 a^{2} c^{2} e^{6}+36 a \,b^{2} c \,e^{6}+30 a b \,c^{2} d \,e^{5}+12 a \,c^{3} d^{2} e^{4}+3 b^{4} e^{6}+10 b^{3} c d \,e^{5}+18 b^{2} c^{2} d^{2} e^{4}+21 b \,c^{3} d^{3} e^{3}+14 c^{4} d^{4} e^{2}\right ) x^{4}}{21 e^{7}}-\frac {\left (63 a^{2} b c \,e^{7}+18 d \,e^{6} a^{2} c^{2}+21 a \,b^{3} e^{7}+36 a \,b^{2} c d \,e^{6}+30 a b \,c^{2} d^{2} e^{5}+12 d^{3} e^{4} c^{3} a +3 b^{4} d \,e^{6}+10 b^{3} c \,d^{2} e^{5}+18 b^{2} c^{2} d^{3} e^{4}+21 b \,c^{3} d^{4} e^{3}+14 d^{5} e^{2} c^{4}\right ) x^{3}}{42 e^{8}}-\frac {\left (56 c \,a^{3} e^{8}+84 a^{2} b^{2} e^{8}+63 a^{2} b c d \,e^{7}+18 a^{2} c^{2} d^{2} e^{6}+21 a \,b^{3} d \,e^{7}+36 a \,b^{2} c \,d^{2} e^{6}+30 a b \,c^{2} d^{3} e^{5}+12 a \,c^{3} d^{4} e^{4}+3 b^{4} d^{2} e^{6}+10 b^{3} c \,d^{3} e^{5}+18 b^{2} c^{2} d^{4} e^{4}+21 b \,c^{3} d^{5} e^{3}+14 c^{4} d^{6} e^{2}\right ) x^{2}}{126 e^{9}}-\frac {\left (252 a^{3} b \,e^{9}+56 a^{3} c d \,e^{8}+84 a^{2} b^{2} d \,e^{8}+63 a^{2} b c \,d^{2} e^{7}+18 a^{2} c^{2} d^{3} e^{6}+21 a \,b^{3} d^{2} e^{7}+36 a \,b^{2} c \,d^{3} e^{6}+30 a b \,c^{2} d^{4} e^{5}+12 a \,c^{3} d^{5} e^{4}+3 b^{4} d^{3} e^{6}+10 b^{3} c \,d^{4} e^{5}+18 b^{2} c^{2} d^{5} e^{4}+21 b \,c^{3} d^{6} e^{3}+14 c^{4} d^{7} e^{2}\right ) x}{630 e^{10}}-\frac {630 a^{4} e^{10}+252 a^{3} b d \,e^{9}+56 a^{3} c \,d^{2} e^{8}+84 a^{2} b^{2} d^{2} e^{8}+63 a^{2} b c \,d^{3} e^{7}+18 a^{2} c^{2} d^{4} e^{6}+21 a \,b^{3} d^{3} e^{7}+36 a \,b^{2} c \,d^{4} e^{6}+30 a b \,c^{2} d^{5} e^{5}+12 a \,c^{3} d^{6} e^{4}+3 b^{4} d^{4} e^{6}+10 b^{3} c \,d^{5} e^{5}+18 b^{2} c^{2} d^{6} e^{4}+21 b \,c^{3} d^{7} e^{3}+14 c^{4} d^{8} e^{2}}{6930 e^{11}}}{\left (e x +d \right )^{11}}\)	\(937\)
gosper	\(\text {Expression too large to display}\)	\(1016\)
parallelrisch	\(\text {Expression too large to display}\)	\(1023\)

[In]

int((c*x^2+b*x+a)^4/(e*x+d)^12,x,method=_RETURNVERBOSE)

[Out]

(-1/3/e*c^4*x^8-1/3/e^2*c^3*(3*b*e+2*c*d)*x^7-1/15*c^2/e^3*(12*a*c*e^2+18*b^2*e^2+21*b*c*d*e+14*c^2*d^2)*x^6-1
/15/e^4*c*(30*a*b*c*e^3+12*a*c^2*d*e^2+10*b^3*e^3+18*b^2*c*d*e^2+21*b*c^2*d^2*e+14*c^3*d^3)*x^5-1/21/e^5*(18*a
^2*c^2*e^4+36*a*b^2*c*e^4+30*a*b*c^2*d*e^3+12*a*c^3*d^2*e^2+3*b^4*e^4+10*b^3*c*d*e^3+18*b^2*c^2*d^2*e^2+21*b*c
^3*d^3*e+14*c^4*d^4)*x^4-1/42/e^6*(63*a^2*b*c*e^5+18*a^2*c^2*d*e^4+21*a*b^3*e^5+36*a*b^2*c*d*e^4+30*a*b*c^2*d^
2*e^3+12*a*c^3*d^3*e^2+3*b^4*d*e^4+10*b^3*c*d^2*e^3+18*b^2*c^2*d^3*e^2+21*b*c^3*d^4*e+14*c^4*d^5)*x^3-1/126/e^
7*(56*a^3*c*e^6+84*a^2*b^2*e^6+63*a^2*b*c*d*e^5+18*a^2*c^2*d^2*e^4+21*a*b^3*d*e^5+36*a*b^2*c*d^2*e^4+30*a*b*c^
2*d^3*e^3+12*a*c^3*d^4*e^2+3*b^4*d^2*e^4+10*b^3*c*d^3*e^3+18*b^2*c^2*d^4*e^2+21*b*c^3*d^5*e+14*c^4*d^6)*x^2-1/
630/e^8*(252*a^3*b*e^7+56*a^3*c*d*e^6+84*a^2*b^2*d*e^6+63*a^2*b*c*d^2*e^5+18*a^2*c^2*d^3*e^4+21*a*b^3*d^2*e^5+
36*a*b^2*c*d^3*e^4+30*a*b*c^2*d^4*e^3+12*a*c^3*d^5*e^2+3*b^4*d^3*e^4+10*b^3*c*d^4*e^3+18*b^2*c^2*d^5*e^2+21*b*
c^3*d^6*e+14*c^4*d^7)*x-1/6930/e^9*(630*a^4*e^8+252*a^3*b*d*e^7+56*a^3*c*d^2*e^6+84*a^2*b^2*d^2*e^6+63*a^2*b*c
*d^3*e^5+18*a^2*c^2*d^4*e^4+21*a*b^3*d^3*e^5+36*a*b^2*c*d^4*e^4+30*a*b*c^2*d^5*e^3+12*a*c^3*d^6*e^2+3*b^4*d^4*
e^4+10*b^3*c*d^5*e^3+18*b^2*c^2*d^6*e^2+21*b*c^3*d^7*e+14*c^4*d^8))/(e*x+d)^11

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (424) = 848\).

Time = 0.37 (sec) , antiderivative size = 924, normalized size of antiderivative = 2.10 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx=-\frac {2310 \, c^{4} e^{8} x^{8} + 14 \, c^{4} d^{8} + 21 \, b c^{3} d^{7} e + 252 \, a^{3} b d e^{7} + 630 \, a^{4} e^{8} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 2310 \, {\left (2 \, c^{4} d e^{7} + 3 \, b c^{3} e^{8}\right )} x^{7} + 462 \, {\left (14 \, c^{4} d^{2} e^{6} + 21 \, b c^{3} d e^{7} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 462 \, {\left (14 \, c^{4} d^{3} e^{5} + 21 \, b c^{3} d^{2} e^{6} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 330 \, {\left (14 \, c^{4} d^{4} e^{4} + 21 \, b c^{3} d^{3} e^{5} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 165 \, {\left (14 \, c^{4} d^{5} e^{3} + 21 \, b c^{3} d^{4} e^{4} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 55 \, {\left (14 \, c^{4} d^{6} e^{2} + 21 \, b c^{3} d^{5} e^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 11 \, {\left (14 \, c^{4} d^{7} e + 21 \, b c^{3} d^{6} e^{2} + 252 \, a^{3} b e^{8} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{6930 \, {\left (e^{20} x^{11} + 11 \, d e^{19} x^{10} + 55 \, d^{2} e^{18} x^{9} + 165 \, d^{3} e^{17} x^{8} + 330 \, d^{4} e^{16} x^{7} + 462 \, d^{5} e^{15} x^{6} + 462 \, d^{6} e^{14} x^{5} + 330 \, d^{7} e^{13} x^{4} + 165 \, d^{8} e^{12} x^{3} + 55 \, d^{9} e^{11} x^{2} + 11 \, d^{10} e^{10} x + d^{11} e^{9}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^12,x, algorithm="fricas")

[Out]

-1/6930*(2310*c^4*e^8*x^8 + 14*c^4*d^8 + 21*b*c^3*d^7*e + 252*a^3*b*d*e^7 + 630*a^4*e^8 + 6*(3*b^2*c^2 + 2*a*c
^3)*d^6*e^2 + 10*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 21*(a*b^3 + 3*a^2*b*
c)*d^3*e^5 + 28*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 2310*(2*c^4*d*e^7 + 3*b*c^3*e^8)*x^7 + 462*(14*c^4*d^2*e^6 + 2
1*b*c^3*d*e^7 + 6*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 462*(14*c^4*d^3*e^5 + 21*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a
*c^3)*d*e^7 + 10*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 330*(14*c^4*d^4*e^4 + 21*b*c^3*d^3*e^5 + 6*(3*b^2*c^2 + 2*a*c^
3)*d^2*e^6 + 10*(b^3*c + 3*a*b*c^2)*d*e^7 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 165*(14*c^4*d^5*e^3 +
21*b*c^3*d^4*e^4 + 6*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 10*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 3*(b^4 + 12*a*b^2*c + 6*
a^2*c^2)*d*e^7 + 21*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 55*(14*c^4*d^6*e^2 + 21*b*c^3*d^5*e^3 + 6*(3*b^2*c^2 + 2*a*
c^3)*d^4*e^4 + 10*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 21*(a*b^3 + 3*a^2*b
*c)*d*e^7 + 28*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 11*(14*c^4*d^7*e + 21*b*c^3*d^6*e^2 + 252*a^3*b*e^8 + 6*(3*b^2
*c^2 + 2*a*c^3)*d^5*e^3 + 10*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 21*(a*b^
3 + 3*a^2*b*c)*d^2*e^6 + 28*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^20*x^11 + 11*d*e^19*x^10 + 55*d^2*e^18*x^9 + 16
5*d^3*e^17*x^8 + 330*d^4*e^16*x^7 + 462*d^5*e^15*x^6 + 462*d^6*e^14*x^5 + 330*d^7*e^13*x^4 + 165*d^8*e^12*x^3
+ 55*d^9*e^11*x^2 + 11*d^10*e^10*x + d^11*e^9)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)**4/(e*x+d)**12,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (424) = 848\).

Time = 0.24 (sec) , antiderivative size = 924, normalized size of antiderivative = 2.10 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx=-\frac {2310 \, c^{4} e^{8} x^{8} + 14 \, c^{4} d^{8} + 21 \, b c^{3} d^{7} e + 252 \, a^{3} b d e^{7} + 630 \, a^{4} e^{8} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 2310 \, {\left (2 \, c^{4} d e^{7} + 3 \, b c^{3} e^{8}\right )} x^{7} + 462 \, {\left (14 \, c^{4} d^{2} e^{6} + 21 \, b c^{3} d e^{7} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 462 \, {\left (14 \, c^{4} d^{3} e^{5} + 21 \, b c^{3} d^{2} e^{6} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 330 \, {\left (14 \, c^{4} d^{4} e^{4} + 21 \, b c^{3} d^{3} e^{5} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 165 \, {\left (14 \, c^{4} d^{5} e^{3} + 21 \, b c^{3} d^{4} e^{4} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 55 \, {\left (14 \, c^{4} d^{6} e^{2} + 21 \, b c^{3} d^{5} e^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 11 \, {\left (14 \, c^{4} d^{7} e + 21 \, b c^{3} d^{6} e^{2} + 252 \, a^{3} b e^{8} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} + 21 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 28 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{6930 \, {\left (e^{20} x^{11} + 11 \, d e^{19} x^{10} + 55 \, d^{2} e^{18} x^{9} + 165 \, d^{3} e^{17} x^{8} + 330 \, d^{4} e^{16} x^{7} + 462 \, d^{5} e^{15} x^{6} + 462 \, d^{6} e^{14} x^{5} + 330 \, d^{7} e^{13} x^{4} + 165 \, d^{8} e^{12} x^{3} + 55 \, d^{9} e^{11} x^{2} + 11 \, d^{10} e^{10} x + d^{11} e^{9}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^12,x, algorithm="maxima")

[Out]

-1/6930*(2310*c^4*e^8*x^8 + 14*c^4*d^8 + 21*b*c^3*d^7*e + 252*a^3*b*d*e^7 + 630*a^4*e^8 + 6*(3*b^2*c^2 + 2*a*c
^3)*d^6*e^2 + 10*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 21*(a*b^3 + 3*a^2*b*
c)*d^3*e^5 + 28*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 2310*(2*c^4*d*e^7 + 3*b*c^3*e^8)*x^7 + 462*(14*c^4*d^2*e^6 + 2
1*b*c^3*d*e^7 + 6*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 462*(14*c^4*d^3*e^5 + 21*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a
*c^3)*d*e^7 + 10*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 330*(14*c^4*d^4*e^4 + 21*b*c^3*d^3*e^5 + 6*(3*b^2*c^2 + 2*a*c^
3)*d^2*e^6 + 10*(b^3*c + 3*a*b*c^2)*d*e^7 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 165*(14*c^4*d^5*e^3 +
21*b*c^3*d^4*e^4 + 6*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 10*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 3*(b^4 + 12*a*b^2*c + 6*
a^2*c^2)*d*e^7 + 21*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 55*(14*c^4*d^6*e^2 + 21*b*c^3*d^5*e^3 + 6*(3*b^2*c^2 + 2*a*
c^3)*d^4*e^4 + 10*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 21*(a*b^3 + 3*a^2*b
*c)*d*e^7 + 28*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 11*(14*c^4*d^7*e + 21*b*c^3*d^6*e^2 + 252*a^3*b*e^8 + 6*(3*b^2
*c^2 + 2*a*c^3)*d^5*e^3 + 10*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 21*(a*b^
3 + 3*a^2*b*c)*d^2*e^6 + 28*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^20*x^11 + 11*d*e^19*x^10 + 55*d^2*e^18*x^9 + 16
5*d^3*e^17*x^8 + 330*d^4*e^16*x^7 + 462*d^5*e^15*x^6 + 462*d^6*e^14*x^5 + 330*d^7*e^13*x^4 + 165*d^8*e^12*x^3
+ 55*d^9*e^11*x^2 + 11*d^10*e^10*x + d^11*e^9)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (424) = 848\).

Time = 0.28 (sec) , antiderivative size = 1015, normalized size of antiderivative = 2.31 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx=-\frac {2310 \, c^{4} e^{8} x^{8} + 4620 \, c^{4} d e^{7} x^{7} + 6930 \, b c^{3} e^{8} x^{7} + 6468 \, c^{4} d^{2} e^{6} x^{6} + 9702 \, b c^{3} d e^{7} x^{6} + 8316 \, b^{2} c^{2} e^{8} x^{6} + 5544 \, a c^{3} e^{8} x^{6} + 6468 \, c^{4} d^{3} e^{5} x^{5} + 9702 \, b c^{3} d^{2} e^{6} x^{5} + 8316 \, b^{2} c^{2} d e^{7} x^{5} + 5544 \, a c^{3} d e^{7} x^{5} + 4620 \, b^{3} c e^{8} x^{5} + 13860 \, a b c^{2} e^{8} x^{5} + 4620 \, c^{4} d^{4} e^{4} x^{4} + 6930 \, b c^{3} d^{3} e^{5} x^{4} + 5940 \, b^{2} c^{2} d^{2} e^{6} x^{4} + 3960 \, a c^{3} d^{2} e^{6} x^{4} + 3300 \, b^{3} c d e^{7} x^{4} + 9900 \, a b c^{2} d e^{7} x^{4} + 990 \, b^{4} e^{8} x^{4} + 11880 \, a b^{2} c e^{8} x^{4} + 5940 \, a^{2} c^{2} e^{8} x^{4} + 2310 \, c^{4} d^{5} e^{3} x^{3} + 3465 \, b c^{3} d^{4} e^{4} x^{3} + 2970 \, b^{2} c^{2} d^{3} e^{5} x^{3} + 1980 \, a c^{3} d^{3} e^{5} x^{3} + 1650 \, b^{3} c d^{2} e^{6} x^{3} + 4950 \, a b c^{2} d^{2} e^{6} x^{3} + 495 \, b^{4} d e^{7} x^{3} + 5940 \, a b^{2} c d e^{7} x^{3} + 2970 \, a^{2} c^{2} d e^{7} x^{3} + 3465 \, a b^{3} e^{8} x^{3} + 10395 \, a^{2} b c e^{8} x^{3} + 770 \, c^{4} d^{6} e^{2} x^{2} + 1155 \, b c^{3} d^{5} e^{3} x^{2} + 990 \, b^{2} c^{2} d^{4} e^{4} x^{2} + 660 \, a c^{3} d^{4} e^{4} x^{2} + 550 \, b^{3} c d^{3} e^{5} x^{2} + 1650 \, a b c^{2} d^{3} e^{5} x^{2} + 165 \, b^{4} d^{2} e^{6} x^{2} + 1980 \, a b^{2} c d^{2} e^{6} x^{2} + 990 \, a^{2} c^{2} d^{2} e^{6} x^{2} + 1155 \, a b^{3} d e^{7} x^{2} + 3465 \, a^{2} b c d e^{7} x^{2} + 4620 \, a^{2} b^{2} e^{8} x^{2} + 3080 \, a^{3} c e^{8} x^{2} + 154 \, c^{4} d^{7} e x + 231 \, b c^{3} d^{6} e^{2} x + 198 \, b^{2} c^{2} d^{5} e^{3} x + 132 \, a c^{3} d^{5} e^{3} x + 110 \, b^{3} c d^{4} e^{4} x + 330 \, a b c^{2} d^{4} e^{4} x + 33 \, b^{4} d^{3} e^{5} x + 396 \, a b^{2} c d^{3} e^{5} x + 198 \, a^{2} c^{2} d^{3} e^{5} x + 231 \, a b^{3} d^{2} e^{6} x + 693 \, a^{2} b c d^{2} e^{6} x + 924 \, a^{2} b^{2} d e^{7} x + 616 \, a^{3} c d e^{7} x + 2772 \, a^{3} b e^{8} x + 14 \, c^{4} d^{8} + 21 \, b c^{3} d^{7} e + 18 \, b^{2} c^{2} d^{6} e^{2} + 12 \, a c^{3} d^{6} e^{2} + 10 \, b^{3} c d^{5} e^{3} + 30 \, a b c^{2} d^{5} e^{3} + 3 \, b^{4} d^{4} e^{4} + 36 \, a b^{2} c d^{4} e^{4} + 18 \, a^{2} c^{2} d^{4} e^{4} + 21 \, a b^{3} d^{3} e^{5} + 63 \, a^{2} b c d^{3} e^{5} + 84 \, a^{2} b^{2} d^{2} e^{6} + 56 \, a^{3} c d^{2} e^{6} + 252 \, a^{3} b d e^{7} + 630 \, a^{4} e^{8}}{6930 \, {\left (e x + d\right )}^{11} e^{9}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^12,x, algorithm="giac")

[Out]

-1/6930*(2310*c^4*e^8*x^8 + 4620*c^4*d*e^7*x^7 + 6930*b*c^3*e^8*x^7 + 6468*c^4*d^2*e^6*x^6 + 9702*b*c^3*d*e^7*
x^6 + 8316*b^2*c^2*e^8*x^6 + 5544*a*c^3*e^8*x^6 + 6468*c^4*d^3*e^5*x^5 + 9702*b*c^3*d^2*e^6*x^5 + 8316*b^2*c^2
*d*e^7*x^5 + 5544*a*c^3*d*e^7*x^5 + 4620*b^3*c*e^8*x^5 + 13860*a*b*c^2*e^8*x^5 + 4620*c^4*d^4*e^4*x^4 + 6930*b
*c^3*d^3*e^5*x^4 + 5940*b^2*c^2*d^2*e^6*x^4 + 3960*a*c^3*d^2*e^6*x^4 + 3300*b^3*c*d*e^7*x^4 + 9900*a*b*c^2*d*e
^7*x^4 + 990*b^4*e^8*x^4 + 11880*a*b^2*c*e^8*x^4 + 5940*a^2*c^2*e^8*x^4 + 2310*c^4*d^5*e^3*x^3 + 3465*b*c^3*d^
4*e^4*x^3 + 2970*b^2*c^2*d^3*e^5*x^3 + 1980*a*c^3*d^3*e^5*x^3 + 1650*b^3*c*d^2*e^6*x^3 + 4950*a*b*c^2*d^2*e^6*
x^3 + 495*b^4*d*e^7*x^3 + 5940*a*b^2*c*d*e^7*x^3 + 2970*a^2*c^2*d*e^7*x^3 + 3465*a*b^3*e^8*x^3 + 10395*a^2*b*c
*e^8*x^3 + 770*c^4*d^6*e^2*x^2 + 1155*b*c^3*d^5*e^3*x^2 + 990*b^2*c^2*d^4*e^4*x^2 + 660*a*c^3*d^4*e^4*x^2 + 55
0*b^3*c*d^3*e^5*x^2 + 1650*a*b*c^2*d^3*e^5*x^2 + 165*b^4*d^2*e^6*x^2 + 1980*a*b^2*c*d^2*e^6*x^2 + 990*a^2*c^2*
d^2*e^6*x^2 + 1155*a*b^3*d*e^7*x^2 + 3465*a^2*b*c*d*e^7*x^2 + 4620*a^2*b^2*e^8*x^2 + 3080*a^3*c*e^8*x^2 + 154*
c^4*d^7*e*x + 231*b*c^3*d^6*e^2*x + 198*b^2*c^2*d^5*e^3*x + 132*a*c^3*d^5*e^3*x + 110*b^3*c*d^4*e^4*x + 330*a*
b*c^2*d^4*e^4*x + 33*b^4*d^3*e^5*x + 396*a*b^2*c*d^3*e^5*x + 198*a^2*c^2*d^3*e^5*x + 231*a*b^3*d^2*e^6*x + 693
*a^2*b*c*d^2*e^6*x + 924*a^2*b^2*d*e^7*x + 616*a^3*c*d*e^7*x + 2772*a^3*b*e^8*x + 14*c^4*d^8 + 21*b*c^3*d^7*e
+ 18*b^2*c^2*d^6*e^2 + 12*a*c^3*d^6*e^2 + 10*b^3*c*d^5*e^3 + 30*a*b*c^2*d^5*e^3 + 3*b^4*d^4*e^4 + 36*a*b^2*c*d
^4*e^4 + 18*a^2*c^2*d^4*e^4 + 21*a*b^3*d^3*e^5 + 63*a^2*b*c*d^3*e^5 + 84*a^2*b^2*d^2*e^6 + 56*a^3*c*d^2*e^6 +
252*a^3*b*d*e^7 + 630*a^4*e^8)/((e*x + d)^11*e^9)

Mupad [B] (verification not implemented)

Time = 10.16 (sec) , antiderivative size = 997, normalized size of antiderivative = 2.27 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx=-\frac {\frac {630\,a^4\,e^8+252\,a^3\,b\,d\,e^7+56\,a^3\,c\,d^2\,e^6+84\,a^2\,b^2\,d^2\,e^6+63\,a^2\,b\,c\,d^3\,e^5+18\,a^2\,c^2\,d^4\,e^4+21\,a\,b^3\,d^3\,e^5+36\,a\,b^2\,c\,d^4\,e^4+30\,a\,b\,c^2\,d^5\,e^3+12\,a\,c^3\,d^6\,e^2+3\,b^4\,d^4\,e^4+10\,b^3\,c\,d^5\,e^3+18\,b^2\,c^2\,d^6\,e^2+21\,b\,c^3\,d^7\,e+14\,c^4\,d^8}{6930\,e^9}+\frac {x^3\,\left (63\,a^2\,b\,c\,e^5+18\,a^2\,c^2\,d\,e^4+21\,a\,b^3\,e^5+36\,a\,b^2\,c\,d\,e^4+30\,a\,b\,c^2\,d^2\,e^3+12\,a\,c^3\,d^3\,e^2+3\,b^4\,d\,e^4+10\,b^3\,c\,d^2\,e^3+18\,b^2\,c^2\,d^3\,e^2+21\,b\,c^3\,d^4\,e+14\,c^4\,d^5\right )}{42\,e^6}+\frac {x^4\,\left (18\,a^2\,c^2\,e^4+36\,a\,b^2\,c\,e^4+30\,a\,b\,c^2\,d\,e^3+12\,a\,c^3\,d^2\,e^2+3\,b^4\,e^4+10\,b^3\,c\,d\,e^3+18\,b^2\,c^2\,d^2\,e^2+21\,b\,c^3\,d^3\,e+14\,c^4\,d^4\right )}{21\,e^5}+\frac {x\,\left (252\,a^3\,b\,e^7+56\,a^3\,c\,d\,e^6+84\,a^2\,b^2\,d\,e^6+63\,a^2\,b\,c\,d^2\,e^5+18\,a^2\,c^2\,d^3\,e^4+21\,a\,b^3\,d^2\,e^5+36\,a\,b^2\,c\,d^3\,e^4+30\,a\,b\,c^2\,d^4\,e^3+12\,a\,c^3\,d^5\,e^2+3\,b^4\,d^3\,e^4+10\,b^3\,c\,d^4\,e^3+18\,b^2\,c^2\,d^5\,e^2+21\,b\,c^3\,d^6\,e+14\,c^4\,d^7\right )}{630\,e^8}+\frac {c^4\,x^8}{3\,e}+\frac {x^2\,\left (56\,a^3\,c\,e^6+84\,a^2\,b^2\,e^6+63\,a^2\,b\,c\,d\,e^5+18\,a^2\,c^2\,d^2\,e^4+21\,a\,b^3\,d\,e^5+36\,a\,b^2\,c\,d^2\,e^4+30\,a\,b\,c^2\,d^3\,e^3+12\,a\,c^3\,d^4\,e^2+3\,b^4\,d^2\,e^4+10\,b^3\,c\,d^3\,e^3+18\,b^2\,c^2\,d^4\,e^2+21\,b\,c^3\,d^5\,e+14\,c^4\,d^6\right )}{126\,e^7}+\frac {c^3\,x^7\,\left (3\,b\,e+2\,c\,d\right )}{3\,e^2}+\frac {c^2\,x^6\,\left (18\,b^2\,e^2+21\,b\,c\,d\,e+14\,c^2\,d^2+12\,a\,c\,e^2\right )}{15\,e^3}+\frac {c\,x^5\,\left (10\,b^3\,e^3+18\,b^2\,c\,d\,e^2+21\,b\,c^2\,d^2\,e+30\,a\,b\,c\,e^3+14\,c^3\,d^3+12\,a\,c^2\,d\,e^2\right )}{15\,e^4}}{d^{11}+11\,d^{10}\,e\,x+55\,d^9\,e^2\,x^2+165\,d^8\,e^3\,x^3+330\,d^7\,e^4\,x^4+462\,d^6\,e^5\,x^5+462\,d^5\,e^6\,x^6+330\,d^4\,e^7\,x^7+165\,d^3\,e^8\,x^8+55\,d^2\,e^9\,x^9+11\,d\,e^{10}\,x^{10}+e^{11}\,x^{11}} \]

[In]

int((a + b*x + c*x^2)^4/(d + e*x)^12,x)

[Out]

-((630*a^4*e^8 + 14*c^4*d^8 + 3*b^4*d^4*e^4 + 21*a*b^3*d^3*e^5 + 12*a*c^3*d^6*e^2 + 56*a^3*c*d^2*e^6 + 10*b^3*
c*d^5*e^3 + 84*a^2*b^2*d^2*e^6 + 18*a^2*c^2*d^4*e^4 + 18*b^2*c^2*d^6*e^2 + 252*a^3*b*d*e^7 + 21*b*c^3*d^7*e +
30*a*b*c^2*d^5*e^3 + 36*a*b^2*c*d^4*e^4 + 63*a^2*b*c*d^3*e^5)/(6930*e^9) + (x^3*(14*c^4*d^5 + 21*a*b^3*e^5 + 3
*b^4*d*e^4 + 12*a*c^3*d^3*e^2 + 18*a^2*c^2*d*e^4 + 10*b^3*c*d^2*e^3 + 18*b^2*c^2*d^3*e^2 + 63*a^2*b*c*e^5 + 21
*b*c^3*d^4*e + 36*a*b^2*c*d*e^4 + 30*a*b*c^2*d^2*e^3))/(42*e^6) + (x^4*(3*b^4*e^4 + 14*c^4*d^4 + 18*a^2*c^2*e^
4 + 12*a*c^3*d^2*e^2 + 18*b^2*c^2*d^2*e^2 + 36*a*b^2*c*e^4 + 21*b*c^3*d^3*e + 10*b^3*c*d*e^3 + 30*a*b*c^2*d*e^
3))/(21*e^5) + (x*(14*c^4*d^7 + 252*a^3*b*e^7 + 3*b^4*d^3*e^4 + 21*a*b^3*d^2*e^5 + 84*a^2*b^2*d*e^6 + 12*a*c^3
*d^5*e^2 + 10*b^3*c*d^4*e^3 + 18*a^2*c^2*d^3*e^4 + 18*b^2*c^2*d^5*e^2 + 56*a^3*c*d*e^6 + 21*b*c^3*d^6*e + 30*a
*b*c^2*d^4*e^3 + 36*a*b^2*c*d^3*e^4 + 63*a^2*b*c*d^2*e^5))/(630*e^8) + (c^4*x^8)/(3*e) + (x^2*(14*c^4*d^6 + 56
*a^3*c*e^6 + 84*a^2*b^2*e^6 + 3*b^4*d^2*e^4 + 12*a*c^3*d^4*e^2 + 10*b^3*c*d^3*e^3 + 18*a^2*c^2*d^2*e^4 + 18*b^
2*c^2*d^4*e^2 + 21*a*b^3*d*e^5 + 21*b*c^3*d^5*e + 63*a^2*b*c*d*e^5 + 30*a*b*c^2*d^3*e^3 + 36*a*b^2*c*d^2*e^4))
/(126*e^7) + (c^3*x^7*(3*b*e + 2*c*d))/(3*e^2) + (c^2*x^6*(18*b^2*e^2 + 14*c^2*d^2 + 12*a*c*e^2 + 21*b*c*d*e))
/(15*e^3) + (c*x^5*(10*b^3*e^3 + 14*c^3*d^3 + 30*a*b*c*e^3 + 12*a*c^2*d*e^2 + 21*b*c^2*d^2*e + 18*b^2*c*d*e^2)
)/(15*e^4))/(d^11 + e^11*x^11 + 11*d*e^10*x^10 + 55*d^9*e^2*x^2 + 165*d^8*e^3*x^3 + 330*d^7*e^4*x^4 + 462*d^6*
e^5*x^5 + 462*d^5*e^6*x^6 + 330*d^4*e^7*x^7 + 165*d^3*e^8*x^8 + 55*d^2*e^9*x^9 + 11*d^10*e*x)