\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^8} \, dx\) [2158]

Optimal result

Integrand size = 20, antiderivative size = 424 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=\frac {c^4 x}{e^8}-\frac {\left (c d^2-b d e+a e^2\right )^4}{7 e^9 (d+e x)^7}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^9 (d+e x)^6}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{5 e^9 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^4}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{3 e^9 (d+e x)^3}+\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^2}-\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)}-\frac {4 c^3 (2 c d-b e) \log (d+e x)}{e^9} \]

[Out]

c^4*x/e^8-1/7*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)^7+2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^6-2/5*(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)^5+(-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)^4+1/3*(-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)^3+2*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)^2-2*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)-4*c^3*(-b*e+
2*c*d)*ln(e*x+d)/e^9

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 424, 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} \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=-\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{3 e^9 (d+e x)^3}-\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}+\frac {2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^4}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right )^4}{7 e^9 (d+e x)^7}-\frac {4 c^3 (2 c d-b e) \log (d+e x)}{e^9}+\frac {c^4 x}{e^8} \]

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^8,x]

[Out]

(c^4*x)/e^8 - (c*d^2 - b*d*e + a*e^2)^4/(7*e^9*(d + e*x)^7) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e
^9*(d + e*x)^6) - (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)))/(5*e^9*(d + e*x
)^5) + ((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)))/(e^9*(d + e*x)^4) -
(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))/(3*e^9*(d + e*x)^3) + (2*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(
e^9*(d + e*x)^2) - (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(e^9*(d + e*x)) - (4*c^3*(2*c*d - b*
e)*Log[d + e*x])/e^9

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 {c^4}{e^8}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^8}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^7}+\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)^6}+\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)^5}+\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)^4}+\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)^3}+\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)^2}-\frac {4 c^3 (2 c d-b e)}{e^8 (d+e x)}\right ) \, dx \\ & = \frac {c^4 x}{e^8}-\frac {\left (c d^2-b d e+a e^2\right )^4}{7 e^9 (d+e x)^7}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^9 (d+e x)^6}-\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 )}{5 e^9 (d+e x)^5}+\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 )}{e^9 (d+e x)^4}-\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 )}{3 e^9 (d+e x)^3}+\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 )}{e^9 (d+e x)^2}-\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^9 (d+e x)}-\frac {4 c^3 (2 c d-b e) \log (d+e x)}{e^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 748, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=-\frac {c^4 \left (1443 d^8+9261 d^7 e x+24843 d^6 e^2 x^2+35525 d^5 e^3 x^3+28175 d^4 e^4 x^4+11025 d^3 e^5 x^5+735 d^2 e^6 x^6-735 d e^7 x^7-105 e^8 x^8\right )+e^4 \left (15 a^4 e^4+10 a^3 b e^3 (d+7 e x)+6 a^2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b^3 e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+c e^3 \left (4 a^3 e^3 \left (d^2+7 d e x+21 e^2 x^2\right )+9 a^2 b e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+12 a b^2 e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+10 b^3 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )+6 c^2 e^2 \left (a^2 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 a b e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+15 b^2 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )+c^3 e \left (60 a e \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )-b d \left (1089 d^6+7203 d^5 e x+20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+13230 d e^5 x^5+2940 e^6 x^6\right )\right )+420 c^3 (2 c d-b e) (d+e x)^7 \log (d+e x)}{105 e^9 (d+e x)^7} \]

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^8,x]

[Out]

-1/105*(c^4*(1443*d^8 + 9261*d^7*e*x + 24843*d^6*e^2*x^2 + 35525*d^5*e^3*x^3 + 28175*d^4*e^4*x^4 + 11025*d^3*e
^5*x^5 + 735*d^2*e^6*x^6 - 735*d*e^7*x^7 - 105*e^8*x^8) + e^4*(15*a^4*e^4 + 10*a^3*b*e^3*(d + 7*e*x) + 6*a^2*b
^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b^3*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + b^4*(d^4 + 7*d
^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + c*e^3*(4*a^3*e^3*(d^2 + 7*d*e*x + 21*e^2*x^2) + 9*a^2*
b*e^2*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 12*a*b^2*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*
x^3 + 35*e^4*x^4) + 10*b^3*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) +
6*c^2*e^2*(a^2*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*a*b*e*(d^5 + 7*d^4*e*x +
21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 15*b^2*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*
d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)) + c^3*e*(60*a*e*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 +
35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6) - b*d*(1089*d^6 + 7203*d^5*e*x + 20139*d^4*e^2*x^2
+ 30625*d^3*e^3*x^3 + 26950*d^2*e^4*x^4 + 13230*d*e^5*x^5 + 2940*e^6*x^6)) + 420*c^3*(2*c*d - b*e)*(d + e*x)^
7*Log[d + e*x])/(e^9*(d + e*x)^7)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(891\) vs. \(2(417)=834\).

Time = 8.69 (sec) , antiderivative size = 892, normalized size of antiderivative = 2.10

method	result	size
norman	\(\frac {\frac {c^{4} x^{8}}{e}-\frac {15 a^{4} e^{8}+10 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}+9 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+3 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+30 a b \,c^{2} d^{5} e^{3}+60 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+10 b^{3} c \,d^{5} e^{3}+90 b^{2} c^{2} d^{6} e^{2}-1089 b \,c^{3} d^{7} e +2178 c^{4} d^{8}}{105 e^{9}}-\frac {\left (4 a \,c^{3} e^{2}+6 b^{2} c^{2} e^{2}-28 d e \,c^{3} b +56 c^{4} d^{2}\right ) x^{6}}{e^{3}}-\frac {\left (6 a b \,c^{2} e^{3}+12 a \,c^{3} d \,e^{2}+2 b^{3} c \,e^{3}+18 b^{2} c^{2} d \,e^{2}-126 b \,c^{3} d^{2} e +252 d^{3} c^{4}\right ) x^{5}}{e^{4}}-\frac {\left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}+30 a b \,c^{2} d \,e^{3}+60 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}+10 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}-770 b \,c^{3} d^{3} e +1540 c^{4} d^{4}\right ) x^{4}}{3 e^{5}}-\frac {\left (9 a^{2} b c \,e^{5}+6 d \,e^{4} a^{2} c^{2}+3 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+30 a b \,c^{2} d^{2} e^{3}+60 d^{3} e^{2} c^{3} a +b^{4} d \,e^{4}+10 b^{3} c \,d^{2} e^{3}+90 b^{2} c^{2} d^{3} e^{2}-875 b \,c^{3} d^{4} e +1750 c^{4} d^{5}\right ) x^{3}}{3 e^{6}}-\frac {\left (4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}+9 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+3 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+30 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} c^{3} a +b^{4} d^{2} e^{4}+10 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-959 b \,c^{3} d^{5} e +1918 d^{6} c^{4}\right ) x^{2}}{5 e^{7}}-\frac {\left (10 a^{3} b \,e^{7}+4 d \,e^{6} c \,a^{3}+6 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}+6 d^{3} e^{4} a^{2} c^{2}+3 a \,b^{3} d^{2} e^{5}+12 a \,b^{2} c \,d^{3} e^{4}+30 a b \,c^{2} d^{4} e^{3}+60 d^{5} e^{2} c^{3} a +b^{4} d^{3} e^{4}+10 b^{3} c \,d^{4} e^{3}+90 b^{2} c^{2} d^{5} e^{2}-1029 b \,c^{3} d^{6} e +2058 d^{7} c^{4}\right ) x}{15 e^{8}}}{\left (e x +d \right )^{7}}+\frac {4 c^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{9}}\)	\(892\)
risch	\(\frac {c^{4} x}{e^{8}}+\frac {\left (-4 e^{7} c^{3} a -6 b^{2} c^{2} e^{7}+28 b \,c^{3} d \,e^{6}-28 d^{2} e^{5} c^{4}\right ) x^{6}-2 c \,e^{4} \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 -63 b \,c^{2} d^{2} e +70 c^{3} d^{3}\right ) x^{5}-\frac {e^{3} \left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}+30 a b \,c^{2} d \,e^{3}+60 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}+10 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}-770 b \,c^{3} d^{3} e +910 c^{4} d^{4}\right ) x^{4}}{3}-\frac {e^{2} \left (9 a^{2} b c \,e^{5}+6 d \,e^{4} a^{2} c^{2}+3 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+30 a b \,c^{2} d^{2} e^{3}+60 d^{3} e^{2} c^{3} a +b^{4} d \,e^{4}+10 b^{3} c \,d^{2} e^{3}+90 b^{2} c^{2} d^{3} e^{2}-875 b \,c^{3} d^{4} e +1078 c^{4} d^{5}\right ) x^{3}}{3}-\frac {e \left (4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}+9 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+3 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+30 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} c^{3} a +b^{4} d^{2} e^{4}+10 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-959 b \,c^{3} d^{5} e +1218 d^{6} c^{4}\right ) x^{2}}{5}+\left (-\frac {2}{3} a^{3} b \,e^{7}-\frac {4}{15} d \,e^{6} c \,a^{3}-\frac {2}{5} a^{2} b^{2} d \,e^{6}-\frac {3}{5} a^{2} b c \,d^{2} e^{5}-\frac {2}{5} d^{3} e^{4} a^{2} c^{2}-\frac {1}{5} a \,b^{3} d^{2} e^{5}-\frac {4}{5} a \,b^{2} c \,d^{3} e^{4}-2 a b \,c^{2} d^{4} e^{3}-4 d^{5} e^{2} c^{3} a -\frac {1}{15} b^{4} d^{3} e^{4}-\frac {2}{3} b^{3} c \,d^{4} e^{3}-6 b^{2} c^{2} d^{5} e^{2}+\frac {343}{5} b \,c^{3} d^{6} e -\frac {446}{5} d^{7} c^{4}\right ) x -\frac {15 a^{4} e^{8}+10 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}+9 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+3 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+30 a b \,c^{2} d^{5} e^{3}+60 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+10 b^{3} c \,d^{5} e^{3}+90 b^{2} c^{2} d^{6} e^{2}-1089 b \,c^{3} d^{7} e +1443 c^{4} d^{8}}{105 e}}{e^{8} \left (e x +d \right )^{7}}+\frac {4 c^{3} \ln \left (e x +d \right ) b}{e^{8}}-\frac {8 c^{4} \ln \left (e x +d \right ) d}{e^{9}}\)	\(892\)
default	\(\frac {c^{4} x}{e^{8}}-\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}}{5 e^{9} \left (e x +d \right )^{5}}-\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 )}{e^{9} \left (e x +d \right )}-\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}}{3 e^{9} \left (e x +d \right )^{3}}-\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}}{7 e^{9} \left (e x +d \right )^{7}}-\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}}{4 e^{9} \left (e x +d \right )^{4}}-\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 )}{e^{9} \left (e x +d \right )^{2}}+\frac {4 c^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{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}}{6 e^{9} \left (e x +d \right )^{6}}\)	\(906\)
parallelrisch	\(\text {Expression too large to display}\)	\(1296\)

[In]

int((c*x^2+b*x+a)^4/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

(1/e*c^4*x^8-1/105*(15*a^4*e^8+10*a^3*b*d*e^7+4*a^3*c*d^2*e^6+6*a^2*b^2*d^2*e^6+9*a^2*b*c*d^3*e^5+6*a^2*c^2*d^
4*e^4+3*a*b^3*d^3*e^5+12*a*b^2*c*d^4*e^4+30*a*b*c^2*d^5*e^3+60*a*c^3*d^6*e^2+b^4*d^4*e^4+10*b^3*c*d^5*e^3+90*b
^2*c^2*d^6*e^2-1089*b*c^3*d^7*e+2178*c^4*d^8)/e^9-(4*a*c^3*e^2+6*b^2*c^2*e^2-28*b*c^3*d*e+56*c^4*d^2)/e^3*x^6-
(6*a*b*c^2*e^3+12*a*c^3*d*e^2+2*b^3*c*e^3+18*b^2*c^2*d*e^2-126*b*c^3*d^2*e+252*c^4*d^3)/e^4*x^5-1/3*(6*a^2*c^2
*e^4+12*a*b^2*c*e^4+30*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4+10*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-770*b*c^3*d^3*
e+1540*c^4*d^4)/e^5*x^4-1/3*(9*a^2*b*c*e^5+6*a^2*c^2*d*e^4+3*a*b^3*e^5+12*a*b^2*c*d*e^4+30*a*b*c^2*d^2*e^3+60*
a*c^3*d^3*e^2+b^4*d*e^4+10*b^3*c*d^2*e^3+90*b^2*c^2*d^3*e^2-875*b*c^3*d^4*e+1750*c^4*d^5)/e^6*x^3-1/5*(4*a^3*c
*e^6+6*a^2*b^2*e^6+9*a^2*b*c*d*e^5+6*a^2*c^2*d^2*e^4+3*a*b^3*d*e^5+12*a*b^2*c*d^2*e^4+30*a*b*c^2*d^3*e^3+60*a*
c^3*d^4*e^2+b^4*d^2*e^4+10*b^3*c*d^3*e^3+90*b^2*c^2*d^4*e^2-959*b*c^3*d^5*e+1918*c^4*d^6)/e^7*x^2-1/15*(10*a^3
*b*e^7+4*a^3*c*d*e^6+6*a^2*b^2*d*e^6+9*a^2*b*c*d^2*e^5+6*a^2*c^2*d^3*e^4+3*a*b^3*d^2*e^5+12*a*b^2*c*d^3*e^4+30
*a*b*c^2*d^4*e^3+60*a*c^3*d^5*e^2+b^4*d^3*e^4+10*b^3*c*d^4*e^3+90*b^2*c^2*d^5*e^2-1029*b*c^3*d^6*e+2058*c^4*d^
7)/e^8*x)/(e*x+d)^7+4*c^3/e^9*(b*e-2*c*d)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1082 vs. \(2 (416) = 832\).

Time = 0.31 (sec) , antiderivative size = 1082, normalized size of antiderivative = 2.55 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^8,x, algorithm="fricas")

[Out]

1/105*(105*c^4*e^8*x^8 + 735*c^4*d*e^7*x^7 - 1443*c^4*d^8 + 1089*b*c^3*d^7*e - 10*a^3*b*d*e^7 - 15*a^4*e^8 - 3
0*(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 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 3*
(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 105*(7*c^4*d^2*e^6 - 28*b*c^3*d*e^7 + 2*(3*b^2
*c^2 + 2*a*c^3)*e^8)*x^6 - 105*(105*c^4*d^3*e^5 - 126*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 2*(b^3*c
+ 3*a*b*c^2)*e^8)*x^5 - 35*(805*c^4*d^4*e^4 - 770*b*c^3*d^3*e^5 + 30*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 - 35*(1015*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 30
*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 3*(a*
b^3 + 3*a^2*b*c)*e^8)*x^3 - 21*(1183*c^4*d^6*e^2 - 959*b*c^3*d^5*e^3 + 30*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 3*(a*b^3 + 3*a^2*b*c)*d*e^7 + 2*(3*a^2*b
^2 + 2*a^3*c)*e^8)*x^2 - 7*(1323*c^4*d^7*e - 1029*b*c^3*d^6*e^2 + 10*a^3*b*e^8 + 30*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 3*(a*b^3 + 3*a^2*b*c)*d^2*e^6
+ 2*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x - 420*(2*c^4*d^8 - b*c^3*d^7*e + (2*c^4*d*e^7 - b*c^3*e^8)*x^7 + 7*(2*c^4*d
^2*e^6 - b*c^3*d*e^7)*x^6 + 21*(2*c^4*d^3*e^5 - b*c^3*d^2*e^6)*x^5 + 35*(2*c^4*d^4*e^4 - b*c^3*d^3*e^5)*x^4 +
35*(2*c^4*d^5*e^3 - b*c^3*d^4*e^4)*x^3 + 21*(2*c^4*d^6*e^2 - b*c^3*d^5*e^3)*x^2 + 7*(2*c^4*d^7*e - b*c^3*d^6*e
^2)*x)*log(e*x + d))/(e^16*x^7 + 7*d*e^15*x^6 + 21*d^2*e^14*x^5 + 35*d^3*e^13*x^4 + 35*d^4*e^12*x^3 + 21*d^5*e
^11*x^2 + 7*d^6*e^10*x + d^7*e^9)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)**4/(e*x+d)**8,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (416) = 832\).

Time = 0.22 (sec) , antiderivative size = 872, normalized size of antiderivative = 2.06 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=-\frac {1443 \, c^{4} d^{8} - 1089 \, b c^{3} d^{7} e + 10 \, a^{3} b d e^{7} + 15 \, a^{4} e^{8} + 30 \, {\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} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 210 \, {\left (14 \, c^{4} d^{2} e^{6} - 14 \, b c^{3} d e^{7} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 210 \, {\left (70 \, c^{4} d^{3} e^{5} - 63 \, b c^{3} d^{2} e^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} + {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 35 \, {\left (910 \, c^{4} d^{4} e^{4} - 770 \, b c^{3} d^{3} e^{5} + 30 \, {\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} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 35 \, {\left (1078 \, c^{4} d^{5} e^{3} - 875 \, b c^{3} d^{4} e^{4} + 30 \, {\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} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 21 \, {\left (1218 \, c^{4} d^{6} e^{2} - 959 \, b c^{3} d^{5} e^{3} + 30 \, {\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} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 7 \, {\left (1338 \, c^{4} d^{7} e - 1029 \, b c^{3} d^{6} e^{2} + 10 \, a^{3} b e^{8} + 30 \, {\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} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{105 \, {\left (e^{16} x^{7} + 7 \, d e^{15} x^{6} + 21 \, d^{2} e^{14} x^{5} + 35 \, d^{3} e^{13} x^{4} + 35 \, d^{4} e^{12} x^{3} + 21 \, d^{5} e^{11} x^{2} + 7 \, d^{6} e^{10} x + d^{7} e^{9}\right )}} + \frac {c^{4} x}{e^{8}} - \frac {4 \, {\left (2 \, c^{4} d - b c^{3} e\right )} \log \left (e x + d\right )}{e^{9}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/105*(1443*c^4*d^8 - 1089*b*c^3*d^7*e + 10*a^3*b*d*e^7 + 15*a^4*e^8 + 30*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 3*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^
2*b^2 + 2*a^3*c)*d^2*e^6 + 210*(14*c^4*d^2*e^6 - 14*b*c^3*d*e^7 + (3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 210*(70*c^4
*d^3*e^5 - 63*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + (b^3*c + 3*a*b*c^2)*e^8)*x^5 + 35*(910*c^4*d^4*e
^4 - 770*b*c^3*d^3*e^5 + 30*(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 + (b^4 + 12*a*b^2*c +
6*a^2*c^2)*e^8)*x^4 + 35*(1078*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 30*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 3*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 21*(1218*c^4*d
^6*e^2 - 959*b*c^3*d^5*e^3 + 30*(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 + (b^4 + 12*a*b
^2*c + 6*a^2*c^2)*d^2*e^6 + 3*(a*b^3 + 3*a^2*b*c)*d*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 7*(1338*c^4*d^7*e
- 1029*b*c^3*d^6*e^2 + 10*a^3*b*e^8 + 30*(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 + (b^
4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 3*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^16*
x^7 + 7*d*e^15*x^6 + 21*d^2*e^14*x^5 + 35*d^3*e^13*x^4 + 35*d^4*e^12*x^3 + 21*d^5*e^11*x^2 + 7*d^6*e^10*x + d^
7*e^9) + c^4*x/e^8 - 4*(2*c^4*d - b*c^3*e)*log(e*x + d)/e^9

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (416) = 832\).

Time = 0.27 (sec) , antiderivative size = 900, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=\frac {c^{4} x}{e^{8}} - \frac {4 \, {\left (2 \, c^{4} d - b c^{3} e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} - \frac {1443 \, c^{4} d^{8} - 1089 \, b c^{3} d^{7} e + 90 \, b^{2} c^{2} d^{6} e^{2} + 60 \, a c^{3} d^{6} e^{2} + 10 \, b^{3} c d^{5} e^{3} + 30 \, a b c^{2} d^{5} e^{3} + b^{4} d^{4} e^{4} + 12 \, a b^{2} c d^{4} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a b^{3} d^{3} e^{5} + 9 \, a^{2} b c d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} + 4 \, a^{3} c d^{2} e^{6} + 10 \, a^{3} b d e^{7} + 15 \, a^{4} e^{8} + 210 \, {\left (14 \, c^{4} d^{2} e^{6} - 14 \, b c^{3} d e^{7} + 3 \, b^{2} c^{2} e^{8} + 2 \, a c^{3} e^{8}\right )} x^{6} + 210 \, {\left (70 \, c^{4} d^{3} e^{5} - 63 \, b c^{3} d^{2} e^{6} + 9 \, b^{2} c^{2} d e^{7} + 6 \, a c^{3} d e^{7} + b^{3} c e^{8} + 3 \, a b c^{2} e^{8}\right )} x^{5} + 35 \, {\left (910 \, c^{4} d^{4} e^{4} - 770 \, b c^{3} d^{3} e^{5} + 90 \, b^{2} c^{2} d^{2} e^{6} + 60 \, a c^{3} d^{2} e^{6} + 10 \, b^{3} c d e^{7} + 30 \, a b c^{2} d e^{7} + b^{4} e^{8} + 12 \, a b^{2} c e^{8} + 6 \, a^{2} c^{2} e^{8}\right )} x^{4} + 35 \, {\left (1078 \, c^{4} d^{5} e^{3} - 875 \, b c^{3} d^{4} e^{4} + 90 \, b^{2} c^{2} d^{3} e^{5} + 60 \, a c^{3} d^{3} e^{5} + 10 \, b^{3} c d^{2} e^{6} + 30 \, a b c^{2} d^{2} e^{6} + b^{4} d e^{7} + 12 \, a b^{2} c d e^{7} + 6 \, a^{2} c^{2} d e^{7} + 3 \, a b^{3} e^{8} + 9 \, a^{2} b c e^{8}\right )} x^{3} + 21 \, {\left (1218 \, c^{4} d^{6} e^{2} - 959 \, b c^{3} d^{5} e^{3} + 90 \, b^{2} c^{2} d^{4} e^{4} + 60 \, a c^{3} d^{4} e^{4} + 10 \, b^{3} c d^{3} e^{5} + 30 \, a b c^{2} d^{3} e^{5} + b^{4} d^{2} e^{6} + 12 \, a b^{2} c d^{2} e^{6} + 6 \, a^{2} c^{2} d^{2} e^{6} + 3 \, a b^{3} d e^{7} + 9 \, a^{2} b c d e^{7} + 6 \, a^{2} b^{2} e^{8} + 4 \, a^{3} c e^{8}\right )} x^{2} + 7 \, {\left (1338 \, c^{4} d^{7} e - 1029 \, b c^{3} d^{6} e^{2} + 90 \, b^{2} c^{2} d^{5} e^{3} + 60 \, a c^{3} d^{5} e^{3} + 10 \, b^{3} c d^{4} e^{4} + 30 \, a b c^{2} d^{4} e^{4} + b^{4} d^{3} e^{5} + 12 \, a b^{2} c d^{3} e^{5} + 6 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a b^{3} d^{2} e^{6} + 9 \, a^{2} b c d^{2} e^{6} + 6 \, a^{2} b^{2} d e^{7} + 4 \, a^{3} c d e^{7} + 10 \, a^{3} b e^{8}\right )} x}{105 \, {\left (e x + d\right )}^{7} e^{9}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^8,x, algorithm="giac")

[Out]

c^4*x/e^8 - 4*(2*c^4*d - b*c^3*e)*log(abs(e*x + d))/e^9 - 1/105*(1443*c^4*d^8 - 1089*b*c^3*d^7*e + 90*b^2*c^2*
d^6*e^2 + 60*a*c^3*d^6*e^2 + 10*b^3*c*d^5*e^3 + 30*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*
c^2*d^4*e^4 + 3*a*b^3*d^3*e^5 + 9*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 + 10*a^3*b*d*e^7 + 15*
a^4*e^8 + 210*(14*c^4*d^2*e^6 - 14*b*c^3*d*e^7 + 3*b^2*c^2*e^8 + 2*a*c^3*e^8)*x^6 + 210*(70*c^4*d^3*e^5 - 63*b
*c^3*d^2*e^6 + 9*b^2*c^2*d*e^7 + 6*a*c^3*d*e^7 + b^3*c*e^8 + 3*a*b*c^2*e^8)*x^5 + 35*(910*c^4*d^4*e^4 - 770*b*
c^3*d^3*e^5 + 90*b^2*c^2*d^2*e^6 + 60*a*c^3*d^2*e^6 + 10*b^3*c*d*e^7 + 30*a*b*c^2*d*e^7 + b^4*e^8 + 12*a*b^2*c
*e^8 + 6*a^2*c^2*e^8)*x^4 + 35*(1078*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 90*b^2*c^2*d^3*e^5 + 60*a*c^3*d^3*e^5 +
10*b^3*c*d^2*e^6 + 30*a*b*c^2*d^2*e^6 + b^4*d*e^7 + 12*a*b^2*c*d*e^7 + 6*a^2*c^2*d*e^7 + 3*a*b^3*e^8 + 9*a^2*
b*c*e^8)*x^3 + 21*(1218*c^4*d^6*e^2 - 959*b*c^3*d^5*e^3 + 90*b^2*c^2*d^4*e^4 + 60*a*c^3*d^4*e^4 + 10*b^3*c*d^3
*e^5 + 30*a*b*c^2*d^3*e^5 + b^4*d^2*e^6 + 12*a*b^2*c*d^2*e^6 + 6*a^2*c^2*d^2*e^6 + 3*a*b^3*d*e^7 + 9*a^2*b*c*d
*e^7 + 6*a^2*b^2*e^8 + 4*a^3*c*e^8)*x^2 + 7*(1338*c^4*d^7*e - 1029*b*c^3*d^6*e^2 + 90*b^2*c^2*d^5*e^3 + 60*a*c
^3*d^5*e^3 + 10*b^3*c*d^4*e^4 + 30*a*b*c^2*d^4*e^4 + b^4*d^3*e^5 + 12*a*b^2*c*d^3*e^5 + 6*a^2*c^2*d^3*e^5 + 3*
a*b^3*d^2*e^6 + 9*a^2*b*c*d^2*e^6 + 6*a^2*b^2*d*e^7 + 4*a^3*c*d*e^7 + 10*a^3*b*e^8)*x)/((e*x + d)^7*e^9)

Mupad [B] (verification not implemented)

Time = 10.08 (sec) , antiderivative size = 1306, normalized size of antiderivative = 3.08 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx=-\frac {\frac {a^4\,e^8}{7}+\frac {481\,c^4\,d^8}{35}+8\,c^4\,d^8\,\ln \left (d+e\,x\right )+\frac {b^4\,d^4\,e^4}{105}+\frac {b^4\,e^8\,x^4}{3}-c^4\,e^8\,x^8+\frac {a\,b^3\,d^3\,e^5}{35}+\frac {4\,a\,c^3\,d^6\,e^2}{7}+\frac {4\,a^3\,c\,d^2\,e^6}{105}+\frac {2\,b^3\,c\,d^5\,e^3}{21}+a\,b^3\,e^8\,x^3+\frac {4\,a^3\,c\,e^8\,x^2}{5}+4\,a\,c^3\,e^8\,x^6+2\,b^3\,c\,e^8\,x^5+\frac {b^4\,d^3\,e^5\,x}{15}+\frac {b^4\,d\,e^7\,x^3}{3}-7\,c^4\,d\,e^7\,x^7+\frac {2\,a^2\,b^2\,d^2\,e^6}{35}+\frac {2\,a^2\,c^2\,d^4\,e^4}{35}+\frac {6\,b^2\,c^2\,d^6\,e^2}{7}+\frac {6\,a^2\,b^2\,e^8\,x^2}{5}+2\,a^2\,c^2\,e^8\,x^4+6\,b^2\,c^2\,e^8\,x^6+\frac {b^4\,d^2\,e^6\,x^2}{5}+\frac {1183\,c^4\,d^6\,e^2\,x^2}{5}+\frac {1015\,c^4\,d^5\,e^3\,x^3}{3}+\frac {805\,c^4\,d^4\,e^4\,x^4}{3}+105\,c^4\,d^3\,e^5\,x^5+7\,c^4\,d^2\,e^6\,x^6+\frac {2\,a^3\,b\,d\,e^7}{21}-\frac {363\,b\,c^3\,d^7\,e}{35}+\frac {2\,a^3\,b\,e^8\,x}{3}+\frac {441\,c^4\,d^7\,e\,x}{5}-4\,b\,c^3\,d^7\,e\,\ln \left (d+e\,x\right )+\frac {4\,a^3\,c\,d\,e^7\,x}{15}+56\,c^4\,d^7\,e\,x\,\ln \left (d+e\,x\right )+\frac {6\,a^2\,c^2\,d^2\,e^6\,x^2}{5}+18\,b^2\,c^2\,d^4\,e^4\,x^2+30\,b^2\,c^2\,d^3\,e^5\,x^3+30\,b^2\,c^2\,d^2\,e^6\,x^4+\frac {2\,a\,b\,c^2\,d^5\,e^3}{7}+\frac {4\,a\,b^2\,c\,d^4\,e^4}{35}+\frac {3\,a^2\,b\,c\,d^3\,e^5}{35}+3\,a^2\,b\,c\,e^8\,x^3+4\,a\,b^2\,c\,e^8\,x^4+6\,a\,b\,c^2\,e^8\,x^5+\frac {a\,b^3\,d^2\,e^6\,x}{5}+\frac {2\,a^2\,b^2\,d\,e^7\,x}{5}+\frac {3\,a\,b^3\,d\,e^7\,x^2}{5}+4\,a\,c^3\,d^5\,e^3\,x+12\,a\,c^3\,d\,e^7\,x^5-\frac {343\,b\,c^3\,d^6\,e^2\,x}{5}+\frac {2\,b^3\,c\,d^4\,e^4\,x}{3}+\frac {10\,b^3\,c\,d\,e^7\,x^4}{3}-28\,b\,c^3\,d\,e^7\,x^6-4\,b\,c^3\,e^8\,x^7\,\ln \left (d+e\,x\right )+8\,c^4\,d\,e^7\,x^7\,\ln \left (d+e\,x\right )+\frac {2\,a^2\,c^2\,d^3\,e^5\,x}{5}+12\,a\,c^3\,d^4\,e^4\,x^2+20\,a\,c^3\,d^3\,e^5\,x^3+2\,a^2\,c^2\,d\,e^7\,x^3+20\,a\,c^3\,d^2\,e^6\,x^4+6\,b^2\,c^2\,d^5\,e^3\,x-\frac {959\,b\,c^3\,d^5\,e^3\,x^2}{5}+2\,b^3\,c\,d^3\,e^5\,x^2-\frac {875\,b\,c^3\,d^4\,e^4\,x^3}{3}+\frac {10\,b^3\,c\,d^2\,e^6\,x^3}{3}-\frac {770\,b\,c^3\,d^3\,e^5\,x^4}{3}-126\,b\,c^3\,d^2\,e^6\,x^5+18\,b^2\,c^2\,d\,e^7\,x^5+168\,c^4\,d^6\,e^2\,x^2\,\ln \left (d+e\,x\right )+280\,c^4\,d^5\,e^3\,x^3\,\ln \left (d+e\,x\right )+280\,c^4\,d^4\,e^4\,x^4\,\ln \left (d+e\,x\right )+168\,c^4\,d^3\,e^5\,x^5\,\ln \left (d+e\,x\right )+56\,c^4\,d^2\,e^6\,x^6\,\ln \left (d+e\,x\right )+6\,a\,b\,c^2\,d^3\,e^5\,x^2+\frac {12\,a\,b^2\,c\,d^2\,e^6\,x^2}{5}+10\,a\,b\,c^2\,d^2\,e^6\,x^3-84\,b\,c^3\,d^5\,e^3\,x^2\,\ln \left (d+e\,x\right )-140\,b\,c^3\,d^4\,e^4\,x^3\,\ln \left (d+e\,x\right )-140\,b\,c^3\,d^3\,e^5\,x^4\,\ln \left (d+e\,x\right )-84\,b\,c^3\,d^2\,e^6\,x^5\,\ln \left (d+e\,x\right )+2\,a\,b\,c^2\,d^4\,e^4\,x+\frac {4\,a\,b^2\,c\,d^3\,e^5\,x}{5}+\frac {3\,a^2\,b\,c\,d^2\,e^6\,x}{5}+\frac {9\,a^2\,b\,c\,d\,e^7\,x^2}{5}+4\,a\,b^2\,c\,d\,e^7\,x^3+10\,a\,b\,c^2\,d\,e^7\,x^4-28\,b\,c^3\,d^6\,e^2\,x\,\ln \left (d+e\,x\right )-28\,b\,c^3\,d\,e^7\,x^6\,\ln \left (d+e\,x\right )}{e^9\,{\left (d+e\,x\right )}^7} \]

[In]

int((a + b*x + c*x^2)^4/(d + e*x)^8,x)

[Out]

-((a^4*e^8)/7 + (481*c^4*d^8)/35 + 8*c^4*d^8*log(d + e*x) + (b^4*d^4*e^4)/105 + (b^4*e^8*x^4)/3 - c^4*e^8*x^8
+ (a*b^3*d^3*e^5)/35 + (4*a*c^3*d^6*e^2)/7 + (4*a^3*c*d^2*e^6)/105 + (2*b^3*c*d^5*e^3)/21 + a*b^3*e^8*x^3 + (4
*a^3*c*e^8*x^2)/5 + 4*a*c^3*e^8*x^6 + 2*b^3*c*e^8*x^5 + (b^4*d^3*e^5*x)/15 + (b^4*d*e^7*x^3)/3 - 7*c^4*d*e^7*x
^7 + (2*a^2*b^2*d^2*e^6)/35 + (2*a^2*c^2*d^4*e^4)/35 + (6*b^2*c^2*d^6*e^2)/7 + (6*a^2*b^2*e^8*x^2)/5 + 2*a^2*c
^2*e^8*x^4 + 6*b^2*c^2*e^8*x^6 + (b^4*d^2*e^6*x^2)/5 + (1183*c^4*d^6*e^2*x^2)/5 + (1015*c^4*d^5*e^3*x^3)/3 + (
805*c^4*d^4*e^4*x^4)/3 + 105*c^4*d^3*e^5*x^5 + 7*c^4*d^2*e^6*x^6 + (2*a^3*b*d*e^7)/21 - (363*b*c^3*d^7*e)/35 +
(2*a^3*b*e^8*x)/3 + (441*c^4*d^7*e*x)/5 - 4*b*c^3*d^7*e*log(d + e*x) + (4*a^3*c*d*e^7*x)/15 + 56*c^4*d^7*e*x*
log(d + e*x) + (6*a^2*c^2*d^2*e^6*x^2)/5 + 18*b^2*c^2*d^4*e^4*x^2 + 30*b^2*c^2*d^3*e^5*x^3 + 30*b^2*c^2*d^2*e^
6*x^4 + (2*a*b*c^2*d^5*e^3)/7 + (4*a*b^2*c*d^4*e^4)/35 + (3*a^2*b*c*d^3*e^5)/35 + 3*a^2*b*c*e^8*x^3 + 4*a*b^2*
c*e^8*x^4 + 6*a*b*c^2*e^8*x^5 + (a*b^3*d^2*e^6*x)/5 + (2*a^2*b^2*d*e^7*x)/5 + (3*a*b^3*d*e^7*x^2)/5 + 4*a*c^3*
d^5*e^3*x + 12*a*c^3*d*e^7*x^5 - (343*b*c^3*d^6*e^2*x)/5 + (2*b^3*c*d^4*e^4*x)/3 + (10*b^3*c*d*e^7*x^4)/3 - 28
*b*c^3*d*e^7*x^6 - 4*b*c^3*e^8*x^7*log(d + e*x) + 8*c^4*d*e^7*x^7*log(d + e*x) + (2*a^2*c^2*d^3*e^5*x)/5 + 12*
a*c^3*d^4*e^4*x^2 + 20*a*c^3*d^3*e^5*x^3 + 2*a^2*c^2*d*e^7*x^3 + 20*a*c^3*d^2*e^6*x^4 + 6*b^2*c^2*d^5*e^3*x -
(959*b*c^3*d^5*e^3*x^2)/5 + 2*b^3*c*d^3*e^5*x^2 - (875*b*c^3*d^4*e^4*x^3)/3 + (10*b^3*c*d^2*e^6*x^3)/3 - (770*
b*c^3*d^3*e^5*x^4)/3 - 126*b*c^3*d^2*e^6*x^5 + 18*b^2*c^2*d*e^7*x^5 + 168*c^4*d^6*e^2*x^2*log(d + e*x) + 280*c
^4*d^5*e^3*x^3*log(d + e*x) + 280*c^4*d^4*e^4*x^4*log(d + e*x) + 168*c^4*d^3*e^5*x^5*log(d + e*x) + 56*c^4*d^2
*e^6*x^6*log(d + e*x) + 6*a*b*c^2*d^3*e^5*x^2 + (12*a*b^2*c*d^2*e^6*x^2)/5 + 10*a*b*c^2*d^2*e^6*x^3 - 84*b*c^3
*d^5*e^3*x^2*log(d + e*x) - 140*b*c^3*d^4*e^4*x^3*log(d + e*x) - 140*b*c^3*d^3*e^5*x^4*log(d + e*x) - 84*b*c^3
*d^2*e^6*x^5*log(d + e*x) + 2*a*b*c^2*d^4*e^4*x + (4*a*b^2*c*d^3*e^5*x)/5 + (3*a^2*b*c*d^2*e^6*x)/5 + (9*a^2*b
*c*d*e^7*x^2)/5 + 4*a*b^2*c*d*e^7*x^3 + 10*a*b*c^2*d*e^7*x^4 - 28*b*c^3*d^6*e^2*x*log(d + e*x) - 28*b*c^3*d*e^
7*x^6*log(d + e*x))/(e^9*(d + e*x)^7)