\(\int \frac {(a+b x+c x^2)^3}{(d+e x)^{10}} \, dx\) [443]

Optimal result

Integrand size = 20, antiderivative size = 272 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^3}{9 e^7 (d+e x)^9}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{8 e^7 (d+e x)^8}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{7 e^7 (d+e x)^7}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{6 e^7 (d+e x)^6}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{5 e^7 (d+e x)^5}+\frac {3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac {c^3}{3 e^7 (d+e x)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {10 c^3 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )+5 e^3 \left (56 a^3 e^3+21 a^2 b e^2 (d+9 e x)+6 a b^2 e \left (d^2+9 d e x+36 e^2 x^2\right )+b^3 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+6 c e^2 \left (5 a^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+5 a b e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+2 b^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+3 c^2 e \left (4 a e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 b \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )}{2520 e^7 (d+e x)^9} \] Input:

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^10,x]
 

Output:

-1/2520*(10*c^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d 
^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6) + 5*e^3*(56*a^3*e^3 + 21*a^2*b*e^ 
2*(d + 9*e*x) + 6*a*b^2*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + b^3*(d^3 + 9*d^2* 
e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 6*c*e^2*(5*a^2*e^2*(d^2 + 9*d*e*x + 36 
*e^2*x^2) + 5*a*b*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 2*b^2* 
(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + 3*c^2*e 
*(4*a*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 
5*b*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 1 
26*e^5*x^5)))/(e^7*(d + e*x)^9)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1140, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx\)

\(\Big \downarrow \) 1140

\(\displaystyle \int \left (\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)^6}+\frac {(2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6 (d+e x)^7}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 (d+e x)^8}+\frac {3 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)^9}+\frac {\left (a e^2-b d e+c d^2\right )^3}{e^6 (d+e x)^{10}}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)^5}+\frac {c^3}{e^6 (d+e x)^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^7 (d+e x)^6}-\frac {3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7 (d+e x)^7}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{8 e^7 (d+e x)^8}-\frac {\left (a e^2-b d e+c d^2\right )^3}{9 e^7 (d+e x)^9}+\frac {3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac {c^3}{3 e^7 (d+e x)^3}\)

Input:

Int[(a + b*x + c*x^2)^3/(d + e*x)^10,x]
 

Output:

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

Defintions of rubi rules used

rule 1140

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] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.62

method	result	size
risch	\(\frac {-\frac {c^{3} x^{6}}{3 e}-\frac {c^{2} \left (3 b e +2 c d \right ) x^{5}}{4 e^{2}}-\frac {c \left (12 a c \,e^{2}+12 b^{2} e^{2}+15 b c d e +10 c^{2} d^{2}\right ) x^{4}}{20 e^{3}}-\frac {\left (30 a b c \,e^{3}+12 d \,e^{2} a \,c^{2}+5 b^{3} e^{3}+12 d \,e^{2} b^{2} c +15 d^{2} e b \,c^{2}+10 d^{3} c^{3}\right ) x^{3}}{30 e^{4}}-\frac {\left (30 e^{4} a^{2} c +30 a \,b^{2} e^{4}+30 a b c d \,e^{3}+12 d^{2} e^{2} a \,c^{2}+5 b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}+15 b \,c^{2} d^{3} e +10 d^{4} c^{3}\right ) x^{2}}{70 e^{5}}-\frac {\left (105 a^{2} b \,e^{5}+30 d \,e^{4} a^{2} c +30 a \,b^{2} d \,e^{4}+30 a b c \,d^{2} e^{3}+12 d^{3} e^{2} a \,c^{2}+5 b^{3} d^{2} e^{3}+12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e +10 d^{5} c^{3}\right ) x}{280 e^{6}}-\frac {280 e^{6} a^{3}+105 a^{2} b d \,e^{5}+30 d^{2} e^{4} a^{2} c +30 a \,b^{2} d^{2} e^{4}+30 a b c \,d^{3} e^{3}+12 d^{4} e^{2} a \,c^{2}+5 b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}+15 b \,c^{2} d^{5} e +10 d^{6} c^{3}}{2520 e^{7}}}{\left (e x +d \right )^{9}}\)	\(442\)
default	\(-\frac {c^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 c^{2} \left (b e -2 c d \right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} a \,c^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{9 e^{7} \left (e x +d \right )^{9}}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} a \,c^{2}-3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 b \,c^{2} d^{3} e +15 d^{4} c^{3}}{7 e^{7} \left (e x +d \right )^{7}}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} a \,c^{2}+3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{8 e^{7} \left (e x +d \right )^{8}}-\frac {6 a b c \,e^{3}-12 d \,e^{2} a \,c^{2}+b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}}{6 e^{7} \left (e x +d \right )^{6}}\)	\(461\)
norman	\(\frac {-\frac {c^{3} x^{6}}{3 e}-\frac {\left (3 e^{3} b \,c^{2}+2 d \,e^{2} c^{3}\right ) x^{5}}{4 e^{4}}-\frac {\left (12 a \,c^{2} e^{4}+12 b^{2} c \,e^{4}+15 d \,e^{3} b \,c^{2}+10 d^{2} e^{2} c^{3}\right ) x^{4}}{20 e^{5}}-\frac {\left (30 a b c \,e^{5}+12 a \,c^{2} d \,e^{4}+5 b^{3} e^{5}+12 b^{2} c d \,e^{4}+15 b \,c^{2} d^{2} e^{3}+10 e^{2} d^{3} c^{3}\right ) x^{3}}{30 e^{6}}-\frac {\left (30 a^{2} c \,e^{6}+30 a \,b^{2} e^{6}+30 a b c d \,e^{5}+12 a \,c^{2} d^{2} e^{4}+5 b^{3} d \,e^{5}+12 b^{2} c \,d^{2} e^{4}+15 b \,c^{2} d^{3} e^{3}+10 c^{3} d^{4} e^{2}\right ) x^{2}}{70 e^{7}}-\frac {\left (105 a^{2} b \,e^{7}+30 a^{2} c d \,e^{6}+30 a \,b^{2} d \,e^{6}+30 a b c \,d^{2} e^{5}+12 a \,c^{2} d^{3} e^{4}+5 b^{3} d^{2} e^{5}+12 b^{2} c \,d^{3} e^{4}+15 b \,c^{2} d^{4} e^{3}+10 c^{3} d^{5} e^{2}\right ) x}{280 e^{8}}-\frac {280 a^{3} e^{8}+105 a^{2} b d \,e^{7}+30 a^{2} c \,d^{2} e^{6}+30 a \,b^{2} d^{2} e^{6}+30 a b c \,d^{3} e^{5}+12 a \,c^{2} d^{4} e^{4}+5 b^{3} d^{3} e^{5}+12 b^{2} c \,d^{4} e^{4}+15 b \,c^{2} d^{5} e^{3}+10 c^{3} d^{6} e^{2}}{2520 e^{9}}}{\left (e x +d \right )^{9}}\)	\(478\)
gosper	\(-\frac {840 c^{3} e^{6} x^{6}+1890 x^{5} b \,c^{2} e^{6}+1260 c^{3} d \,e^{5} x^{5}+1512 x^{4} a \,c^{2} e^{6}+1512 x^{4} b^{2} c \,e^{6}+1890 x^{4} b \,c^{2} d \,e^{5}+1260 c^{3} d^{2} e^{4} x^{4}+2520 x^{3} a b c \,e^{6}+1008 x^{3} a \,c^{2} d \,e^{5}+420 x^{3} b^{3} e^{6}+1008 x^{3} b^{2} c d \,e^{5}+1260 x^{3} b \,c^{2} d^{2} e^{4}+840 c^{3} d^{3} e^{3} x^{3}+1080 x^{2} a^{2} c \,e^{6}+1080 x^{2} a \,b^{2} e^{6}+1080 x^{2} a b c d \,e^{5}+432 x^{2} a \,c^{2} d^{2} e^{4}+180 x^{2} b^{3} d \,e^{5}+432 x^{2} b^{2} c \,d^{2} e^{4}+540 x^{2} b \,c^{2} d^{3} e^{3}+360 c^{3} d^{4} e^{2} x^{2}+945 x \,a^{2} b \,e^{6}+270 x \,a^{2} c d \,e^{5}+270 x a \,b^{2} d \,e^{5}+270 x a b c \,d^{2} e^{4}+108 x a \,c^{2} d^{3} e^{3}+45 x \,b^{3} d^{2} e^{4}+108 x \,b^{2} c \,d^{3} e^{3}+135 x b \,c^{2} d^{4} e^{2}+90 c^{3} d^{5} e x +280 e^{6} a^{3}+105 a^{2} b d \,e^{5}+30 d^{2} e^{4} a^{2} c +30 a \,b^{2} d^{2} e^{4}+30 a b c \,d^{3} e^{3}+12 d^{4} e^{2} a \,c^{2}+5 b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}+15 b \,c^{2} d^{5} e +10 d^{6} c^{3}}{2520 e^{7} \left (e x +d \right )^{9}}\)	\(495\)
orering	\(-\frac {840 c^{3} e^{6} x^{6}+1890 x^{5} b \,c^{2} e^{6}+1260 c^{3} d \,e^{5} x^{5}+1512 x^{4} a \,c^{2} e^{6}+1512 x^{4} b^{2} c \,e^{6}+1890 x^{4} b \,c^{2} d \,e^{5}+1260 c^{3} d^{2} e^{4} x^{4}+2520 x^{3} a b c \,e^{6}+1008 x^{3} a \,c^{2} d \,e^{5}+420 x^{3} b^{3} e^{6}+1008 x^{3} b^{2} c d \,e^{5}+1260 x^{3} b \,c^{2} d^{2} e^{4}+840 c^{3} d^{3} e^{3} x^{3}+1080 x^{2} a^{2} c \,e^{6}+1080 x^{2} a \,b^{2} e^{6}+1080 x^{2} a b c d \,e^{5}+432 x^{2} a \,c^{2} d^{2} e^{4}+180 x^{2} b^{3} d \,e^{5}+432 x^{2} b^{2} c \,d^{2} e^{4}+540 x^{2} b \,c^{2} d^{3} e^{3}+360 c^{3} d^{4} e^{2} x^{2}+945 x \,a^{2} b \,e^{6}+270 x \,a^{2} c d \,e^{5}+270 x a \,b^{2} d \,e^{5}+270 x a b c \,d^{2} e^{4}+108 x a \,c^{2} d^{3} e^{3}+45 x \,b^{3} d^{2} e^{4}+108 x \,b^{2} c \,d^{3} e^{3}+135 x b \,c^{2} d^{4} e^{2}+90 c^{3} d^{5} e x +280 e^{6} a^{3}+105 a^{2} b d \,e^{5}+30 d^{2} e^{4} a^{2} c +30 a \,b^{2} d^{2} e^{4}+30 a b c \,d^{3} e^{3}+12 d^{4} e^{2} a \,c^{2}+5 b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}+15 b \,c^{2} d^{5} e +10 d^{6} c^{3}}{2520 e^{7} \left (e x +d \right )^{9}}\)	\(495\)
parallelrisch	\(\frac {-840 c^{3} x^{6} e^{8}-1890 b \,c^{2} e^{8} x^{5}-1260 c^{3} d \,e^{7} x^{5}-1512 a \,c^{2} e^{8} x^{4}-1512 b^{2} c \,e^{8} x^{4}-1890 b \,c^{2} d \,e^{7} x^{4}-1260 c^{3} d^{2} e^{6} x^{4}-2520 a b c \,e^{8} x^{3}-1008 a \,c^{2} d \,e^{7} x^{3}-420 b^{3} e^{8} x^{3}-1008 b^{2} c d \,e^{7} x^{3}-1260 b \,c^{2} d^{2} e^{6} x^{3}-840 c^{3} d^{3} e^{5} x^{3}-1080 a^{2} c \,e^{8} x^{2}-1080 a \,b^{2} e^{8} x^{2}-1080 a b c d \,e^{7} x^{2}-432 a \,c^{2} d^{2} e^{6} x^{2}-180 b^{3} d \,e^{7} x^{2}-432 b^{2} c \,d^{2} e^{6} x^{2}-540 b \,c^{2} d^{3} e^{5} x^{2}-360 c^{3} d^{4} e^{4} x^{2}-945 a^{2} b \,e^{8} x -270 a^{2} c d \,e^{7} x -270 a \,b^{2} d \,e^{7} x -270 a b c \,d^{2} e^{6} x -108 a \,c^{2} d^{3} e^{5} x -45 b^{3} d^{2} e^{6} x -108 b^{2} c \,d^{3} e^{5} x -135 b \,c^{2} d^{4} e^{4} x -90 c^{3} d^{5} e^{3} x -280 a^{3} e^{8}-105 a^{2} b d \,e^{7}-30 a^{2} c \,d^{2} e^{6}-30 a \,b^{2} d^{2} e^{6}-30 a b c \,d^{3} e^{5}-12 a \,c^{2} d^{4} e^{4}-5 b^{3} d^{3} e^{5}-12 b^{2} c \,d^{4} e^{4}-15 b \,c^{2} d^{5} e^{3}-10 c^{3} d^{6} e^{2}}{2520 e^{9} \left (e x +d \right )^{9}}\)	\(502\)

Input:

int((c*x^2+b*x+a)^3/(e*x+d)^10,x,method=_RETURNVERBOSE)
 

Output:

(-1/3*c^3*x^6/e-1/4*c^2/e^2*(3*b*e+2*c*d)*x^5-1/20*c/e^3*(12*a*c*e^2+12*b^ 
2*e^2+15*b*c*d*e+10*c^2*d^2)*x^4-1/30/e^4*(30*a*b*c*e^3+12*a*c^2*d*e^2+5*b 
^3*e^3+12*b^2*c*d*e^2+15*b*c^2*d^2*e+10*c^3*d^3)*x^3-1/70/e^5*(30*a^2*c*e^ 
4+30*a*b^2*e^4+30*a*b*c*d*e^3+12*a*c^2*d^2*e^2+5*b^3*d*e^3+12*b^2*c*d^2*e^ 
2+15*b*c^2*d^3*e+10*c^3*d^4)*x^2-1/280/e^6*(105*a^2*b*e^5+30*a^2*c*d*e^4+3 
0*a*b^2*d*e^4+30*a*b*c*d^2*e^3+12*a*c^2*d^3*e^2+5*b^3*d^2*e^3+12*b^2*c*d^3 
*e^2+15*b*c^2*d^4*e+10*c^3*d^5)*x-1/2520/e^7*(280*a^3*e^6+105*a^2*b*d*e^5+ 
30*a^2*c*d^2*e^4+30*a*b^2*d^2*e^4+30*a*b*c*d^3*e^3+12*a*c^2*d^4*e^2+5*b^3* 
d^3*e^3+12*b^2*c*d^4*e^2+15*b*c^2*d^5*e+10*c^3*d^6))/(e*x+d)^9
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 105 \, a^{2} b d e^{5} + 280 \, a^{3} e^{6} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 5 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 30 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 630 \, {\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 126 \, {\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 84 \, {\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + 5 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 36 \, {\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + 5 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 30 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 9 \, {\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 105 \, a^{2} b e^{6} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + 5 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 30 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{2520 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \] Input:

integrate((c*x^2+b*x+a)^3/(e*x+d)^10,x, algorithm="fricas")
 

Output:

-1/2520*(840*c^3*e^6*x^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 105*a^2*b*d*e^5 + 
 280*a^3*e^6 + 12*(b^2*c + a*c^2)*d^4*e^2 + 5*(b^3 + 6*a*b*c)*d^3*e^3 + 30 
*(a*b^2 + a^2*c)*d^2*e^4 + 630*(2*c^3*d*e^5 + 3*b*c^2*e^6)*x^5 + 126*(10*c 
^3*d^2*e^4 + 15*b*c^2*d*e^5 + 12*(b^2*c + a*c^2)*e^6)*x^4 + 84*(10*c^3*d^3 
*e^3 + 15*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 + 5*(b^3 + 6*a*b*c)*e^6 
)*x^3 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2*e^4 
 + 5*(b^3 + 6*a*b*c)*d*e^5 + 30*(a*b^2 + a^2*c)*e^6)*x^2 + 9*(10*c^3*d^5*e 
 + 15*b*c^2*d^4*e^2 + 105*a^2*b*e^6 + 12*(b^2*c + a*c^2)*d^3*e^3 + 5*(b^3 
+ 6*a*b*c)*d^2*e^4 + 30*(a*b^2 + a^2*c)*d*e^5)*x)/(e^16*x^9 + 9*d*e^15*x^8 
 + 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 
 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8*e^8*x + d^9*e^7)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((c*x**2+b*x+a)**3/(e*x+d)**10,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 105 \, a^{2} b d e^{5} + 280 \, a^{3} e^{6} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 5 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 30 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 630 \, {\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 126 \, {\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 84 \, {\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + 5 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 36 \, {\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + 5 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 30 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 9 \, {\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 105 \, a^{2} b e^{6} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + 5 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 30 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{2520 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \] Input:

integrate((c*x^2+b*x+a)^3/(e*x+d)^10,x, algorithm="maxima")
 

Output:

-1/2520*(840*c^3*e^6*x^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 105*a^2*b*d*e^5 + 
 280*a^3*e^6 + 12*(b^2*c + a*c^2)*d^4*e^2 + 5*(b^3 + 6*a*b*c)*d^3*e^3 + 30 
*(a*b^2 + a^2*c)*d^2*e^4 + 630*(2*c^3*d*e^5 + 3*b*c^2*e^6)*x^5 + 126*(10*c 
^3*d^2*e^4 + 15*b*c^2*d*e^5 + 12*(b^2*c + a*c^2)*e^6)*x^4 + 84*(10*c^3*d^3 
*e^3 + 15*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 + 5*(b^3 + 6*a*b*c)*e^6 
)*x^3 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2*e^4 
 + 5*(b^3 + 6*a*b*c)*d*e^5 + 30*(a*b^2 + a^2*c)*e^6)*x^2 + 9*(10*c^3*d^5*e 
 + 15*b*c^2*d^4*e^2 + 105*a^2*b*e^6 + 12*(b^2*c + a*c^2)*d^3*e^3 + 5*(b^3 
+ 6*a*b*c)*d^2*e^4 + 30*(a*b^2 + a^2*c)*d*e^5)*x)/(e^16*x^9 + 9*d*e^15*x^8 
 + 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 
 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8*e^8*x + d^9*e^7)
 

Giac [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.82 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {840 \, c^{3} e^{6} x^{6} + 1260 \, c^{3} d e^{5} x^{5} + 1890 \, b c^{2} e^{6} x^{5} + 1260 \, c^{3} d^{2} e^{4} x^{4} + 1890 \, b c^{2} d e^{5} x^{4} + 1512 \, b^{2} c e^{6} x^{4} + 1512 \, a c^{2} e^{6} x^{4} + 840 \, c^{3} d^{3} e^{3} x^{3} + 1260 \, b c^{2} d^{2} e^{4} x^{3} + 1008 \, b^{2} c d e^{5} x^{3} + 1008 \, a c^{2} d e^{5} x^{3} + 420 \, b^{3} e^{6} x^{3} + 2520 \, a b c e^{6} x^{3} + 360 \, c^{3} d^{4} e^{2} x^{2} + 540 \, b c^{2} d^{3} e^{3} x^{2} + 432 \, b^{2} c d^{2} e^{4} x^{2} + 432 \, a c^{2} d^{2} e^{4} x^{2} + 180 \, b^{3} d e^{5} x^{2} + 1080 \, a b c d e^{5} x^{2} + 1080 \, a b^{2} e^{6} x^{2} + 1080 \, a^{2} c e^{6} x^{2} + 90 \, c^{3} d^{5} e x + 135 \, b c^{2} d^{4} e^{2} x + 108 \, b^{2} c d^{3} e^{3} x + 108 \, a c^{2} d^{3} e^{3} x + 45 \, b^{3} d^{2} e^{4} x + 270 \, a b c d^{2} e^{4} x + 270 \, a b^{2} d e^{5} x + 270 \, a^{2} c d e^{5} x + 945 \, a^{2} b e^{6} x + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 12 \, a c^{2} d^{4} e^{2} + 5 \, b^{3} d^{3} e^{3} + 30 \, a b c d^{3} e^{3} + 30 \, a b^{2} d^{2} e^{4} + 30 \, a^{2} c d^{2} e^{4} + 105 \, a^{2} b d e^{5} + 280 \, a^{3} e^{6}}{2520 \, {\left (e x + d\right )}^{9} e^{7}} \] Input:

integrate((c*x^2+b*x+a)^3/(e*x+d)^10,x, algorithm="giac")
 

Output:

-1/2520*(840*c^3*e^6*x^6 + 1260*c^3*d*e^5*x^5 + 1890*b*c^2*e^6*x^5 + 1260* 
c^3*d^2*e^4*x^4 + 1890*b*c^2*d*e^5*x^4 + 1512*b^2*c*e^6*x^4 + 1512*a*c^2*e 
^6*x^4 + 840*c^3*d^3*e^3*x^3 + 1260*b*c^2*d^2*e^4*x^3 + 1008*b^2*c*d*e^5*x 
^3 + 1008*a*c^2*d*e^5*x^3 + 420*b^3*e^6*x^3 + 2520*a*b*c*e^6*x^3 + 360*c^3 
*d^4*e^2*x^2 + 540*b*c^2*d^3*e^3*x^2 + 432*b^2*c*d^2*e^4*x^2 + 432*a*c^2*d 
^2*e^4*x^2 + 180*b^3*d*e^5*x^2 + 1080*a*b*c*d*e^5*x^2 + 1080*a*b^2*e^6*x^2 
 + 1080*a^2*c*e^6*x^2 + 90*c^3*d^5*e*x + 135*b*c^2*d^4*e^2*x + 108*b^2*c*d 
^3*e^3*x + 108*a*c^2*d^3*e^3*x + 45*b^3*d^2*e^4*x + 270*a*b*c*d^2*e^4*x + 
270*a*b^2*d*e^5*x + 270*a^2*c*d*e^5*x + 945*a^2*b*e^6*x + 10*c^3*d^6 + 15* 
b*c^2*d^5*e + 12*b^2*c*d^4*e^2 + 12*a*c^2*d^4*e^2 + 5*b^3*d^3*e^3 + 30*a*b 
*c*d^3*e^3 + 30*a*b^2*d^2*e^4 + 30*a^2*c*d^2*e^4 + 105*a^2*b*d*e^5 + 280*a 
^3*e^6)/((e*x + d)^9*e^7)
 

Mupad [B] (verification not implemented)

Time = 5.40 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {\frac {280\,a^3\,e^6+105\,a^2\,b\,d\,e^5+30\,a^2\,c\,d^2\,e^4+30\,a\,b^2\,d^2\,e^4+30\,a\,b\,c\,d^3\,e^3+12\,a\,c^2\,d^4\,e^2+5\,b^3\,d^3\,e^3+12\,b^2\,c\,d^4\,e^2+15\,b\,c^2\,d^5\,e+10\,c^3\,d^6}{2520\,e^7}+\frac {x^3\,\left (5\,b^3\,e^3+12\,b^2\,c\,d\,e^2+15\,b\,c^2\,d^2\,e+30\,a\,b\,c\,e^3+10\,c^3\,d^3+12\,a\,c^2\,d\,e^2\right )}{30\,e^4}+\frac {x^2\,\left (30\,a^2\,c\,e^4+30\,a\,b^2\,e^4+30\,a\,b\,c\,d\,e^3+12\,a\,c^2\,d^2\,e^2+5\,b^3\,d\,e^3+12\,b^2\,c\,d^2\,e^2+15\,b\,c^2\,d^3\,e+10\,c^3\,d^4\right )}{70\,e^5}+\frac {c^3\,x^6}{3\,e}+\frac {x\,\left (105\,a^2\,b\,e^5+30\,a^2\,c\,d\,e^4+30\,a\,b^2\,d\,e^4+30\,a\,b\,c\,d^2\,e^3+12\,a\,c^2\,d^3\,e^2+5\,b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2+15\,b\,c^2\,d^4\,e+10\,c^3\,d^5\right )}{280\,e^6}+\frac {c\,x^4\,\left (12\,b^2\,e^2+15\,b\,c\,d\,e+10\,c^2\,d^2+12\,a\,c\,e^2\right )}{20\,e^3}+\frac {c^2\,x^5\,\left (3\,b\,e+2\,c\,d\right )}{4\,e^2}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \] Input:

int((a + b*x + c*x^2)^3/(d + e*x)^10,x)
 

Output:

-((280*a^3*e^6 + 10*c^3*d^6 + 5*b^3*d^3*e^3 + 30*a*b^2*d^2*e^4 + 12*a*c^2* 
d^4*e^2 + 30*a^2*c*d^2*e^4 + 12*b^2*c*d^4*e^2 + 105*a^2*b*d*e^5 + 15*b*c^2 
*d^5*e + 30*a*b*c*d^3*e^3)/(2520*e^7) + (x^3*(5*b^3*e^3 + 10*c^3*d^3 + 30* 
a*b*c*e^3 + 12*a*c^2*d*e^2 + 15*b*c^2*d^2*e + 12*b^2*c*d*e^2))/(30*e^4) + 
(x^2*(10*c^3*d^4 + 30*a*b^2*e^4 + 30*a^2*c*e^4 + 5*b^3*d*e^3 + 12*a*c^2*d^ 
2*e^2 + 12*b^2*c*d^2*e^2 + 15*b*c^2*d^3*e + 30*a*b*c*d*e^3))/(70*e^5) + (c 
^3*x^6)/(3*e) + (x*(10*c^3*d^5 + 105*a^2*b*e^5 + 5*b^3*d^2*e^3 + 12*a*c^2* 
d^3*e^2 + 12*b^2*c*d^3*e^2 + 30*a*b^2*d*e^4 + 30*a^2*c*d*e^4 + 15*b*c^2*d^ 
4*e + 30*a*b*c*d^2*e^3))/(280*e^6) + (c*x^4*(12*b^2*e^2 + 10*c^2*d^2 + 12* 
a*c*e^2 + 15*b*c*d*e))/(20*e^3) + (c^2*x^5*(3*b*e + 2*c*d))/(4*e^2))/(d^9 
+ e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84*d^6*e^3*x^3 + 126*d^5*e^4*x^ 
4 + 126*d^4*e^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.14 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=\frac {-840 c^{3} e^{6} x^{6}-1890 b \,c^{2} e^{6} x^{5}-1260 c^{3} d \,e^{5} x^{5}-1512 a \,c^{2} e^{6} x^{4}-1512 b^{2} c \,e^{6} x^{4}-1890 b \,c^{2} d \,e^{5} x^{4}-1260 c^{3} d^{2} e^{4} x^{4}-2520 a b c \,e^{6} x^{3}-1008 a \,c^{2} d \,e^{5} x^{3}-420 b^{3} e^{6} x^{3}-1008 b^{2} c d \,e^{5} x^{3}-1260 b \,c^{2} d^{2} e^{4} x^{3}-840 c^{3} d^{3} e^{3} x^{3}-1080 a^{2} c \,e^{6} x^{2}-1080 a \,b^{2} e^{6} x^{2}-1080 a b c d \,e^{5} x^{2}-432 a \,c^{2} d^{2} e^{4} x^{2}-180 b^{3} d \,e^{5} x^{2}-432 b^{2} c \,d^{2} e^{4} x^{2}-540 b \,c^{2} d^{3} e^{3} x^{2}-360 c^{3} d^{4} e^{2} x^{2}-945 a^{2} b \,e^{6} x -270 a^{2} c d \,e^{5} x -270 a \,b^{2} d \,e^{5} x -270 a b c \,d^{2} e^{4} x -108 a \,c^{2} d^{3} e^{3} x -45 b^{3} d^{2} e^{4} x -108 b^{2} c \,d^{3} e^{3} x -135 b \,c^{2} d^{4} e^{2} x -90 c^{3} d^{5} e x -280 a^{3} e^{6}-105 a^{2} b d \,e^{5}-30 a^{2} c \,d^{2} e^{4}-30 a \,b^{2} d^{2} e^{4}-30 a b c \,d^{3} e^{3}-12 a \,c^{2} d^{4} e^{2}-5 b^{3} d^{3} e^{3}-12 b^{2} c \,d^{4} e^{2}-15 b \,c^{2} d^{5} e -10 c^{3} d^{6}}{2520 e^{7} \left (e^{9} x^{9}+9 d \,e^{8} x^{8}+36 d^{2} e^{7} x^{7}+84 d^{3} e^{6} x^{6}+126 d^{4} e^{5} x^{5}+126 d^{5} e^{4} x^{4}+84 d^{6} e^{3} x^{3}+36 d^{7} e^{2} x^{2}+9 d^{8} e x +d^{9}\right )} \] Input:

int((c*x^2+b*x+a)^3/(e*x+d)^10,x)
 

Output:

( - 280*a**3*e**6 - 105*a**2*b*d*e**5 - 945*a**2*b*e**6*x - 30*a**2*c*d**2 
*e**4 - 270*a**2*c*d*e**5*x - 1080*a**2*c*e**6*x**2 - 30*a*b**2*d**2*e**4 
- 270*a*b**2*d*e**5*x - 1080*a*b**2*e**6*x**2 - 30*a*b*c*d**3*e**3 - 270*a 
*b*c*d**2*e**4*x - 1080*a*b*c*d*e**5*x**2 - 2520*a*b*c*e**6*x**3 - 12*a*c* 
*2*d**4*e**2 - 108*a*c**2*d**3*e**3*x - 432*a*c**2*d**2*e**4*x**2 - 1008*a 
*c**2*d*e**5*x**3 - 1512*a*c**2*e**6*x**4 - 5*b**3*d**3*e**3 - 45*b**3*d** 
2*e**4*x - 180*b**3*d*e**5*x**2 - 420*b**3*e**6*x**3 - 12*b**2*c*d**4*e**2 
 - 108*b**2*c*d**3*e**3*x - 432*b**2*c*d**2*e**4*x**2 - 1008*b**2*c*d*e**5 
*x**3 - 1512*b**2*c*e**6*x**4 - 15*b*c**2*d**5*e - 135*b*c**2*d**4*e**2*x 
- 540*b*c**2*d**3*e**3*x**2 - 1260*b*c**2*d**2*e**4*x**3 - 1890*b*c**2*d*e 
**5*x**4 - 1890*b*c**2*e**6*x**5 - 10*c**3*d**6 - 90*c**3*d**5*e*x - 360*c 
**3*d**4*e**2*x**2 - 840*c**3*d**3*e**3*x**3 - 1260*c**3*d**2*e**4*x**4 - 
1260*c**3*d*e**5*x**5 - 840*c**3*e**6*x**6)/(2520*e**7*(d**9 + 9*d**8*e*x 
+ 36*d**7*e**2*x**2 + 84*d**6*e**3*x**3 + 126*d**5*e**4*x**4 + 126*d**4*e* 
*5*x**5 + 84*d**3*e**6*x**6 + 36*d**2*e**7*x**7 + 9*d*e**8*x**8 + e**9*x** 
9))