\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^{10}} \, dx\) [2160]

Optimal result

Integrand size = 20, antiderivative size = 436 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^4}{9 e^9 (d+e x)^9}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{2 e^9 (d+e x)^8}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{7 e^9 (d+e x)^7}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (d+e x)^6}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{5 e^9 (d+e x)^5}+\frac {c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^4}-\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{3 e^9 (d+e x)^3}+\frac {2 c^3 (2 c d-b e)}{e^9 (d+e x)^2}-\frac {c^4}{e^9 (d+e x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 436, 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} \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=-\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{5 e^9 (d+e x)^5}-\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}+\frac {c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^4}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^6}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^9 (d+e x)^7}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{2 e^9 (d+e x)^8}-\frac {\left (a e^2-b d e+c d^2\right )^4}{9 e^9 (d+e x)^9}+\frac {2 c^3 (2 c d-b e)}{e^9 (d+e x)^2}-\frac {c^4}{e^9 (d+e x)} \]

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^10,x]

[Out]

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

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)^{10}}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^9}+\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)^8}+\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)^7}+\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)^6}+\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)^5}+\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)^4}-\frac {4 c^3 (2 c d-b e)}{e^8 (d+e x)^3}+\frac {c^4}{e^8 (d+e x)^2}\right ) \, dx \\ & = -\frac {\left (c d^2-b d e+a e^2\right )^4}{9 e^9 (d+e x)^9}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{2 e^9 (d+e x)^8}-\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 )}{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 ) \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 {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 )}{5 e^9 (d+e x)^5}+\frac {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)^4}-\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{3 e^9 (d+e x)^3}+\frac {2 c^3 (2 c d-b e)}{e^9 (d+e x)^2}-\frac {c^4}{e^9 (d+e x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=-\frac {70 c^4 \left (d^8+9 d^7 e x+36 d^6 e^2 x^2+84 d^5 e^3 x^3+126 d^4 e^4 x^4+126 d^3 e^5 x^5+84 d^2 e^6 x^6+36 d e^7 x^7+9 e^8 x^8\right )+e^4 \left (70 a^4 e^4+35 a^3 b e^3 (d+9 e x)+15 a^2 b^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+5 a b^3 e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+b^4 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+c e^3 \left (10 a^3 e^3 \left (d^2+9 d e x+36 e^2 x^2\right )+15 a^2 b e^2 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+12 a b^2 e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 b^3 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+3 c^2 e^2 \left (2 a^2 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 a b e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+5 b^2 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )+5 c^3 e \left (2 a e \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )+7 b \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )\right )}{630 e^9 (d+e x)^9} \]

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^10,x]

[Out]

-1/630*(70*c^4*(d^8 + 9*d^7*e*x + 36*d^6*e^2*x^2 + 84*d^5*e^3*x^3 + 126*d^4*e^4*x^4 + 126*d^3*e^5*x^5 + 84*d^2
*e^6*x^6 + 36*d*e^7*x^7 + 9*e^8*x^8) + e^4*(70*a^4*e^4 + 35*a^3*b*e^3*(d + 9*e*x) + 15*a^2*b^2*e^2*(d^2 + 9*d*
e*x + 36*e^2*x^2) + 5*a*b^3*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + b^4*(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)) + c*e^3*(10*a^3*e^3*(d^2 + 9*d*e*x + 36*e^2*x^2) + 15*a^2*b*e^2*(d^3 + 9*
d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 12*a*b^2*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^3*(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)) + 3*c^2*e^2*(2*
a^2*e^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) + 5*a*b*e*(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) + 5*b^2*(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*c^3*e*(2*a*e*(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) + 7*b*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3
*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^7)))/(e^9*(d + e*x)^9)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(879\) vs. \(2(425)=850\).

Time = 57.20 (sec) , antiderivative size = 880, normalized size of antiderivative = 2.02

method	result	size
risch	\(\frac {-\frac {c^{4} x^{8}}{e}-\frac {2 c^{3} \left (b e +2 c d \right ) x^{7}}{e^{2}}-\frac {2 c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}+7 b c d e +14 c^{2} d^{2}\right ) x^{6}}{3 e^{3}}-\frac {c \left (3 a b c \,e^{3}+2 c^{2} a d \,e^{2}+b^{3} e^{3}+3 b^{2} d \,e^{2} c +7 b \,c^{2} d^{2} e +14 c^{3} d^{3}\right ) x^{5}}{e^{4}}-\frac {\left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}+15 a b \,c^{2} d \,e^{3}+10 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}+5 b^{3} c d \,e^{3}+15 b^{2} c^{2} d^{2} e^{2}+35 b \,c^{3} d^{3} e +70 c^{4} d^{4}\right ) x^{4}}{5 e^{5}}-\frac {2 \left (15 a^{2} b c \,e^{5}+6 d \,e^{4} a^{2} c^{2}+5 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+15 a b \,c^{2} d^{2} e^{3}+10 d^{3} e^{2} c^{3} a +b^{4} d \,e^{4}+5 b^{3} c \,d^{2} e^{3}+15 b^{2} c^{2} d^{3} e^{2}+35 b \,c^{3} d^{4} e +70 c^{4} d^{5}\right ) x^{3}}{15 e^{6}}-\frac {2 \left (10 e^{6} c \,a^{3}+15 a^{2} b^{2} e^{6}+15 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+5 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+15 a b \,c^{2} d^{3} e^{3}+10 d^{4} e^{2} c^{3} a +b^{4} d^{2} e^{4}+5 b^{3} c \,d^{3} e^{3}+15 b^{2} c^{2} d^{4} e^{2}+35 b \,c^{3} d^{5} e +70 d^{6} c^{4}\right ) x^{2}}{35 e^{7}}-\frac {\left (35 a^{3} b \,e^{7}+10 d \,e^{6} c \,a^{3}+15 a^{2} b^{2} d \,e^{6}+15 a^{2} b c \,d^{2} e^{5}+6 d^{3} e^{4} a^{2} c^{2}+5 a \,b^{3} d^{2} e^{5}+12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}+10 d^{5} e^{2} c^{3} a +b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}+15 b^{2} c^{2} d^{5} e^{2}+35 b \,c^{3} d^{6} e +70 d^{7} c^{4}\right ) x}{70 e^{8}}-\frac {70 a^{4} e^{8}+35 a^{3} b d \,e^{7}+10 a^{3} c \,d^{2} e^{6}+15 a^{2} b^{2} d^{2} e^{6}+15 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+5 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+15 a b \,c^{2} d^{5} e^{3}+10 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+5 b^{3} c \,d^{5} e^{3}+15 b^{2} c^{2} d^{6} e^{2}+35 b \,c^{3} d^{7} e +70 c^{4} d^{8}}{630 e^{9}}}{\left (e x +d \right )^{9}}\)	\(880\)
norman	\(\frac {-\frac {c^{4} x^{8}}{e}-\frac {2 \left (b \,c^{3} e +2 c^{4} d \right ) x^{7}}{e^{2}}-\frac {2 \left (2 a \,c^{3} e^{2}+3 b^{2} c^{2} e^{2}+7 d e \,c^{3} b +14 c^{4} d^{2}\right ) x^{6}}{3 e^{3}}-\frac {\left (3 a b \,c^{2} e^{3}+2 a \,c^{3} d \,e^{2}+b^{3} c \,e^{3}+3 b^{2} c^{2} d \,e^{2}+7 b \,c^{3} d^{2} e +14 d^{3} c^{4}\right ) x^{5}}{e^{4}}-\frac {\left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}+15 a b \,c^{2} d \,e^{3}+10 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}+5 b^{3} c d \,e^{3}+15 b^{2} c^{2} d^{2} e^{2}+35 b \,c^{3} d^{3} e +70 c^{4} d^{4}\right ) x^{4}}{5 e^{5}}-\frac {2 \left (15 a^{2} b c \,e^{5}+6 d \,e^{4} a^{2} c^{2}+5 a \,b^{3} e^{5}+12 a \,b^{2} c d \,e^{4}+15 a b \,c^{2} d^{2} e^{3}+10 d^{3} e^{2} c^{3} a +b^{4} d \,e^{4}+5 b^{3} c \,d^{2} e^{3}+15 b^{2} c^{2} d^{3} e^{2}+35 b \,c^{3} d^{4} e +70 c^{4} d^{5}\right ) x^{3}}{15 e^{6}}-\frac {2 \left (10 e^{6} c \,a^{3}+15 a^{2} b^{2} e^{6}+15 a^{2} b c d \,e^{5}+6 d^{2} e^{4} a^{2} c^{2}+5 a \,b^{3} d \,e^{5}+12 a \,b^{2} c \,d^{2} e^{4}+15 a b \,c^{2} d^{3} e^{3}+10 d^{4} e^{2} c^{3} a +b^{4} d^{2} e^{4}+5 b^{3} c \,d^{3} e^{3}+15 b^{2} c^{2} d^{4} e^{2}+35 b \,c^{3} d^{5} e +70 d^{6} c^{4}\right ) x^{2}}{35 e^{7}}-\frac {\left (35 a^{3} b \,e^{7}+10 d \,e^{6} c \,a^{3}+15 a^{2} b^{2} d \,e^{6}+15 a^{2} b c \,d^{2} e^{5}+6 d^{3} e^{4} a^{2} c^{2}+5 a \,b^{3} d^{2} e^{5}+12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}+10 d^{5} e^{2} c^{3} a +b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}+15 b^{2} c^{2} d^{5} e^{2}+35 b \,c^{3} d^{6} e +70 d^{7} c^{4}\right ) x}{70 e^{8}}-\frac {70 a^{4} e^{8}+35 a^{3} b d \,e^{7}+10 a^{3} c \,d^{2} e^{6}+15 a^{2} b^{2} d^{2} e^{6}+15 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}+5 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}+15 a b \,c^{2} d^{5} e^{3}+10 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}+5 b^{3} c \,d^{5} e^{3}+15 b^{2} c^{2} d^{6} e^{2}+35 b \,c^{3} d^{7} e +70 c^{4} d^{8}}{630 e^{9}}}{\left (e x +d \right )^{9}}\)	\(890\)
default	\(-\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}}{9 e^{9} \left (e x +d \right )^{9}}-\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}}{5 e^{9} \left (e x +d \right )^{5}}-\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}}{8 e^{9} \left (e x +d \right )^{8}}-\frac {c^{4}}{e^{9} \left (e x +d \right )}-\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 )}{3 e^{9} \left (e x +d \right )^{3}}-\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}}{7 e^{9} \left (e x +d \right )^{7}}-\frac {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 )^{4}}-\frac {2 c^{3} \left (b e -2 c d \right )}{e^{9} \left (e x +d \right )^{2}}-\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}}{6 e^{9} \left (e x +d \right )^{6}}\)	\(914\)
gosper	\(\text {Expression too large to display}\)	\(1015\)
parallelrisch	\(\text {Expression too large to display}\)	\(1016\)

[In]

int((c*x^2+b*x+a)^4/(e*x+d)^10,x,method=_RETURNVERBOSE)

[Out]

(-1/e*c^4*x^8-2*c^3*(b*e+2*c*d)/e^2*x^7-2/3*c^2*(2*a*c*e^2+3*b^2*e^2+7*b*c*d*e+14*c^2*d^2)/e^3*x^6-c*(3*a*b*c*
e^3+2*a*c^2*d*e^2+b^3*e^3+3*b^2*c*d*e^2+7*b*c^2*d^2*e+14*c^3*d^3)/e^4*x^5-1/5*(6*a^2*c^2*e^4+12*a*b^2*c*e^4+15
*a*b*c^2*d*e^3+10*a*c^3*d^2*e^2+b^4*e^4+5*b^3*c*d*e^3+15*b^2*c^2*d^2*e^2+35*b*c^3*d^3*e+70*c^4*d^4)/e^5*x^4-2/
15*(15*a^2*b*c*e^5+6*a^2*c^2*d*e^4+5*a*b^3*e^5+12*a*b^2*c*d*e^4+15*a*b*c^2*d^2*e^3+10*a*c^3*d^3*e^2+b^4*d*e^4+
5*b^3*c*d^2*e^3+15*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e+70*c^4*d^5)/e^6*x^3-2/35*(10*a^3*c*e^6+15*a^2*b^2*e^6+15*a^2
*b*c*d*e^5+6*a^2*c^2*d^2*e^4+5*a*b^3*d*e^5+12*a*b^2*c*d^2*e^4+15*a*b*c^2*d^3*e^3+10*a*c^3*d^4*e^2+b^4*d^2*e^4+
5*b^3*c*d^3*e^3+15*b^2*c^2*d^4*e^2+35*b*c^3*d^5*e+70*c^4*d^6)/e^7*x^2-1/70*(35*a^3*b*e^7+10*a^3*c*d*e^6+15*a^2
*b^2*d*e^6+15*a^2*b*c*d^2*e^5+6*a^2*c^2*d^3*e^4+5*a*b^3*d^2*e^5+12*a*b^2*c*d^3*e^4+15*a*b*c^2*d^4*e^3+10*a*c^3
*d^5*e^2+b^4*d^3*e^4+5*b^3*c*d^4*e^3+15*b^2*c^2*d^5*e^2+35*b*c^3*d^6*e+70*c^4*d^7)/e^8*x-1/630*(70*a^4*e^8+35*
a^3*b*d*e^7+10*a^3*c*d^2*e^6+15*a^2*b^2*d^2*e^6+15*a^2*b*c*d^3*e^5+6*a^2*c^2*d^4*e^4+5*a*b^3*d^3*e^5+12*a*b^2*
c*d^4*e^4+15*a*b*c^2*d^5*e^3+10*a*c^3*d^6*e^2+b^4*d^4*e^4+5*b^3*c*d^5*e^3+15*b^2*c^2*d^6*e^2+35*b*c^3*d^7*e+70
*c^4*d^8)/e^9)/(e*x+d)^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 893 vs. \(2 (424) = 848\).

Time = 0.38 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=-\frac {630 \, c^{4} e^{8} x^{8} + 70 \, c^{4} d^{8} + 35 \, b c^{3} d^{7} e + 35 \, a^{3} b d e^{7} + 70 \, a^{4} e^{8} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 5 \, {\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} + 5 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 5 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 1260 \, {\left (2 \, c^{4} d e^{7} + b c^{3} e^{8}\right )} x^{7} + 420 \, {\left (14 \, c^{4} d^{2} e^{6} + 7 \, b c^{3} d e^{7} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 630 \, {\left (14 \, c^{4} d^{3} e^{5} + 7 \, b c^{3} d^{2} e^{6} + {\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} + 126 \, {\left (70 \, c^{4} d^{4} e^{4} + 35 \, b c^{3} d^{3} e^{5} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} + 5 \, {\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} + 84 \, {\left (70 \, c^{4} d^{5} e^{3} + 35 \, b c^{3} d^{4} e^{4} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} + 5 \, {\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} + 5 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 36 \, {\left (70 \, c^{4} d^{6} e^{2} + 35 \, b c^{3} d^{5} e^{3} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} + 5 \, {\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} + 5 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 5 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 9 \, {\left (70 \, c^{4} d^{7} e + 35 \, b c^{3} d^{6} e^{2} + 35 \, a^{3} b e^{8} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} + 5 \, {\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} + 5 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 5 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{630 \, {\left (e^{18} x^{9} + 9 \, d e^{17} x^{8} + 36 \, d^{2} e^{16} x^{7} + 84 \, d^{3} e^{15} x^{6} + 126 \, d^{4} e^{14} x^{5} + 126 \, d^{5} e^{13} x^{4} + 84 \, d^{6} e^{12} x^{3} + 36 \, d^{7} e^{11} x^{2} + 9 \, d^{8} e^{10} x + d^{9} e^{9}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/630*(630*c^4*e^8*x^8 + 70*c^4*d^8 + 35*b*c^3*d^7*e + 35*a^3*b*d*e^7 + 70*a^4*e^8 + 5*(3*b^2*c^2 + 2*a*c^3)*
d^6*e^2 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c)*d^3*e
^5 + 5*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 1260*(2*c^4*d*e^7 + b*c^3*e^8)*x^7 + 420*(14*c^4*d^2*e^6 + 7*b*c^3*d*e^
7 + (3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 630*(14*c^4*d^3*e^5 + 7*b*c^3*d^2*e^6 + (3*b^2*c^2 + 2*a*c^3)*d*e^7 + (b^
3*c + 3*a*b*c^2)*e^8)*x^5 + 126*(70*c^4*d^4*e^4 + 35*b*c^3*d^3*e^5 + 5*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 5*(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 + 84*(70*c^4*d^5*e^3 + 35*b*c^3*d^4*e^4 + 5*(3*
b^2*c^2 + 2*a*c^3)*d^3*e^5 + 5*(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 + 5*(a*b^3 +
3*a^2*b*c)*e^8)*x^3 + 36*(70*c^4*d^6*e^2 + 35*b*c^3*d^5*e^3 + 5*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c)*d*e^7 + 5*(3*a^2*b^2 + 2*a^3
*c)*e^8)*x^2 + 9*(70*c^4*d^7*e + 35*b*c^3*d^6*e^2 + 35*a^3*b*e^8 + 5*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 5*(3*a^2*b^2 +
2*a^3*c)*d*e^7)*x)/(e^18*x^9 + 9*d*e^17*x^8 + 36*d^2*e^16*x^7 + 84*d^3*e^15*x^6 + 126*d^4*e^14*x^5 + 126*d^5*
e^13*x^4 + 84*d^6*e^12*x^3 + 36*d^7*e^11*x^2 + 9*d^8*e^10*x + d^9*e^9)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)**4/(e*x+d)**10,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 893 vs. \(2 (424) = 848\).

Time = 0.23 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=-\frac {630 \, c^{4} e^{8} x^{8} + 70 \, c^{4} d^{8} + 35 \, b c^{3} d^{7} e + 35 \, a^{3} b d e^{7} + 70 \, a^{4} e^{8} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 5 \, {\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} + 5 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 5 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 1260 \, {\left (2 \, c^{4} d e^{7} + b c^{3} e^{8}\right )} x^{7} + 420 \, {\left (14 \, c^{4} d^{2} e^{6} + 7 \, b c^{3} d e^{7} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 630 \, {\left (14 \, c^{4} d^{3} e^{5} + 7 \, b c^{3} d^{2} e^{6} + {\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} + 126 \, {\left (70 \, c^{4} d^{4} e^{4} + 35 \, b c^{3} d^{3} e^{5} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} + 5 \, {\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} + 84 \, {\left (70 \, c^{4} d^{5} e^{3} + 35 \, b c^{3} d^{4} e^{4} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} + 5 \, {\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} + 5 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 36 \, {\left (70 \, c^{4} d^{6} e^{2} + 35 \, b c^{3} d^{5} e^{3} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} + 5 \, {\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} + 5 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 5 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 9 \, {\left (70 \, c^{4} d^{7} e + 35 \, b c^{3} d^{6} e^{2} + 35 \, a^{3} b e^{8} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} + 5 \, {\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} + 5 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 5 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{630 \, {\left (e^{18} x^{9} + 9 \, d e^{17} x^{8} + 36 \, d^{2} e^{16} x^{7} + 84 \, d^{3} e^{15} x^{6} + 126 \, d^{4} e^{14} x^{5} + 126 \, d^{5} e^{13} x^{4} + 84 \, d^{6} e^{12} x^{3} + 36 \, d^{7} e^{11} x^{2} + 9 \, d^{8} e^{10} x + d^{9} e^{9}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^10,x, algorithm="maxima")

[Out]

-1/630*(630*c^4*e^8*x^8 + 70*c^4*d^8 + 35*b*c^3*d^7*e + 35*a^3*b*d*e^7 + 70*a^4*e^8 + 5*(3*b^2*c^2 + 2*a*c^3)*
d^6*e^2 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c)*d^3*e
^5 + 5*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 1260*(2*c^4*d*e^7 + b*c^3*e^8)*x^7 + 420*(14*c^4*d^2*e^6 + 7*b*c^3*d*e^
7 + (3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 630*(14*c^4*d^3*e^5 + 7*b*c^3*d^2*e^6 + (3*b^2*c^2 + 2*a*c^3)*d*e^7 + (b^
3*c + 3*a*b*c^2)*e^8)*x^5 + 126*(70*c^4*d^4*e^4 + 35*b*c^3*d^3*e^5 + 5*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 5*(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 + 84*(70*c^4*d^5*e^3 + 35*b*c^3*d^4*e^4 + 5*(3*
b^2*c^2 + 2*a*c^3)*d^3*e^5 + 5*(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 + 5*(a*b^3 +
3*a^2*b*c)*e^8)*x^3 + 36*(70*c^4*d^6*e^2 + 35*b*c^3*d^5*e^3 + 5*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c)*d*e^7 + 5*(3*a^2*b^2 + 2*a^3
*c)*e^8)*x^2 + 9*(70*c^4*d^7*e + 35*b*c^3*d^6*e^2 + 35*a^3*b*e^8 + 5*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 5*(3*a^2*b^2 +
2*a^3*c)*d*e^7)*x)/(e^18*x^9 + 9*d*e^17*x^8 + 36*d^2*e^16*x^7 + 84*d^3*e^15*x^6 + 126*d^4*e^14*x^5 + 126*d^5*
e^13*x^4 + 84*d^6*e^12*x^3 + 36*d^7*e^11*x^2 + 9*d^8*e^10*x + d^9*e^9)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1014 vs. \(2 (424) = 848\).

Time = 0.26 (sec) , antiderivative size = 1014, normalized size of antiderivative = 2.33 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=-\frac {630 \, c^{4} e^{8} x^{8} + 2520 \, c^{4} d e^{7} x^{7} + 1260 \, b c^{3} e^{8} x^{7} + 5880 \, c^{4} d^{2} e^{6} x^{6} + 2940 \, b c^{3} d e^{7} x^{6} + 1260 \, b^{2} c^{2} e^{8} x^{6} + 840 \, a c^{3} e^{8} x^{6} + 8820 \, c^{4} d^{3} e^{5} x^{5} + 4410 \, b c^{3} d^{2} e^{6} x^{5} + 1890 \, b^{2} c^{2} d e^{7} x^{5} + 1260 \, a c^{3} d e^{7} x^{5} + 630 \, b^{3} c e^{8} x^{5} + 1890 \, a b c^{2} e^{8} x^{5} + 8820 \, c^{4} d^{4} e^{4} x^{4} + 4410 \, b c^{3} d^{3} e^{5} x^{4} + 1890 \, b^{2} c^{2} d^{2} e^{6} x^{4} + 1260 \, a c^{3} d^{2} e^{6} x^{4} + 630 \, b^{3} c d e^{7} x^{4} + 1890 \, a b c^{2} d e^{7} x^{4} + 126 \, b^{4} e^{8} x^{4} + 1512 \, a b^{2} c e^{8} x^{4} + 756 \, a^{2} c^{2} e^{8} x^{4} + 5880 \, c^{4} d^{5} e^{3} x^{3} + 2940 \, b c^{3} d^{4} e^{4} x^{3} + 1260 \, b^{2} c^{2} d^{3} e^{5} x^{3} + 840 \, a c^{3} d^{3} e^{5} x^{3} + 420 \, b^{3} c d^{2} e^{6} x^{3} + 1260 \, a b c^{2} d^{2} e^{6} x^{3} + 84 \, b^{4} d e^{7} x^{3} + 1008 \, a b^{2} c d e^{7} x^{3} + 504 \, a^{2} c^{2} d e^{7} x^{3} + 420 \, a b^{3} e^{8} x^{3} + 1260 \, a^{2} b c e^{8} x^{3} + 2520 \, c^{4} d^{6} e^{2} x^{2} + 1260 \, b c^{3} d^{5} e^{3} x^{2} + 540 \, b^{2} c^{2} d^{4} e^{4} x^{2} + 360 \, a c^{3} d^{4} e^{4} x^{2} + 180 \, b^{3} c d^{3} e^{5} x^{2} + 540 \, a b c^{2} d^{3} e^{5} x^{2} + 36 \, b^{4} d^{2} e^{6} x^{2} + 432 \, a b^{2} c d^{2} e^{6} x^{2} + 216 \, a^{2} c^{2} d^{2} e^{6} x^{2} + 180 \, a b^{3} d e^{7} x^{2} + 540 \, a^{2} b c d e^{7} x^{2} + 540 \, a^{2} b^{2} e^{8} x^{2} + 360 \, a^{3} c e^{8} x^{2} + 630 \, c^{4} d^{7} e x + 315 \, b c^{3} d^{6} e^{2} x + 135 \, b^{2} c^{2} d^{5} e^{3} x + 90 \, a c^{3} d^{5} e^{3} x + 45 \, b^{3} c d^{4} e^{4} x + 135 \, a b c^{2} d^{4} e^{4} x + 9 \, b^{4} d^{3} e^{5} x + 108 \, a b^{2} c d^{3} e^{5} x + 54 \, a^{2} c^{2} d^{3} e^{5} x + 45 \, a b^{3} d^{2} e^{6} x + 135 \, a^{2} b c d^{2} e^{6} x + 135 \, a^{2} b^{2} d e^{7} x + 90 \, a^{3} c d e^{7} x + 315 \, a^{3} b e^{8} x + 70 \, c^{4} d^{8} + 35 \, b c^{3} d^{7} e + 15 \, b^{2} c^{2} d^{6} e^{2} + 10 \, a c^{3} d^{6} e^{2} + 5 \, b^{3} c d^{5} e^{3} + 15 \, 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} + 5 \, a b^{3} d^{3} e^{5} + 15 \, a^{2} b c d^{3} e^{5} + 15 \, a^{2} b^{2} d^{2} e^{6} + 10 \, a^{3} c d^{2} e^{6} + 35 \, a^{3} b d e^{7} + 70 \, a^{4} e^{8}}{630 \, {\left (e x + d\right )}^{9} e^{9}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/630*(630*c^4*e^8*x^8 + 2520*c^4*d*e^7*x^7 + 1260*b*c^3*e^8*x^7 + 5880*c^4*d^2*e^6*x^6 + 2940*b*c^3*d*e^7*x^
6 + 1260*b^2*c^2*e^8*x^6 + 840*a*c^3*e^8*x^6 + 8820*c^4*d^3*e^5*x^5 + 4410*b*c^3*d^2*e^6*x^5 + 1890*b^2*c^2*d*
e^7*x^5 + 1260*a*c^3*d*e^7*x^5 + 630*b^3*c*e^8*x^5 + 1890*a*b*c^2*e^8*x^5 + 8820*c^4*d^4*e^4*x^4 + 4410*b*c^3*
d^3*e^5*x^4 + 1890*b^2*c^2*d^2*e^6*x^4 + 1260*a*c^3*d^2*e^6*x^4 + 630*b^3*c*d*e^7*x^4 + 1890*a*b*c^2*d*e^7*x^4
+ 126*b^4*e^8*x^4 + 1512*a*b^2*c*e^8*x^4 + 756*a^2*c^2*e^8*x^4 + 5880*c^4*d^5*e^3*x^3 + 2940*b*c^3*d^4*e^4*x^
3 + 1260*b^2*c^2*d^3*e^5*x^3 + 840*a*c^3*d^3*e^5*x^3 + 420*b^3*c*d^2*e^6*x^3 + 1260*a*b*c^2*d^2*e^6*x^3 + 84*b
^4*d*e^7*x^3 + 1008*a*b^2*c*d*e^7*x^3 + 504*a^2*c^2*d*e^7*x^3 + 420*a*b^3*e^8*x^3 + 1260*a^2*b*c*e^8*x^3 + 252
0*c^4*d^6*e^2*x^2 + 1260*b*c^3*d^5*e^3*x^2 + 540*b^2*c^2*d^4*e^4*x^2 + 360*a*c^3*d^4*e^4*x^2 + 180*b^3*c*d^3*e
^5*x^2 + 540*a*b*c^2*d^3*e^5*x^2 + 36*b^4*d^2*e^6*x^2 + 432*a*b^2*c*d^2*e^6*x^2 + 216*a^2*c^2*d^2*e^6*x^2 + 18
0*a*b^3*d*e^7*x^2 + 540*a^2*b*c*d*e^7*x^2 + 540*a^2*b^2*e^8*x^2 + 360*a^3*c*e^8*x^2 + 630*c^4*d^7*e*x + 315*b*
c^3*d^6*e^2*x + 135*b^2*c^2*d^5*e^3*x + 90*a*c^3*d^5*e^3*x + 45*b^3*c*d^4*e^4*x + 135*a*b*c^2*d^4*e^4*x + 9*b^
4*d^3*e^5*x + 108*a*b^2*c*d^3*e^5*x + 54*a^2*c^2*d^3*e^5*x + 45*a*b^3*d^2*e^6*x + 135*a^2*b*c*d^2*e^6*x + 135*
a^2*b^2*d*e^7*x + 90*a^3*c*d*e^7*x + 315*a^3*b*e^8*x + 70*c^4*d^8 + 35*b*c^3*d^7*e + 15*b^2*c^2*d^6*e^2 + 10*a
*c^3*d^6*e^2 + 5*b^3*c*d^5*e^3 + 15*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 + 5
*a*b^3*d^3*e^5 + 15*a^2*b*c*d^3*e^5 + 15*a^2*b^2*d^2*e^6 + 10*a^3*c*d^2*e^6 + 35*a^3*b*d*e^7 + 70*a^4*e^8)/((e
*x + d)^9*e^9)

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 966, normalized size of antiderivative = 2.22 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx=-\frac {\frac {70\,a^4\,e^8+35\,a^3\,b\,d\,e^7+10\,a^3\,c\,d^2\,e^6+15\,a^2\,b^2\,d^2\,e^6+15\,a^2\,b\,c\,d^3\,e^5+6\,a^2\,c^2\,d^4\,e^4+5\,a\,b^3\,d^3\,e^5+12\,a\,b^2\,c\,d^4\,e^4+15\,a\,b\,c^2\,d^5\,e^3+10\,a\,c^3\,d^6\,e^2+b^4\,d^4\,e^4+5\,b^3\,c\,d^5\,e^3+15\,b^2\,c^2\,d^6\,e^2+35\,b\,c^3\,d^7\,e+70\,c^4\,d^8}{630\,e^9}+\frac {2\,x^3\,\left (15\,a^2\,b\,c\,e^5+6\,a^2\,c^2\,d\,e^4+5\,a\,b^3\,e^5+12\,a\,b^2\,c\,d\,e^4+15\,a\,b\,c^2\,d^2\,e^3+10\,a\,c^3\,d^3\,e^2+b^4\,d\,e^4+5\,b^3\,c\,d^2\,e^3+15\,b^2\,c^2\,d^3\,e^2+35\,b\,c^3\,d^4\,e+70\,c^4\,d^5\right )}{15\,e^6}+\frac {x^4\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4+15\,a\,b\,c^2\,d\,e^3+10\,a\,c^3\,d^2\,e^2+b^4\,e^4+5\,b^3\,c\,d\,e^3+15\,b^2\,c^2\,d^2\,e^2+35\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{5\,e^5}+\frac {x\,\left (35\,a^3\,b\,e^7+10\,a^3\,c\,d\,e^6+15\,a^2\,b^2\,d\,e^6+15\,a^2\,b\,c\,d^2\,e^5+6\,a^2\,c^2\,d^3\,e^4+5\,a\,b^3\,d^2\,e^5+12\,a\,b^2\,c\,d^3\,e^4+15\,a\,b\,c^2\,d^4\,e^3+10\,a\,c^3\,d^5\,e^2+b^4\,d^3\,e^4+5\,b^3\,c\,d^4\,e^3+15\,b^2\,c^2\,d^5\,e^2+35\,b\,c^3\,d^6\,e+70\,c^4\,d^7\right )}{70\,e^8}+\frac {c^4\,x^8}{e}+\frac {2\,x^2\,\left (10\,a^3\,c\,e^6+15\,a^2\,b^2\,e^6+15\,a^2\,b\,c\,d\,e^5+6\,a^2\,c^2\,d^2\,e^4+5\,a\,b^3\,d\,e^5+12\,a\,b^2\,c\,d^2\,e^4+15\,a\,b\,c^2\,d^3\,e^3+10\,a\,c^3\,d^4\,e^2+b^4\,d^2\,e^4+5\,b^3\,c\,d^3\,e^3+15\,b^2\,c^2\,d^4\,e^2+35\,b\,c^3\,d^5\,e+70\,c^4\,d^6\right )}{35\,e^7}+\frac {2\,c^3\,x^7\,\left (b\,e+2\,c\,d\right )}{e^2}+\frac {2\,c^2\,x^6\,\left (3\,b^2\,e^2+7\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{3\,e^3}+\frac {c\,x^5\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+7\,b\,c^2\,d^2\,e+3\,a\,b\,c\,e^3+14\,c^3\,d^3+2\,a\,c^2\,d\,e^2\right )}{e^4}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \]

[In]

int((a + b*x + c*x^2)^4/(d + e*x)^10,x)

[Out]

-((70*a^4*e^8 + 70*c^4*d^8 + b^4*d^4*e^4 + 5*a*b^3*d^3*e^5 + 10*a*c^3*d^6*e^2 + 10*a^3*c*d^2*e^6 + 5*b^3*c*d^5
*e^3 + 15*a^2*b^2*d^2*e^6 + 6*a^2*c^2*d^4*e^4 + 15*b^2*c^2*d^6*e^2 + 35*a^3*b*d*e^7 + 35*b*c^3*d^7*e + 15*a*b*
c^2*d^5*e^3 + 12*a*b^2*c*d^4*e^4 + 15*a^2*b*c*d^3*e^5)/(630*e^9) + (2*x^3*(70*c^4*d^5 + 5*a*b^3*e^5 + b^4*d*e^
4 + 10*a*c^3*d^3*e^2 + 6*a^2*c^2*d*e^4 + 5*b^3*c*d^2*e^3 + 15*b^2*c^2*d^3*e^2 + 15*a^2*b*c*e^5 + 35*b*c^3*d^4*
e + 12*a*b^2*c*d*e^4 + 15*a*b*c^2*d^2*e^3))/(15*e^6) + (x^4*(b^4*e^4 + 70*c^4*d^4 + 6*a^2*c^2*e^4 + 10*a*c^3*d
^2*e^2 + 15*b^2*c^2*d^2*e^2 + 12*a*b^2*c*e^4 + 35*b*c^3*d^3*e + 5*b^3*c*d*e^3 + 15*a*b*c^2*d*e^3))/(5*e^5) + (
x*(70*c^4*d^7 + 35*a^3*b*e^7 + b^4*d^3*e^4 + 5*a*b^3*d^2*e^5 + 15*a^2*b^2*d*e^6 + 10*a*c^3*d^5*e^2 + 5*b^3*c*d
^4*e^3 + 6*a^2*c^2*d^3*e^4 + 15*b^2*c^2*d^5*e^2 + 10*a^3*c*d*e^6 + 35*b*c^3*d^6*e + 15*a*b*c^2*d^4*e^3 + 12*a*
b^2*c*d^3*e^4 + 15*a^2*b*c*d^2*e^5))/(70*e^8) + (c^4*x^8)/e + (2*x^2*(70*c^4*d^6 + 10*a^3*c*e^6 + 15*a^2*b^2*e
^6 + b^4*d^2*e^4 + 10*a*c^3*d^4*e^2 + 5*b^3*c*d^3*e^3 + 6*a^2*c^2*d^2*e^4 + 15*b^2*c^2*d^4*e^2 + 5*a*b^3*d*e^5
+ 35*b*c^3*d^5*e + 15*a^2*b*c*d*e^5 + 15*a*b*c^2*d^3*e^3 + 12*a*b^2*c*d^2*e^4))/(35*e^7) + (2*c^3*x^7*(b*e +
2*c*d))/e^2 + (2*c^2*x^6*(3*b^2*e^2 + 14*c^2*d^2 + 2*a*c*e^2 + 7*b*c*d*e))/(3*e^3) + (c*x^5*(b^3*e^3 + 14*c^3*
d^3 + 3*a*b*c*e^3 + 2*a*c^2*d*e^2 + 7*b*c^2*d^2*e + 3*b^2*c*d*e^2))/e^4)/(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)