3.2162 \(\int \frac{(a+b x+c x^2)^4}{(d+e x)^{12}} \, dx\)

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

[Out]

-(c*d^2 - b*d*e + a*e^2)^4/(11*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)

________________________________________________________________________________________

Rubi [A]  time = 0.396922, antiderivative size = 440, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{7 e^9 (d+e x)^7}-\frac{2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac{2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^6}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9 (d+e x)^8}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^9 (d+e x)^9}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9 (d+e x)^{10}}-\frac{\left (a e^2-b d e+c d^2\right )^4}{11 e^9 (d+e x)^{11}}+\frac{c^3 (2 c d-b e)}{e^9 (d+e x)^4}-\frac{c^4}{3 e^9 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*d^2 - b*d*e + a*e^2)^4/(11*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 698

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*} \int \frac{\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx &=\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]  time = 0.324083, size = 731, normalized size = 1.66 \[ -\frac{6 c^2 e^2 \left (3 a^2 e^2 \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )+5 a b e \left (55 d^3 e^2 x^2+165 d^2 e^3 x^3+11 d^4 e x+d^5+330 d e^4 x^4+462 e^5 x^5\right )+3 b^2 \left (55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+11 d^5 e x+d^6+462 d e^5 x^5+462 e^6 x^6\right )\right )+c e^3 \left (63 a^2 b e^2 \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )+56 a^3 e^3 \left (d^2+11 d e x+55 e^2 x^2\right )+36 a b^2 e \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )+10 b^3 \left (55 d^3 e^2 x^2+165 d^2 e^3 x^3+11 d^4 e x+d^5+330 d e^4 x^4+462 e^5 x^5\right )\right )+3 e^4 \left (28 a^2 b^2 e^2 \left (d^2+11 d e x+55 e^2 x^2\right )+84 a^3 b e^3 (d+11 e x)+210 a^4 e^4+7 a b^3 e \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )+b^4 \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )\right )+3 c^3 e \left (4 a e \left (55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+11 d^5 e x+d^6+462 d e^5 x^5+462 e^6 x^6\right )+7 b \left (55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+11 d^6 e x+d^7+462 d e^6 x^6+330 e^7 x^7\right )\right )+14 c^4 \left (55 d^6 e^2 x^2+165 d^5 e^3 x^3+330 d^4 e^4 x^4+462 d^3 e^5 x^5+462 d^2 e^6 x^6+11 d^7 e x+d^8+330 d e^7 x^7+165 e^8 x^8\right )}{6930 e^9 (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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 + 5
5*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)))/(6930*e
^9*(d + e*x)^11)

________________________________________________________________________________________

Maple [B]  time = 0.05, size = 914, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(6*a^2*c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*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)/e^9/(e*x+d)^7-c^3*(b*e-2*c*d)/e^9/(e*x+d)^4-1/10*(4*a^3*b*e^7-8*a^3*c*d*e^6-12*a^
2*b^2*d*e^6+36*a^2*b*c*d^2*e^5-24*a^2*c^2*d^3*e^4+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*a*
c^3*d^5*e^2-4*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*c^4*d^7)/e^9/(e*x+d)^10-1/3*c^4
/e^9/(e*x+d)^3-1/8*(12*a^2*b*c*e^5-24*a^2*c^2*d*e^4+4*a*b^3*e^5-48*a*b^2*c*d*e^4+120*a*b*c^2*d^2*e^3-80*a*c^3*
d^3*e^2-4*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)/e^9/(e*x+d)^8-1/9*(4*a^3*
c*e^6+6*a^2*b^2*e^6-36*a^2*b*c*d*e^5+36*a^2*c^2*d^2*e^4-12*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*a*c^3*d^4*e^2+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*c^4*d^6)/e^9/(e*x+d)^9-2/
3*c*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/e^9/(e*x+d)^6-2/5*c^2*(2*a*c*e
^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^2)/e^9/(e*x+d)^5-1/11*(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)/e^9/(e*x+d)^11

________________________________________________________________________________________

Maxima [B]  time = 1.19663, size = 1247, normalized size = 2.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [B]  time = 1.77729, size = 1995, normalized size = 4.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.11203, size = 1276, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6930*(2310*c^4*x^8*e^8 + 4620*c^4*d*x^7*e^7 + 6468*c^4*d^2*x^6*e^6 + 6468*c^4*d^3*x^5*e^5 + 4620*c^4*d^4*x^
4*e^4 + 2310*c^4*d^5*x^3*e^3 + 770*c^4*d^6*x^2*e^2 + 154*c^4*d^7*x*e + 14*c^4*d^8 + 6930*b*c^3*x^7*e^8 + 9702*
b*c^3*d*x^6*e^7 + 9702*b*c^3*d^2*x^5*e^6 + 6930*b*c^3*d^3*x^4*e^5 + 3465*b*c^3*d^4*x^3*e^4 + 1155*b*c^3*d^5*x^
2*e^3 + 231*b*c^3*d^6*x*e^2 + 21*b*c^3*d^7*e + 8316*b^2*c^2*x^6*e^8 + 5544*a*c^3*x^6*e^8 + 8316*b^2*c^2*d*x^5*
e^7 + 5544*a*c^3*d*x^5*e^7 + 5940*b^2*c^2*d^2*x^4*e^6 + 3960*a*c^3*d^2*x^4*e^6 + 2970*b^2*c^2*d^3*x^3*e^5 + 19
80*a*c^3*d^3*x^3*e^5 + 990*b^2*c^2*d^4*x^2*e^4 + 660*a*c^3*d^4*x^2*e^4 + 198*b^2*c^2*d^5*x*e^3 + 132*a*c^3*d^5
*x*e^3 + 18*b^2*c^2*d^6*e^2 + 12*a*c^3*d^6*e^2 + 4620*b^3*c*x^5*e^8 + 13860*a*b*c^2*x^5*e^8 + 3300*b^3*c*d*x^4
*e^7 + 9900*a*b*c^2*d*x^4*e^7 + 1650*b^3*c*d^2*x^3*e^6 + 4950*a*b*c^2*d^2*x^3*e^6 + 550*b^3*c*d^3*x^2*e^5 + 16
50*a*b*c^2*d^3*x^2*e^5 + 110*b^3*c*d^4*x*e^4 + 330*a*b*c^2*d^4*x*e^4 + 10*b^3*c*d^5*e^3 + 30*a*b*c^2*d^5*e^3 +
990*b^4*x^4*e^8 + 11880*a*b^2*c*x^4*e^8 + 5940*a^2*c^2*x^4*e^8 + 495*b^4*d*x^3*e^7 + 5940*a*b^2*c*d*x^3*e^7 +
2970*a^2*c^2*d*x^3*e^7 + 165*b^4*d^2*x^2*e^6 + 1980*a*b^2*c*d^2*x^2*e^6 + 990*a^2*c^2*d^2*x^2*e^6 + 33*b^4*d^
3*x*e^5 + 396*a*b^2*c*d^3*x*e^5 + 198*a^2*c^2*d^3*x*e^5 + 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 + 3465*a*b^3*x^3*e^8 + 10395*a^2*b*c*x^3*e^8 + 1155*a*b^3*d*x^2*e^7 + 3465*a^2*b*c*d*x^2*e^7 + 231*a*b^3*d
^2*x*e^6 + 693*a^2*b*c*d^2*x*e^6 + 21*a*b^3*d^3*e^5 + 63*a^2*b*c*d^3*e^5 + 4620*a^2*b^2*x^2*e^8 + 3080*a^3*c*x
^2*e^8 + 924*a^2*b^2*d*x*e^7 + 616*a^3*c*d*x*e^7 + 84*a^2*b^2*d^2*e^6 + 56*a^3*c*d^2*e^6 + 2772*a^3*b*x*e^8 +
252*a^3*b*d*e^7 + 630*a^4*e^8)*e^(-9)/(x*e + d)^11