3.2161 \(\int \frac{(a+b x+c x^2)^4}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=443 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{6 e^9 (d+e x)^6}-\frac{c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9 (d+e x)^4}+\frac{4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^9 (d+e x)^7}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^9 (d+e x)^8}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{9 e^9 (d+e x)^9}-\frac{\left (a e^2-b d e+c d^2\right )^4}{10 e^9 (d+e x)^{10}}+\frac{4 c^3 (2 c d-b e)}{3 e^9 (d+e x)^3}-\frac{c^4}{2 e^9 (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.40089, antiderivative size = 443, 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} \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{6 e^9 (d+e x)^6}-\frac{c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9 (d+e x)^4}+\frac{4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^9 (d+e x)^7}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^9 (d+e x)^8}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{9 e^9 (d+e x)^9}-\frac{\left (a e^2-b d e+c d^2\right )^4}{10 e^9 (d+e x)^{10}}+\frac{4 c^3 (2 c d-b e)}{3 e^9 (d+e x)^3}-\frac{c^4}{2 e^9 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^11,x]

[Out]

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

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

Mathematica [A]  time = 0.296293, size = 731, normalized size = 1.65 \[ -\frac{3 c^2 e^2 \left (2 a^2 e^2 \left (45 d^2 e^2 x^2+10 d^3 e x+d^4+120 d e^3 x^3+210 e^4 x^4\right )+4 a b e \left (45 d^3 e^2 x^2+120 d^2 e^3 x^3+10 d^4 e x+d^5+210 d e^4 x^4+252 e^5 x^5\right )+3 b^2 \left (45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+10 d^5 e x+d^6+252 d e^5 x^5+210 e^6 x^6\right )\right )+2 c e^3 \left (9 a^2 b e^2 \left (10 d^2 e x+d^3+45 d e^2 x^2+120 e^3 x^3\right )+7 a^3 e^3 \left (d^2+10 d e x+45 e^2 x^2\right )+6 a b^2 e \left (45 d^2 e^2 x^2+10 d^3 e x+d^4+120 d e^3 x^3+210 e^4 x^4\right )+2 b^3 \left (45 d^3 e^2 x^2+120 d^2 e^3 x^3+10 d^4 e x+d^5+210 d e^4 x^4+252 e^5 x^5\right )\right )+e^4 \left (21 a^2 b^2 e^2 \left (d^2+10 d e x+45 e^2 x^2\right )+56 a^3 b e^3 (d+10 e x)+126 a^4 e^4+6 a b^3 e \left (10 d^2 e x+d^3+45 d e^2 x^2+120 e^3 x^3\right )+b^4 \left (45 d^2 e^2 x^2+10 d^3 e x+d^4+120 d e^3 x^3+210 e^4 x^4\right )\right )+2 c^3 e \left (3 a e \left (45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+10 d^5 e x+d^6+252 d e^5 x^5+210 e^6 x^6\right )+7 b \left (45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+10 d^6 e x+d^7+210 d e^6 x^6+120 e^7 x^7\right )\right )+14 c^4 \left (45 d^6 e^2 x^2+120 d^5 e^3 x^3+210 d^4 e^4 x^4+252 d^3 e^5 x^5+210 d^2 e^6 x^6+10 d^7 e x+d^8+120 d e^7 x^7+45 e^8 x^8\right )}{1260 e^9 (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^11,x]

[Out]

-(14*c^4*(d^8 + 10*d^7*e*x + 45*d^6*e^2*x^2 + 120*d^5*e^3*x^3 + 210*d^4*e^4*x^4 + 252*d^3*e^5*x^5 + 210*d^2*e^
6*x^6 + 120*d*e^7*x^7 + 45*e^8*x^8) + e^4*(126*a^4*e^4 + 56*a^3*b*e^3*(d + 10*e*x) + 21*a^2*b^2*e^2*(d^2 + 10*
d*e*x + 45*e^2*x^2) + 6*a*b^3*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + b^4*(d^4 + 10*d^3*e*x + 45*d
^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4)) + 2*c*e^3*(7*a^3*e^3*(d^2 + 10*d*e*x + 45*e^2*x^2) + 9*a^2*b*e^2*(d
^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 6*a*b^2*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 +
210*e^4*x^4) + 2*b^3*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5)) + 3
*c^2*e^2*(2*a^2*e^2*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + 4*a*b*e*(d^5 + 10*d^4*
e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + 3*b^2*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x
^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6)) + 2*c^3*e*(3*a*e*(d^6 + 10*d^5*e*x + 45
*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6) + 7*b*(d^7 + 10*d^6*e*x + 45*d
^5*e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^3*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6*x^6 + 120*e^7*x^7)))/(1260*e^9*(d
+ e*x)^10)

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Maple [B]  time = 0.048, 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)^11,x)

[Out]

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

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Maxima [B]  time = 1.18858, size = 1223, normalized size = 2.76 \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)^11,x, algorithm="maxima")

[Out]

-1/1260*(630*c^4*e^8*x^8 + 14*c^4*d^8 + 14*b*c^3*d^7*e + 56*a^3*b*d*e^7 + 126*a^4*e^8 + 3*(3*b^2*c^2 + 2*a*c^3
)*d^6*e^2 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*d^3
*e^5 + 7*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 1680*(c^4*d*e^7 + b*c^3*e^8)*x^7 + 210*(14*c^4*d^2*e^6 + 14*b*c^3*d*e
^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 252*(14*c^4*d^3*e^5 + 14*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7
+ 4*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 210*(14*c^4*d^4*e^4 + 14*b*c^3*d^3*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 +
4*(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 + 120*(14*c^4*d^5*e^3 + 14*b*c^3*d^4*e^
4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 4*(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 + 6
*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 45*(14*c^4*d^6*e^2 + 14*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*d*e^7 + 7*(3*a^2*b^
2 + 2*a^3*c)*e^8)*x^2 + 10*(14*c^4*d^7*e + 14*b*c^3*d^6*e^2 + 56*a^3*b*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 +
4*(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 + 6*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 7*(3
*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^19*x^10 + 10*d*e^18*x^9 + 45*d^2*e^17*x^8 + 120*d^3*e^16*x^7 + 210*d^4*e^15*x
^6 + 252*d^5*e^14*x^5 + 210*d^6*e^13*x^4 + 120*d^7*e^12*x^3 + 45*d^8*e^11*x^2 + 10*d^9*e^10*x + d^10*e^9)

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Fricas [B]  time = 1.83915, size = 1926, normalized size = 4.35 \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)^11,x, algorithm="fricas")

[Out]

-1/1260*(630*c^4*e^8*x^8 + 14*c^4*d^8 + 14*b*c^3*d^7*e + 56*a^3*b*d*e^7 + 126*a^4*e^8 + 3*(3*b^2*c^2 + 2*a*c^3
)*d^6*e^2 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*d^3
*e^5 + 7*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 1680*(c^4*d*e^7 + b*c^3*e^8)*x^7 + 210*(14*c^4*d^2*e^6 + 14*b*c^3*d*e
^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 252*(14*c^4*d^3*e^5 + 14*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7
+ 4*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 210*(14*c^4*d^4*e^4 + 14*b*c^3*d^3*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 +
4*(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 + 120*(14*c^4*d^5*e^3 + 14*b*c^3*d^4*e^
4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 4*(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 + 6
*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 45*(14*c^4*d^6*e^2 + 14*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*d*e^7 + 7*(3*a^2*b^
2 + 2*a^3*c)*e^8)*x^2 + 10*(14*c^4*d^7*e + 14*b*c^3*d^6*e^2 + 56*a^3*b*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 +
4*(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 + 6*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 7*(3
*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^19*x^10 + 10*d*e^18*x^9 + 45*d^2*e^17*x^8 + 120*d^3*e^16*x^7 + 210*d^4*e^15*x
^6 + 252*d^5*e^14*x^5 + 210*d^6*e^13*x^4 + 120*d^7*e^12*x^3 + 45*d^8*e^11*x^2 + 10*d^9*e^10*x + d^10*e^9)

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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)**11,x)

[Out]

Timed out

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Giac [B]  time = 1.1216, size = 1274, normalized size = 2.88 \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)^11,x, algorithm="giac")

[Out]

-1/1260*(630*c^4*x^8*e^8 + 1680*c^4*d*x^7*e^7 + 2940*c^4*d^2*x^6*e^6 + 3528*c^4*d^3*x^5*e^5 + 2940*c^4*d^4*x^4
*e^4 + 1680*c^4*d^5*x^3*e^3 + 630*c^4*d^6*x^2*e^2 + 140*c^4*d^7*x*e + 14*c^4*d^8 + 1680*b*c^3*x^7*e^8 + 2940*b
*c^3*d*x^6*e^7 + 3528*b*c^3*d^2*x^5*e^6 + 2940*b*c^3*d^3*x^4*e^5 + 1680*b*c^3*d^4*x^3*e^4 + 630*b*c^3*d^5*x^2*
e^3 + 140*b*c^3*d^6*x*e^2 + 14*b*c^3*d^7*e + 1890*b^2*c^2*x^6*e^8 + 1260*a*c^3*x^6*e^8 + 2268*b^2*c^2*d*x^5*e^
7 + 1512*a*c^3*d*x^5*e^7 + 1890*b^2*c^2*d^2*x^4*e^6 + 1260*a*c^3*d^2*x^4*e^6 + 1080*b^2*c^2*d^3*x^3*e^5 + 720*
a*c^3*d^3*x^3*e^5 + 405*b^2*c^2*d^4*x^2*e^4 + 270*a*c^3*d^4*x^2*e^4 + 90*b^2*c^2*d^5*x*e^3 + 60*a*c^3*d^5*x*e^
3 + 9*b^2*c^2*d^6*e^2 + 6*a*c^3*d^6*e^2 + 1008*b^3*c*x^5*e^8 + 3024*a*b*c^2*x^5*e^8 + 840*b^3*c*d*x^4*e^7 + 25
20*a*b*c^2*d*x^4*e^7 + 480*b^3*c*d^2*x^3*e^6 + 1440*a*b*c^2*d^2*x^3*e^6 + 180*b^3*c*d^3*x^2*e^5 + 540*a*b*c^2*
d^3*x^2*e^5 + 40*b^3*c*d^4*x*e^4 + 120*a*b*c^2*d^4*x*e^4 + 4*b^3*c*d^5*e^3 + 12*a*b*c^2*d^5*e^3 + 210*b^4*x^4*
e^8 + 2520*a*b^2*c*x^4*e^8 + 1260*a^2*c^2*x^4*e^8 + 120*b^4*d*x^3*e^7 + 1440*a*b^2*c*d*x^3*e^7 + 720*a^2*c^2*d
*x^3*e^7 + 45*b^4*d^2*x^2*e^6 + 540*a*b^2*c*d^2*x^2*e^6 + 270*a^2*c^2*d^2*x^2*e^6 + 10*b^4*d^3*x*e^5 + 120*a*b
^2*c*d^3*x*e^5 + 60*a^2*c^2*d^3*x*e^5 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 + 720*a*b^3*x^3*e
^8 + 2160*a^2*b*c*x^3*e^8 + 270*a*b^3*d*x^2*e^7 + 810*a^2*b*c*d*x^2*e^7 + 60*a*b^3*d^2*x*e^6 + 180*a^2*b*c*d^2
*x*e^6 + 6*a*b^3*d^3*e^5 + 18*a^2*b*c*d^3*e^5 + 945*a^2*b^2*x^2*e^8 + 630*a^3*c*x^2*e^8 + 210*a^2*b^2*d*x*e^7
+ 140*a^3*c*d*x*e^7 + 21*a^2*b^2*d^2*e^6 + 14*a^3*c*d^2*e^6 + 560*a^3*b*x*e^8 + 56*a^3*b*d*e^7 + 126*a^4*e^8)*
e^(-9)/(x*e + d)^10