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

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

[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)

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Rubi [A]
time = 0.28, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \begin {gather*} -\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)} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx &=\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*}

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Mathematica [A]
time = 0.21, size = 730, normalized size = 1.67 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(913\) vs. \(2(425)=850\).
time = 0.85, size = 914, normalized size = 2.10 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 897 vs. \(2 (432) = 864\).
time = 0.32, size = 897, normalized size = 2.06 \begin {gather*} -\frac {630 \, c^{4} x^{8} e^{8} + 70 \, c^{4} d^{8} + 35 \, b c^{3} d^{7} e + 1260 \, {\left (2 \, c^{4} d e^{7} + b c^{3} e^{8}\right )} x^{7} + 5 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{6} + 420 \, {\left (14 \, c^{4} d^{2} e^{6} + 7 \, b c^{3} d e^{7} + 3 \, b^{2} c^{2} e^{8} + 2 \, a c^{3} e^{8}\right )} x^{6} + 5 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{5} + 630 \, {\left (14 \, c^{4} d^{3} e^{5} + 7 \, b c^{3} d^{2} e^{6} + b^{3} c e^{8} + 3 \, a b c^{2} e^{8} + {\left (3 \, b^{2} c^{2} e^{7} + 2 \, a c^{3} e^{7}\right )} d\right )} x^{5} + 35 \, a^{3} b d e^{7} + {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{4} + 126 \, {\left (70 \, c^{4} d^{4} e^{4} + 35 \, b c^{3} d^{3} e^{5} + b^{4} e^{8} + 12 \, a b^{2} c e^{8} + 6 \, a^{2} c^{2} e^{8} + 5 \, {\left (3 \, b^{2} c^{2} e^{6} + 2 \, a c^{3} e^{6}\right )} d^{2} + 5 \, {\left (b^{3} c e^{7} + 3 \, a b c^{2} e^{7}\right )} d\right )} x^{4} + 70 \, a^{4} e^{8} + 5 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d^{3} + 84 \, {\left (70 \, c^{4} d^{5} e^{3} + 35 \, b c^{3} d^{4} e^{4} + 5 \, a b^{3} e^{8} + 15 \, a^{2} b c e^{8} + 5 \, {\left (3 \, b^{2} c^{2} e^{5} + 2 \, a c^{3} e^{5}\right )} d^{3} + 5 \, {\left (b^{3} c e^{6} + 3 \, a b c^{2} e^{6}\right )} d^{2} + {\left (b^{4} e^{7} + 12 \, a b^{2} c e^{7} + 6 \, a^{2} c^{2} e^{7}\right )} d\right )} x^{3} + 5 \, {\left (3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} d^{2} + 36 \, {\left (70 \, c^{4} d^{6} e^{2} + 35 \, b c^{3} d^{5} e^{3} + 5 \, {\left (3 \, b^{2} c^{2} e^{4} + 2 \, a c^{3} e^{4}\right )} d^{4} + 15 \, a^{2} b^{2} e^{8} + 10 \, a^{3} c e^{8} + 5 \, {\left (b^{3} c e^{5} + 3 \, a b c^{2} e^{5}\right )} d^{3} + {\left (b^{4} e^{6} + 12 \, a b^{2} c e^{6} + 6 \, a^{2} c^{2} e^{6}\right )} d^{2} + 5 \, {\left (a b^{3} e^{7} + 3 \, a^{2} b c e^{7}\right )} d\right )} x^{2} + 9 \, {\left (70 \, c^{4} d^{7} e + 35 \, b c^{3} d^{6} e^{2} + 5 \, {\left (3 \, b^{2} c^{2} e^{3} + 2 \, a c^{3} e^{3}\right )} d^{5} + 5 \, {\left (b^{3} c e^{4} + 3 \, a b c^{2} e^{4}\right )} d^{4} + 35 \, a^{3} b e^{8} + {\left (b^{4} e^{5} + 12 \, a b^{2} c e^{5} + 6 \, a^{2} c^{2} e^{5}\right )} d^{3} + 5 \, {\left (a b^{3} e^{6} + 3 \, a^{2} b c e^{6}\right )} d^{2} + 5 \, {\left (3 \, a^{2} b^{2} e^{7} + 2 \, a^{3} c e^{7}\right )} d\right )} x}{630 \, {\left (x^{9} e^{18} + 9 \, d x^{8} e^{17} + 36 \, d^{2} x^{7} e^{16} + 84 \, d^{3} x^{6} e^{15} + 126 \, d^{4} x^{5} e^{14} + 126 \, d^{5} x^{4} e^{13} + 84 \, d^{6} x^{3} e^{12} + 36 \, d^{7} x^{2} e^{11} + 9 \, d^{8} x e^{10} + d^{9} e^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.99, size = 844, normalized size = 1.94 \begin {gather*} -\frac {70 \, c^{4} d^{8} + {\left (630 \, c^{4} x^{8} + 1260 \, b c^{3} x^{7} + 420 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{6} + 630 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{5} + 315 \, a^{3} b x + 126 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{4} + 70 \, a^{4} + 420 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{3} + 180 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{2}\right )} e^{8} + {\left (2520 \, c^{4} d x^{7} + 2940 \, b c^{3} d x^{6} + 630 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{5} + 630 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{4} + 35 \, a^{3} b d + 84 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{3} + 180 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{2} + 45 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x\right )} e^{7} + {\left (5880 \, c^{4} d^{2} x^{6} + 4410 \, b c^{3} d^{2} x^{5} + 630 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{4} + 420 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{3} + 36 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{2} + 45 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x + 5 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} e^{6} + {\left (8820 \, c^{4} d^{3} x^{5} + 4410 \, b c^{3} d^{3} x^{4} + 420 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{3} + 180 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{2} + 9 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x + 5 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3}\right )} e^{5} + {\left (8820 \, c^{4} d^{4} x^{4} + 2940 \, b c^{3} d^{4} x^{3} + 180 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2} + 45 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} x + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4}\right )} e^{4} + 5 \, {\left (1176 \, c^{4} d^{5} x^{3} + 252 \, b c^{3} d^{5} x^{2} + 9 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x + {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5}\right )} e^{3} + 5 \, {\left (504 \, c^{4} d^{6} x^{2} + 63 \, b c^{3} d^{6} x + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6}\right )} e^{2} + 35 \, {\left (18 \, c^{4} d^{7} x + b c^{3} d^{7}\right )} e}{630 \, {\left (x^{9} e^{18} + 9 \, d x^{8} e^{17} + 36 \, d^{2} x^{7} e^{16} + 84 \, d^{3} x^{6} e^{15} + 126 \, d^{4} x^{5} e^{14} + 126 \, d^{5} x^{4} e^{13} + 84 \, d^{6} x^{3} e^{12} + 36 \, d^{7} x^{2} e^{11} + 9 \, d^{8} x e^{10} + d^{9} e^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/630*(70*c^4*d^8 + (630*c^4*x^8 + 1260*b*c^3*x^7 + 420*(3*b^2*c^2 + 2*a*c^3)*x^6 + 630*(b^3*c + 3*a*b*c^2)*x
^5 + 315*a^3*b*x + 126*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*x^4 + 70*a^4 + 420*(a*b^3 + 3*a^2*b*c)*x^3 + 180*(3*a^2*
b^2 + 2*a^3*c)*x^2)*e^8 + (2520*c^4*d*x^7 + 2940*b*c^3*d*x^6 + 630*(3*b^2*c^2 + 2*a*c^3)*d*x^5 + 630*(b^3*c +
3*a*b*c^2)*d*x^4 + 35*a^3*b*d + 84*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*x^3 + 180*(a*b^3 + 3*a^2*b*c)*d*x^2 + 45*(
3*a^2*b^2 + 2*a^3*c)*d*x)*e^7 + (5880*c^4*d^2*x^6 + 4410*b*c^3*d^2*x^5 + 630*(3*b^2*c^2 + 2*a*c^3)*d^2*x^4 + 4
20*(b^3*c + 3*a*b*c^2)*d^2*x^3 + 36*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*x^2 + 45*(a*b^3 + 3*a^2*b*c)*d^2*x + 5*
(3*a^2*b^2 + 2*a^3*c)*d^2)*e^6 + (8820*c^4*d^3*x^5 + 4410*b*c^3*d^3*x^4 + 420*(3*b^2*c^2 + 2*a*c^3)*d^3*x^3 +
180*(b^3*c + 3*a*b*c^2)*d^3*x^2 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*x + 5*(a*b^3 + 3*a^2*b*c)*d^3)*e^5 + (8
820*c^4*d^4*x^4 + 2940*b*c^3*d^4*x^3 + 180*(3*b^2*c^2 + 2*a*c^3)*d^4*x^2 + 45*(b^3*c + 3*a*b*c^2)*d^4*x + (b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d^4)*e^4 + 5*(1176*c^4*d^5*x^3 + 252*b*c^3*d^5*x^2 + 9*(3*b^2*c^2 + 2*a*c^3)*d^5*x
+ (b^3*c + 3*a*b*c^2)*d^5)*e^3 + 5*(504*c^4*d^6*x^2 + 63*b*c^3*d^6*x + (3*b^2*c^2 + 2*a*c^3)*d^6)*e^2 + 35*(18
*c^4*d^7*x + b*c^3*d^7)*e)/(x^9*e^18 + 9*d*x^8*e^17 + 36*d^2*x^7*e^16 + 84*d^3*x^6*e^15 + 126*d^4*x^5*e^14 + 1
26*d^5*x^4*e^13 + 84*d^6*x^3*e^12 + 36*d^7*x^2*e^11 + 9*d^8*x*e^10 + d^9*e^9)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 944 vs. \(2 (432) = 864\).
time = 1.18, size = 944, normalized size = 2.17 \begin {gather*} -\frac {{\left (630 \, c^{4} x^{8} e^{8} + 2520 \, c^{4} d x^{7} e^{7} + 5880 \, c^{4} d^{2} x^{6} e^{6} + 8820 \, c^{4} d^{3} x^{5} e^{5} + 8820 \, c^{4} d^{4} x^{4} e^{4} + 5880 \, c^{4} d^{5} x^{3} e^{3} + 2520 \, c^{4} d^{6} x^{2} e^{2} + 630 \, c^{4} d^{7} x e + 70 \, c^{4} d^{8} + 1260 \, b c^{3} x^{7} e^{8} + 2940 \, b c^{3} d x^{6} e^{7} + 4410 \, b c^{3} d^{2} x^{5} e^{6} + 4410 \, b c^{3} d^{3} x^{4} e^{5} + 2940 \, b c^{3} d^{4} x^{3} e^{4} + 1260 \, b c^{3} d^{5} x^{2} e^{3} + 315 \, b c^{3} d^{6} x e^{2} + 35 \, b c^{3} d^{7} e + 1260 \, b^{2} c^{2} x^{6} e^{8} + 840 \, a c^{3} x^{6} e^{8} + 1890 \, b^{2} c^{2} d x^{5} e^{7} + 1260 \, 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} + 1260 \, b^{2} c^{2} d^{3} x^{3} e^{5} + 840 \, a c^{3} d^{3} x^{3} e^{5} + 540 \, b^{2} c^{2} d^{4} x^{2} e^{4} + 360 \, a c^{3} d^{4} x^{2} e^{4} + 135 \, b^{2} c^{2} d^{5} x e^{3} + 90 \, a c^{3} d^{5} x e^{3} + 15 \, b^{2} c^{2} d^{6} e^{2} + 10 \, a c^{3} d^{6} e^{2} + 630 \, b^{3} c x^{5} e^{8} + 1890 \, a b c^{2} x^{5} e^{8} + 630 \, b^{3} c d x^{4} e^{7} + 1890 \, a b c^{2} d x^{4} e^{7} + 420 \, b^{3} c d^{2} x^{3} e^{6} + 1260 \, 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} + 45 \, b^{3} c d^{4} x e^{4} + 135 \, a b c^{2} d^{4} x e^{4} + 5 \, b^{3} c d^{5} e^{3} + 15 \, a b c^{2} d^{5} e^{3} + 126 \, b^{4} x^{4} e^{8} + 1512 \, a b^{2} c x^{4} e^{8} + 756 \, a^{2} c^{2} x^{4} e^{8} + 84 \, b^{4} d x^{3} e^{7} + 1008 \, a b^{2} c d x^{3} e^{7} + 504 \, a^{2} c^{2} d x^{3} e^{7} + 36 \, b^{4} d^{2} x^{2} e^{6} + 432 \, a b^{2} c d^{2} x^{2} e^{6} + 216 \, a^{2} c^{2} d^{2} x^{2} e^{6} + 9 \, b^{4} d^{3} x e^{5} + 108 \, a b^{2} c d^{3} x e^{5} + 54 \, 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} + 420 \, a b^{3} x^{3} e^{8} + 1260 \, a^{2} b c x^{3} e^{8} + 180 \, a b^{3} d x^{2} e^{7} + 540 \, a^{2} b c d x^{2} e^{7} + 45 \, a b^{3} d^{2} x e^{6} + 135 \, a^{2} b c d^{2} x e^{6} + 5 \, a b^{3} d^{3} e^{5} + 15 \, a^{2} b c d^{3} e^{5} + 540 \, a^{2} b^{2} x^{2} e^{8} + 360 \, a^{3} c x^{2} e^{8} + 135 \, a^{2} b^{2} d x e^{7} + 90 \, a^{3} c d x e^{7} + 15 \, a^{2} b^{2} d^{2} e^{6} + 10 \, a^{3} c d^{2} e^{6} + 315 \, a^{3} b x e^{8} + 35 \, a^{3} b d e^{7} + 70 \, a^{4} e^{8}\right )} e^{\left (-9\right )}}{630 \, {\left (x e + d\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.26, size = 966, normalized size = 2.22 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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