3.22.54 \(\int \frac {(a+b x+c x^2)^4}{(d+e x)^4} \, dx\) [2154]

Optimal. Leaf size=417 \[ \frac {\left (35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x}{e^8}-\frac {2 c \left (5 c^3 d^3-b^3 e^3-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)\right ) x^2}{e^7}+\frac {2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^3}{3 e^6}-\frac {c^3 (c d-b e) x^4}{e^5}+\frac {c^4 x^5}{5 e^4}-\frac {\left (c d^2-b d e+a e^2\right )^4}{3 e^9 (d+e x)^3}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^2}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \log (d+e x)}{e^9} \]

[Out]

(35*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+4*b*d)-40*c^3*d^2*e*(-a*e+2*b*d)+6*c^2*e^2*(a^2*e^2-8*a*b*d*e+10*b^2*d
^2))*x/e^8-2*c*(5*c^3*d^3-b^3*e^3-2*c^2*d*e*(-2*a*e+5*b*d)+3*b*c*e^2*(-a*e+2*b*d))*x^2/e^7+2/3*c^2*(5*c^2*d^2+
3*b^2*e^2-2*c*e*(-a*e+4*b*d))*x^3/e^6-c^3*(-b*e+c*d)*x^4/e^5+1/5*c^4*x^5/e^4-1/3*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*
x+d)^3+2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^2-2*(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)-4*(-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))*ln(e*x+d)/
e^9

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Rubi [A]
time = 0.45, antiderivative size = 417, 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 {x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4\right )}{e^8}-\frac {2 c x^2 \left (-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)-b^3 e^3+5 c^3 d^3\right )}{e^7}-\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 )}{e^9 (d+e x)}-\frac {4 (2 c d-b e) \log (d+e x) \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 )}{e^9}+\frac {2 c^2 x^3 \left (-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right )}{3 e^6}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right )^4}{3 e^9 (d+e x)^3}-\frac {c^3 x^4 (c d-b e)}{e^5}+\frac {c^4 x^5}{5 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) - 40*c^3*d^2*e*(2*b*d - a*e) + 6*c^2*e^2*(10*b^2*d^2 - 8*
a*b*d*e + a^2*e^2))*x)/e^8 - (2*c*(5*c^3*d^3 - b^3*e^3 - 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(2*b*d - a*e))*
x^2)/e^7 + (2*c^2*(5*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(4*b*d - a*e))*x^3)/(3*e^6) - (c^3*(c*d - b*e)*x^4)/e^5 + (c^
4*x^5)/(5*e^4) - (c*d^2 - b*d*e + a*e^2)^4/(3*e^9*(d + e*x)^3) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(
e^9*(d + e*x)^2) - (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)))/(e^9*(d + e*x)
) - (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))*Log[d + e*x])/e^9

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)^4} \, dx &=\int \left (\frac {35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )}{e^8}+\frac {4 c \left (-5 c^3 d^3+b^3 e^3+2 c^2 d e (5 b d-2 a e)-3 b c e^2 (2 b d-a e)\right ) x}{e^7}+\frac {2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^2}{e^6}-\frac {4 c^3 (c d-b e) x^3}{e^5}+\frac {c^4 x^4}{e^4}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^4}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^3}+\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)^2}+\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)}\right ) \, dx\\ &=\frac {\left (35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x}{e^8}-\frac {2 c \left (5 c^3 d^3-b^3 e^3-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)\right ) x^2}{e^7}+\frac {2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^3}{3 e^6}-\frac {c^3 (c d-b e) x^4}{e^5}+\frac {c^4 x^5}{5 e^4}-\frac {\left (c d^2-b d e+a e^2\right )^4}{3 e^9 (d+e x)^3}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^2}-\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^9 (d+e x)}-\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 ) \log (d+e x)}{e^9}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 425, normalized size = 1.02 \begin {gather*} \frac {15 e \left (35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)+40 c^3 d^2 e (-2 b d+a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x+30 c e^2 \left (-5 c^3 d^3+b^3 e^3+2 c^2 d e (5 b d-2 a e)+3 b c e^2 (-2 b d+a e)\right ) x^2+10 c^2 e^3 \left (5 c^2 d^2+3 b^2 e^2+2 c e (-4 b d+a e)\right ) x^3+15 c^3 e^4 (-c d+b e) x^4+3 c^4 e^5 x^5-\frac {5 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^3}+\frac {30 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^2}-\frac {30 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2}{d+e x}-60 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right ) \log (d+e x)}{15 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15*e*(35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) + 40*c^3*d^2*e*(-2*b*d + a*e) + 6*c^2*e^2*(10*b^2*d^
2 - 8*a*b*d*e + a^2*e^2))*x + 30*c*e^2*(-5*c^3*d^3 + b^3*e^3 + 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(-2*b*d +
a*e))*x^2 + 10*c^2*e^3*(5*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-4*b*d + a*e))*x^3 + 15*c^3*e^4*(-(c*d) + b*e)*x^4 + 3
*c^4*e^5*x^5 - (5*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^3 + (30*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/
(d + e*x)^2 - (30*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x) - 60
*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d
*e + 3*a^2*e^2))*Log[d + e*x])/(15*e^9)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(929\) vs. \(2(411)=822\).
time = 0.72, size = 930, normalized size = 2.23 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)^4,x,method=_RETURNVERBOSE)

[Out]

1/e^8*(1/5*c^4*x^5*e^4+b*c^3*e^4*x^4-c^4*d*e^3*x^4+4/3*a*c^3*e^4*x^3+2*b^2*c^2*e^4*x^3-16/3*b*c^3*d*e^3*x^3+10
/3*c^4*d^2*e^2*x^3+6*a*b*c^2*e^4*x^2-8*a*c^3*d*e^3*x^2+2*b^3*c*e^4*x^2-12*b^2*c^2*d*e^3*x^2+20*b*c^3*d^2*e^2*x
^2-10*c^4*d^3*e*x^2+6*e^4*a^2*c^2*x+12*a*b^2*c*e^4*x-48*a*b*c^2*d*e^3*x+40*d^2*e^2*c^3*a*x+b^4*e^4*x-16*b^3*c*
d*e^3*x+60*b^2*c^2*d^2*e^2*x-80*b*c^3*d^3*e*x+35*c^4*d^4*x)-1/3*(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)^3-(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)+(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*ln(e*x+d)-1/2*(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)^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 845 vs. \(2 (413) = 826\).
time = 0.32, size = 845, normalized size = 2.03 \begin {gather*} -4 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e - a b^{3} e^{5} - 3 \, a^{2} b c e^{5} + 10 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{3} - 10 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{2} + {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d\right )} e^{\left (-9\right )} \log \left (x e + d\right ) + \frac {1}{15} \, {\left (3 \, c^{4} x^{5} e^{4} - 15 \, {\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{4} + 10 \, {\left (5 \, c^{4} d^{2} e^{2} - 8 \, b c^{3} d e^{3} + 3 \, b^{2} c^{2} e^{4} + 2 \, a c^{3} e^{4}\right )} x^{3} - 30 \, {\left (5 \, c^{4} d^{3} e - 10 \, b c^{3} d^{2} e^{2} - b^{3} c e^{4} - 3 \, a b c^{2} e^{4} + 2 \, {\left (3 \, b^{2} c^{2} e^{3} + 2 \, a c^{3} e^{3}\right )} d\right )} x^{2} + 15 \, {\left (35 \, c^{4} d^{4} - 80 \, b c^{3} d^{3} e + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4} + 20 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{2} - 16 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d\right )} x\right )} e^{\left (-8\right )} - \frac {73 \, c^{4} d^{8} - 214 \, b c^{3} d^{7} e + 74 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{6} - 94 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{5} + 2 \, a^{3} b d e^{7} + 13 \, {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{4} + a^{4} e^{8} - 22 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d^{3} + 2 \, {\left (3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} d^{2} + 6 \, {\left (14 \, c^{4} d^{6} e^{2} - 42 \, b c^{3} d^{5} e^{3} + 15 \, {\left (3 \, b^{2} c^{2} e^{4} + 2 \, a c^{3} e^{4}\right )} d^{4} + 3 \, a^{2} b^{2} e^{8} + 2 \, a^{3} c e^{8} - 20 \, {\left (b^{3} c e^{5} + 3 \, a b c^{2} e^{5}\right )} d^{3} + 3 \, {\left (b^{4} e^{6} + 12 \, a b^{2} c e^{6} + 6 \, a^{2} c^{2} e^{6}\right )} d^{2} - 6 \, {\left (a b^{3} e^{7} + 3 \, a^{2} b c e^{7}\right )} d\right )} x^{2} + 6 \, {\left (26 \, c^{4} d^{7} e - 77 \, b c^{3} d^{6} e^{2} + 27 \, {\left (3 \, b^{2} c^{2} e^{3} + 2 \, a c^{3} e^{3}\right )} d^{5} - 35 \, {\left (b^{3} c e^{4} + 3 \, a b c^{2} e^{4}\right )} d^{4} + a^{3} b e^{8} + 5 \, {\left (b^{4} e^{5} + 12 \, a b^{2} c e^{5} + 6 \, a^{2} c^{2} e^{5}\right )} d^{3} - 9 \, {\left (a b^{3} e^{6} + 3 \, a^{2} b c e^{6}\right )} d^{2} + {\left (3 \, a^{2} b^{2} e^{7} + 2 \, a^{3} c e^{7}\right )} d\right )} x}{3 \, {\left (x^{3} e^{12} + 3 \, d x^{2} e^{11} + 3 \, d^{2} x e^{10} + d^{3} 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)^4,x, algorithm="maxima")

[Out]

-4*(14*c^4*d^5 - 35*b*c^3*d^4*e - a*b^3*e^5 - 3*a^2*b*c*e^5 + 10*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^3 - 10*(b^3*c
*e^3 + 3*a*b*c^2*e^3)*d^2 + (b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d)*e^(-9)*log(x*e + d) + 1/15*(3*c^4*x^
5*e^4 - 15*(c^4*d*e^3 - b*c^3*e^4)*x^4 + 10*(5*c^4*d^2*e^2 - 8*b*c^3*d*e^3 + 3*b^2*c^2*e^4 + 2*a*c^3*e^4)*x^3
- 30*(5*c^4*d^3*e - 10*b*c^3*d^2*e^2 - b^3*c*e^4 - 3*a*b*c^2*e^4 + 2*(3*b^2*c^2*e^3 + 2*a*c^3*e^3)*d)*x^2 + 15
*(35*c^4*d^4 - 80*b*c^3*d^3*e + b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4 + 20*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^
2 - 16*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d)*x)*e^(-8) - 1/3*(73*c^4*d^8 - 214*b*c^3*d^7*e + 74*(3*b^2*c^2*e^2 + 2*a*
c^3*e^2)*d^6 - 94*(b^3*c*e^3 + 3*a*b*c^2*e^3)*d^5 + 2*a^3*b*d*e^7 + 13*(b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e
^4)*d^4 + a^4*e^8 - 22*(a*b^3*e^5 + 3*a^2*b*c*e^5)*d^3 + 2*(3*a^2*b^2*e^6 + 2*a^3*c*e^6)*d^2 + 6*(14*c^4*d^6*e
^2 - 42*b*c^3*d^5*e^3 + 15*(3*b^2*c^2*e^4 + 2*a*c^3*e^4)*d^4 + 3*a^2*b^2*e^8 + 2*a^3*c*e^8 - 20*(b^3*c*e^5 + 3
*a*b*c^2*e^5)*d^3 + 3*(b^4*e^6 + 12*a*b^2*c*e^6 + 6*a^2*c^2*e^6)*d^2 - 6*(a*b^3*e^7 + 3*a^2*b*c*e^7)*d)*x^2 +
6*(26*c^4*d^7*e - 77*b*c^3*d^6*e^2 + 27*(3*b^2*c^2*e^3 + 2*a*c^3*e^3)*d^5 - 35*(b^3*c*e^4 + 3*a*b*c^2*e^4)*d^4
+ a^3*b*e^8 + 5*(b^4*e^5 + 12*a*b^2*c*e^5 + 6*a^2*c^2*e^5)*d^3 - 9*(a*b^3*e^6 + 3*a^2*b*c*e^6)*d^2 + (3*a^2*b
^2*e^7 + 2*a^3*c*e^7)*d)*x)/(x^3*e^12 + 3*d*x^2*e^11 + 3*d^2*x*e^10 + d^3*e^9)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1238 vs. \(2 (413) = 826\).
time = 3.01, size = 1238, normalized size = 2.97 \begin {gather*} -\frac {365 \, c^{4} d^{8} - {\left (3 \, c^{4} x^{8} + 15 \, b c^{3} x^{7} + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{6} + 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{5} - 30 \, a^{3} b x + 15 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{4} - 5 \, a^{4} - 30 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{2}\right )} e^{8} + {\left (6 \, c^{4} d x^{7} + 35 \, b c^{3} d x^{6} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{5} + 150 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{4} + 10 \, a^{3} b d - 45 \, {\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} + 30 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x\right )} e^{7} - {\left (14 \, c^{4} d^{2} x^{6} + 105 \, b c^{3} d^{2} x^{5} + 150 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{4} - 630 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{3} - 45 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{2} + 270 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x - 10 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} e^{6} + {\left (42 \, c^{4} d^{3} x^{5} + 525 \, b c^{3} d^{3} x^{4} - 730 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{3} + 90 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{2} + 135 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x - 110 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3}\right )} e^{5} - 5 \, {\left (42 \, c^{4} d^{4} x^{4} - 556 \, b c^{3} d^{4} x^{3} + 78 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2} + 162 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} x - 13 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4}\right )} e^{4} - 5 \, {\left (235 \, c^{4} d^{5} x^{3} - 408 \, b c^{3} d^{5} x^{2} - 102 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x + 94 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5}\right )} e^{3} - 5 \, {\left (201 \, c^{4} d^{6} x^{2} + 222 \, b c^{3} d^{6} x - 74 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6}\right )} e^{2} + 5 \, {\left (51 \, c^{4} d^{7} x - 214 \, b c^{3} d^{7}\right )} e + 60 \, {\left (14 \, c^{4} d^{8} - {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{3} e^{8} + {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{3} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{2}\right )} e^{7} - {\left (10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{3} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{2} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x\right )} e^{6} + {\left (10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{3} - 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{2} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x - {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3}\right )} e^{5} - {\left (35 \, b c^{3} d^{4} x^{3} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2} + 30 \, {\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} + {\left (14 \, c^{4} d^{5} x^{3} - 105 \, b c^{3} d^{5} x^{2} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5}\right )} e^{3} + {\left (42 \, c^{4} d^{6} x^{2} - 105 \, b c^{3} d^{6} x + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6}\right )} e^{2} + 7 \, {\left (6 \, c^{4} d^{7} x - 5 \, b c^{3} d^{7}\right )} e\right )} \log \left (x e + d\right )}{15 \, {\left (x^{3} e^{12} + 3 \, d x^{2} e^{11} + 3 \, d^{2} x e^{10} + d^{3} 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)^4,x, algorithm="fricas")

[Out]

-1/15*(365*c^4*d^8 - (3*c^4*x^8 + 15*b*c^3*x^7 + 10*(3*b^2*c^2 + 2*a*c^3)*x^6 + 30*(b^3*c + 3*a*b*c^2)*x^5 - 3
0*a^3*b*x + 15*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*x^4 - 5*a^4 - 30*(3*a^2*b^2 + 2*a^3*c)*x^2)*e^8 + (6*c^4*d*x^7 +
35*b*c^3*d*x^6 + 30*(3*b^2*c^2 + 2*a*c^3)*d*x^5 + 150*(b^3*c + 3*a*b*c^2)*d*x^4 + 10*a^3*b*d - 45*(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 + 30*(3*a^2*b^2 + 2*a^3*c)*d*x)*e^7 - (14*c^4*d^2*x^
6 + 105*b*c^3*d^2*x^5 + 150*(3*b^2*c^2 + 2*a*c^3)*d^2*x^4 - 630*(b^3*c + 3*a*b*c^2)*d^2*x^3 - 45*(b^4 + 12*a*b
^2*c + 6*a^2*c^2)*d^2*x^2 + 270*(a*b^3 + 3*a^2*b*c)*d^2*x - 10*(3*a^2*b^2 + 2*a^3*c)*d^2)*e^6 + (42*c^4*d^3*x^
5 + 525*b*c^3*d^3*x^4 - 730*(3*b^2*c^2 + 2*a*c^3)*d^3*x^3 + 90*(b^3*c + 3*a*b*c^2)*d^3*x^2 + 135*(b^4 + 12*a*b
^2*c + 6*a^2*c^2)*d^3*x - 110*(a*b^3 + 3*a^2*b*c)*d^3)*e^5 - 5*(42*c^4*d^4*x^4 - 556*b*c^3*d^4*x^3 + 78*(3*b^2
*c^2 + 2*a*c^3)*d^4*x^2 + 162*(b^3*c + 3*a*b*c^2)*d^4*x - 13*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4)*e^4 - 5*(235*
c^4*d^5*x^3 - 408*b*c^3*d^5*x^2 - 102*(3*b^2*c^2 + 2*a*c^3)*d^5*x + 94*(b^3*c + 3*a*b*c^2)*d^5)*e^3 - 5*(201*c
^4*d^6*x^2 + 222*b*c^3*d^6*x - 74*(3*b^2*c^2 + 2*a*c^3)*d^6)*e^2 + 5*(51*c^4*d^7*x - 214*b*c^3*d^7)*e + 60*(14
*c^4*d^8 - (a*b^3 + 3*a^2*b*c)*x^3*e^8 + ((b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*x^3 - 3*(a*b^3 + 3*a^2*b*c)*d*x^2)*
e^7 - (10*(b^3*c + 3*a*b*c^2)*d^2*x^3 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*x^2 + 3*(a*b^3 + 3*a^2*b*c)*d^2*x
)*e^6 + (10*(3*b^2*c^2 + 2*a*c^3)*d^3*x^3 - 30*(b^3*c + 3*a*b*c^2)*d^3*x^2 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*
d^3*x - (a*b^3 + 3*a^2*b*c)*d^3)*e^5 - (35*b*c^3*d^4*x^3 - 30*(3*b^2*c^2 + 2*a*c^3)*d^4*x^2 + 30*(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 + (14*c^4*d^5*x^3 - 105*b*c^3*d^5*x^2 + 30*(3*b^2*c^2 +
2*a*c^3)*d^5*x - 10*(b^3*c + 3*a*b*c^2)*d^5)*e^3 + (42*c^4*d^6*x^2 - 105*b*c^3*d^6*x + 10*(3*b^2*c^2 + 2*a*c^
3)*d^6)*e^2 + 7*(6*c^4*d^7*x - 5*b*c^3*d^7)*e)*log(x*e + d))/(x^3*e^12 + 3*d*x^2*e^11 + 3*d^2*x*e^10 + d^3*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)**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 865 vs. \(2 (413) = 826\).
time = 1.79, size = 865, normalized size = 2.07 \begin {gather*} -4 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 30 \, b^{2} c^{2} d^{3} e^{2} + 20 \, a c^{3} d^{3} e^{2} - 10 \, b^{3} c d^{2} e^{3} - 30 \, a b c^{2} d^{2} e^{3} + b^{4} d e^{4} + 12 \, a b^{2} c d e^{4} + 6 \, a^{2} c^{2} d e^{4} - a b^{3} e^{5} - 3 \, a^{2} b c e^{5}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{15} \, {\left (3 \, c^{4} x^{5} e^{16} - 15 \, c^{4} d x^{4} e^{15} + 50 \, c^{4} d^{2} x^{3} e^{14} - 150 \, c^{4} d^{3} x^{2} e^{13} + 525 \, c^{4} d^{4} x e^{12} + 15 \, b c^{3} x^{4} e^{16} - 80 \, b c^{3} d x^{3} e^{15} + 300 \, b c^{3} d^{2} x^{2} e^{14} - 1200 \, b c^{3} d^{3} x e^{13} + 30 \, b^{2} c^{2} x^{3} e^{16} + 20 \, a c^{3} x^{3} e^{16} - 180 \, b^{2} c^{2} d x^{2} e^{15} - 120 \, a c^{3} d x^{2} e^{15} + 900 \, b^{2} c^{2} d^{2} x e^{14} + 600 \, a c^{3} d^{2} x e^{14} + 30 \, b^{3} c x^{2} e^{16} + 90 \, a b c^{2} x^{2} e^{16} - 240 \, b^{3} c d x e^{15} - 720 \, a b c^{2} d x e^{15} + 15 \, b^{4} x e^{16} + 180 \, a b^{2} c x e^{16} + 90 \, a^{2} c^{2} x e^{16}\right )} e^{\left (-20\right )} - \frac {{\left (73 \, c^{4} d^{8} - 214 \, b c^{3} d^{7} e + 222 \, b^{2} c^{2} d^{6} e^{2} + 148 \, a c^{3} d^{6} e^{2} - 94 \, b^{3} c d^{5} e^{3} - 282 \, a b c^{2} d^{5} e^{3} + 13 \, b^{4} d^{4} e^{4} + 156 \, a b^{2} c d^{4} e^{4} + 78 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a b^{3} d^{3} e^{5} - 66 \, a^{2} b c d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} + 4 \, a^{3} c d^{2} e^{6} + 2 \, a^{3} b d e^{7} + a^{4} e^{8} + 6 \, {\left (14 \, c^{4} d^{6} e^{2} - 42 \, b c^{3} d^{5} e^{3} + 45 \, b^{2} c^{2} d^{4} e^{4} + 30 \, a c^{3} d^{4} e^{4} - 20 \, b^{3} c d^{3} e^{5} - 60 \, a b c^{2} d^{3} e^{5} + 3 \, b^{4} d^{2} e^{6} + 36 \, a b^{2} c d^{2} e^{6} + 18 \, a^{2} c^{2} d^{2} e^{6} - 6 \, a b^{3} d e^{7} - 18 \, a^{2} b c d e^{7} + 3 \, a^{2} b^{2} e^{8} + 2 \, a^{3} c e^{8}\right )} x^{2} + 6 \, {\left (26 \, c^{4} d^{7} e - 77 \, b c^{3} d^{6} e^{2} + 81 \, b^{2} c^{2} d^{5} e^{3} + 54 \, a c^{3} d^{5} e^{3} - 35 \, b^{3} c d^{4} e^{4} - 105 \, a b c^{2} d^{4} e^{4} + 5 \, b^{4} d^{3} e^{5} + 60 \, a b^{2} c d^{3} e^{5} + 30 \, a^{2} c^{2} d^{3} e^{5} - 9 \, a b^{3} d^{2} e^{6} - 27 \, a^{2} b c d^{2} e^{6} + 3 \, a^{2} b^{2} d e^{7} + 2 \, a^{3} c d e^{7} + a^{3} b e^{8}\right )} x\right )} e^{\left (-9\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(14*c^4*d^5 - 35*b*c^3*d^4*e + 30*b^2*c^2*d^3*e^2 + 20*a*c^3*d^3*e^2 - 10*b^3*c*d^2*e^3 - 30*a*b*c^2*d^2*e^
3 + b^4*d*e^4 + 12*a*b^2*c*d*e^4 + 6*a^2*c^2*d*e^4 - a*b^3*e^5 - 3*a^2*b*c*e^5)*e^(-9)*log(abs(x*e + d)) + 1/1
5*(3*c^4*x^5*e^16 - 15*c^4*d*x^4*e^15 + 50*c^4*d^2*x^3*e^14 - 150*c^4*d^3*x^2*e^13 + 525*c^4*d^4*x*e^12 + 15*b
*c^3*x^4*e^16 - 80*b*c^3*d*x^3*e^15 + 300*b*c^3*d^2*x^2*e^14 - 1200*b*c^3*d^3*x*e^13 + 30*b^2*c^2*x^3*e^16 + 2
0*a*c^3*x^3*e^16 - 180*b^2*c^2*d*x^2*e^15 - 120*a*c^3*d*x^2*e^15 + 900*b^2*c^2*d^2*x*e^14 + 600*a*c^3*d^2*x*e^
14 + 30*b^3*c*x^2*e^16 + 90*a*b*c^2*x^2*e^16 - 240*b^3*c*d*x*e^15 - 720*a*b*c^2*d*x*e^15 + 15*b^4*x*e^16 + 180
*a*b^2*c*x*e^16 + 90*a^2*c^2*x*e^16)*e^(-20) - 1/3*(73*c^4*d^8 - 214*b*c^3*d^7*e + 222*b^2*c^2*d^6*e^2 + 148*a
*c^3*d^6*e^2 - 94*b^3*c*d^5*e^3 - 282*a*b*c^2*d^5*e^3 + 13*b^4*d^4*e^4 + 156*a*b^2*c*d^4*e^4 + 78*a^2*c^2*d^4*
e^4 - 22*a*b^3*d^3*e^5 - 66*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 + 2*a^3*b*d*e^7 + a^4*e^8 +
6*(14*c^4*d^6*e^2 - 42*b*c^3*d^5*e^3 + 45*b^2*c^2*d^4*e^4 + 30*a*c^3*d^4*e^4 - 20*b^3*c*d^3*e^5 - 60*a*b*c^2*d
^3*e^5 + 3*b^4*d^2*e^6 + 36*a*b^2*c*d^2*e^6 + 18*a^2*c^2*d^2*e^6 - 6*a*b^3*d*e^7 - 18*a^2*b*c*d*e^7 + 3*a^2*b^
2*e^8 + 2*a^3*c*e^8)*x^2 + 6*(26*c^4*d^7*e - 77*b*c^3*d^6*e^2 + 81*b^2*c^2*d^5*e^3 + 54*a*c^3*d^5*e^3 - 35*b^3
*c*d^4*e^4 - 105*a*b*c^2*d^4*e^4 + 5*b^4*d^3*e^5 + 60*a*b^2*c*d^3*e^5 + 30*a^2*c^2*d^3*e^5 - 9*a*b^3*d^2*e^6 -
27*a^2*b*c*d^2*e^6 + 3*a^2*b^2*d*e^7 + 2*a^3*c*d*e^7 + a^3*b*e^8)*x)*e^(-9)/(x*e + d)^3

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Mupad [B]
time = 0.84, size = 1143, normalized size = 2.74 \begin {gather*} x^4\,\left (\frac {b\,c^3}{e^4}-\frac {c^4\,d}{e^5}\right )-x^2\,\left (\frac {2\,c^4\,d^3}{e^7}+\frac {3\,d^2\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e^2}-\frac {2\,d\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^4}+\frac {6\,c^4\,d^2}{e^6}\right )}{e}-\frac {2\,b\,c\,\left (b^2+3\,a\,c\right )}{e^4}\right )-x^3\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{3\,e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{3\,e^4}+\frac {2\,c^4\,d^2}{e^6}\right )+x\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e^4}+\frac {6\,d^2\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^4}+\frac {6\,c^4\,d^2}{e^6}\right )}{e^2}+\frac {4\,d\,\left (\frac {4\,c^4\,d^3}{e^7}+\frac {6\,d^2\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e^2}-\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^4}+\frac {6\,c^4\,d^2}{e^6}\right )}{e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^4}\right )}{e}-\frac {c^4\,d^4}{e^8}-\frac {4\,d^3\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e^3}\right )-\frac {x\,\left (2\,a^3\,b\,e^7+4\,a^3\,c\,d\,e^6+6\,a^2\,b^2\,d\,e^6-54\,a^2\,b\,c\,d^2\,e^5+60\,a^2\,c^2\,d^3\,e^4-18\,a\,b^3\,d^2\,e^5+120\,a\,b^2\,c\,d^3\,e^4-210\,a\,b\,c^2\,d^4\,e^3+108\,a\,c^3\,d^5\,e^2+10\,b^4\,d^3\,e^4-70\,b^3\,c\,d^4\,e^3+162\,b^2\,c^2\,d^5\,e^2-154\,b\,c^3\,d^6\,e+52\,c^4\,d^7\right )+\frac {a^4\,e^8+2\,a^3\,b\,d\,e^7+4\,a^3\,c\,d^2\,e^6+6\,a^2\,b^2\,d^2\,e^6-66\,a^2\,b\,c\,d^3\,e^5+78\,a^2\,c^2\,d^4\,e^4-22\,a\,b^3\,d^3\,e^5+156\,a\,b^2\,c\,d^4\,e^4-282\,a\,b\,c^2\,d^5\,e^3+148\,a\,c^3\,d^6\,e^2+13\,b^4\,d^4\,e^4-94\,b^3\,c\,d^5\,e^3+222\,b^2\,c^2\,d^6\,e^2-214\,b\,c^3\,d^7\,e+73\,c^4\,d^8}{3\,e}+x^2\,\left (4\,a^3\,c\,e^7+6\,a^2\,b^2\,e^7-36\,a^2\,b\,c\,d\,e^6+36\,a^2\,c^2\,d^2\,e^5-12\,a\,b^3\,d\,e^6+72\,a\,b^2\,c\,d^2\,e^5-120\,a\,b\,c^2\,d^3\,e^4+60\,a\,c^3\,d^4\,e^3+6\,b^4\,d^2\,e^5-40\,b^3\,c\,d^3\,e^4+90\,b^2\,c^2\,d^4\,e^3-84\,b\,c^3\,d^5\,e^2+28\,c^4\,d^6\,e\right )}{d^3\,e^8+3\,d^2\,e^9\,x+3\,d\,e^{10}\,x^2+e^{11}\,x^3}+\frac {c^4\,x^5}{5\,e^4}-\frac {\ln \left (d+e\,x\right )\,\left (-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\right )}{e^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)^4,x)

[Out]

x^4*((b*c^3)/e^4 - (c^4*d)/e^5) - x^2*((2*c^4*d^3)/e^7 + (3*d^2*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e^2 - (2*d*((
4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - (4*a*c^3 + 6*b^2*c^2)/e^4 + (6*c^4*d^2)/e^6))/e - (2*b*c*(3*a*c + b^2
))/e^4) - x^3*((4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/(3*e) - (4*a*c^3 + 6*b^2*c^2)/(3*e^4) + (2*c^4*d^2)/e^6)
+ x*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^4 + (6*d^2*((4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - (4*a*c^3 + 6*b^2*c
^2)/e^4 + (6*c^4*d^2)/e^6))/e^2 + (4*d*((4*c^4*d^3)/e^7 + (6*d^2*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e^2 - (4*d*(
(4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - (4*a*c^3 + 6*b^2*c^2)/e^4 + (6*c^4*d^2)/e^6))/e - (4*b*c*(3*a*c + b^
2))/e^4))/e - (c^4*d^4)/e^8 - (4*d^3*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e^3) - (x*(52*c^4*d^7 + 2*a^3*b*e^7 + 10
*b^4*d^3*e^4 - 18*a*b^3*d^2*e^5 + 6*a^2*b^2*d*e^6 + 108*a*c^3*d^5*e^2 - 70*b^3*c*d^4*e^3 + 60*a^2*c^2*d^3*e^4
+ 162*b^2*c^2*d^5*e^2 + 4*a^3*c*d*e^6 - 154*b*c^3*d^6*e - 210*a*b*c^2*d^4*e^3 + 120*a*b^2*c*d^3*e^4 - 54*a^2*b
*c*d^2*e^5) + (a^4*e^8 + 73*c^4*d^8 + 13*b^4*d^4*e^4 - 22*a*b^3*d^3*e^5 + 148*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6
- 94*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 78*a^2*c^2*d^4*e^4 + 222*b^2*c^2*d^6*e^2 + 2*a^3*b*d*e^7 - 214*b*c^3*
d^7*e - 282*a*b*c^2*d^5*e^3 + 156*a*b^2*c*d^4*e^4 - 66*a^2*b*c*d^3*e^5)/(3*e) + x^2*(4*a^3*c*e^7 + 28*c^4*d^6*
e + 6*a^2*b^2*e^7 + 6*b^4*d^2*e^5 + 60*a*c^3*d^4*e^3 - 84*b*c^3*d^5*e^2 - 40*b^3*c*d^3*e^4 + 36*a^2*c^2*d^2*e^
5 + 90*b^2*c^2*d^4*e^3 - 12*a*b^3*d*e^6 - 36*a^2*b*c*d*e^6 - 120*a*b*c^2*d^3*e^4 + 72*a*b^2*c*d^2*e^5))/(d^3*e
^8 + e^11*x^3 + 3*d^2*e^9*x + 3*d*e^10*x^2) + (c^4*x^5)/(5*e^4) - (log(d + e*x)*(56*c^4*d^5 - 4*a*b^3*e^5 + 4*
b^4*d*e^4 + 80*a*c^3*d^3*e^2 + 24*a^2*c^2*d*e^4 - 40*b^3*c*d^2*e^3 + 120*b^2*c^2*d^3*e^2 - 12*a^2*b*c*e^5 - 14
0*b*c^3*d^4*e + 48*a*b^2*c*d*e^4 - 120*a*b*c^2*d^2*e^3))/e^9

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