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

Optimal. Leaf size=426 \[ \frac {(d+e x)^3 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^9}+\frac {2 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac {c (d+e x)^4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}-\frac {2 (d+e x)^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {2 x \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}-\frac {\left (a e^2-b d e+c d^2\right )^4}{e^9 (d+e x)}-\frac {4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^9}-\frac {2 c^3 (d+e x)^6 (2 c d-b e)}{3 e^9}+\frac {c^4 (d+e x)^7}{7 e^9} \]

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Rubi [A]  time = 0.72, antiderivative size = 426, 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 = {698} \begin {gather*} \frac {(d+e x)^3 \left (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\right )}{3 e^9}+\frac {2 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac {c (d+e x)^4 (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}-\frac {2 (d+e x)^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 )}{e^9}+\frac {2 x \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^8}-\frac {\left (a e^2-b d e+c d^2\right )^4}{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 )^3}{e^9}-\frac {2 c^3 (d+e x)^6 (2 c d-b e)}{3 e^9}+\frac {c^4 (d+e x)^7}{7 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.35, size = 780, normalized size = 1.83 \begin {gather*} \frac {21 c^2 e^2 \left (10 a^2 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+5 a b e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )+b^2 \left (-30 d^6+150 d^5 e x+90 d^4 e^2 x^2-30 d^3 e^3 x^3+15 d^2 e^4 x^4-9 d e^5 x^5+6 e^6 x^6\right )\right )+35 c e^3 \left (12 a^3 e^3 \left (-d^2+d e x+e^2 x^2\right )+18 a^2 b e^2 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+12 a b^2 e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+b^3 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )+35 e^4 \left (-3 a^4 e^4+12 a^3 b d e^3+18 a^2 b^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 a b^3 e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+b^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+7 c^3 e \left (6 a e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+b \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )\right )-420 (d+e x) (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^3+c^4 \left (-105 d^8+735 d^7 e x+420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6-20 d e^7 x^7+15 e^8 x^8\right )}{105 e^9 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^4*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^4 - 42*d^3*e^5*x^5 + 28*d^2*e^
6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) + 35*e^4*(12*a^3*b*d*e^3 - 3*a^4*e^4 + 18*a^2*b^2*e^2*(-d^2 + d*e*x + e^2*x
^2) + 6*a*b^3*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + b^4*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^
3*x^3 + e^4*x^4)) + 35*c*e^3*(12*a^3*e^3*(-d^2 + d*e*x + e^2*x^2) + 18*a^2*b*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*
x^2 + e^3*x^3) + 12*a*b^2*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + b^3*(12*d^5 - 48*d^
4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 21*c^2*e^2*(10*a^2*e^2*(-3*d^4 + 9*d^3*e
*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 5*a*b*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 -
5*d*e^4*x^4 + 3*e^5*x^5) + b^2*(-30*d^6 + 150*d^5*e*x + 90*d^4*e^2*x^2 - 30*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 9*
d*e^5*x^5 + 6*e^6*x^6)) + 7*c^3*e*(6*a*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x
^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + b*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 +
21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7)) - 420*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3*(d + e*x)*Log[d
+ e*x])/(105*e^9*(d + e*x))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 1095, normalized size = 2.57

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(15*c^4*e^8*x^8 - 105*c^4*d^8 + 420*b*c^3*d^7*e + 420*a^3*b*d*e^7 - 105*a^4*e^8 - 210*(3*b^2*c^2 + 2*a*c
^3)*d^6*e^2 + 420*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 105*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 420*(a*b^3 + 3*a^
2*b*c)*d^3*e^5 - 210*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 10*(2*c^4*d*e^7 - 7*b*c^3*e^8)*x^7 + 14*(2*c^4*d^2*e^6 -
7*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 21*(2*c^4*d^3*e^5 - 7*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^
3)*d*e^7 - 5*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 35*(2*c^4*d^4*e^4 - 7*b*c^3*d^3*e^5 + 3*(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 - 70*(2*c^4*d^5*e^3 - 7*b*c^3*d^4*
e^4 + 3*(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 -
3*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 210*(2*c^4*d^6*e^2 - 7*b*c^3*d^5*e^3 + 3*(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 - 3*(a*b^3 + 3*a^2*b*c)*d*e^7 + (3*a^2*b^2
+ 2*a^3*c)*e^8)*x^2 + 105*(7*c^4*d^7*e - 24*b*c^3*d^6*e^2 + 10*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 16*(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 - 8*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 2*(3*a^2*b^2 + 2
*a^3*c)*d*e^7)*x - 420*(2*c^4*d^8 - 7*b*c^3*d^7*e - a^3*b*d*e^7 + 3*(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 - 3*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + (3*a^2*b^2 + 2*
a^3*c)*d^2*e^6 + (2*c^4*d^7*e - 7*b*c^3*d^6*e^2 - a^3*b*e^8 + 3*(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 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + (3*a^2*b^2 + 2*a^3*
c)*d*e^7)*x)*log(e*x + d))/(e^10*x + d*e^9)

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giac [B]  time = 0.20, size = 1013, normalized size = 2.38 \begin {gather*} \frac {1}{105} \, {\left (15 \, c^{4} - \frac {70 \, {\left (2 \, c^{4} d e - b c^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {42 \, {\left (14 \, c^{4} d^{2} e^{2} - 14 \, b c^{3} d e^{3} + 3 \, b^{2} c^{2} e^{4} + 2 \, a c^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {105 \, {\left (14 \, c^{4} d^{3} e^{3} - 21 \, b c^{3} d^{2} e^{4} + 9 \, b^{2} c^{2} d e^{5} + 6 \, a c^{3} d e^{5} - b^{3} c e^{6} - 3 \, a b c^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {35 \, {\left (70 \, c^{4} d^{4} e^{4} - 140 \, b c^{3} d^{3} e^{5} + 90 \, b^{2} c^{2} d^{2} e^{6} + 60 \, a c^{3} d^{2} e^{6} - 20 \, b^{3} c d e^{7} - 60 \, a b c^{2} d e^{7} + b^{4} e^{8} + 12 \, a b^{2} c e^{8} + 6 \, a^{2} c^{2} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac {210 \, {\left (14 \, c^{4} d^{5} e^{5} - 35 \, b c^{3} d^{4} e^{6} + 30 \, b^{2} c^{2} d^{3} e^{7} + 20 \, a c^{3} d^{3} e^{7} - 10 \, b^{3} c d^{2} e^{8} - 30 \, a b c^{2} d^{2} e^{8} + b^{4} d e^{9} + 12 \, a b^{2} c d e^{9} + 6 \, a^{2} c^{2} d e^{9} - a b^{3} e^{10} - 3 \, a^{2} b c e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} + \frac {210 \, {\left (14 \, c^{4} d^{6} e^{6} - 42 \, b c^{3} d^{5} e^{7} + 45 \, b^{2} c^{2} d^{4} e^{8} + 30 \, a c^{3} d^{4} e^{8} - 20 \, b^{3} c d^{3} e^{9} - 60 \, a b c^{2} d^{3} e^{9} + 3 \, b^{4} d^{2} e^{10} + 36 \, a b^{2} c d^{2} e^{10} + 18 \, a^{2} c^{2} d^{2} e^{10} - 6 \, a b^{3} d e^{11} - 18 \, a^{2} b c d e^{11} + 3 \, a^{2} b^{2} e^{12} + 2 \, a^{3} c e^{12}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{6}}\right )} {\left (x e + d\right )}^{7} e^{\left (-9\right )} + 4 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e + 9 \, b^{2} c^{2} d^{5} e^{2} + 6 \, a c^{3} d^{5} e^{2} - 5 \, b^{3} c d^{4} e^{3} - 15 \, a b c^{2} d^{4} e^{3} + b^{4} d^{3} e^{4} + 12 \, a b^{2} c d^{3} e^{4} + 6 \, a^{2} c^{2} d^{3} e^{4} - 3 \, a b^{3} d^{2} e^{5} - 9 \, a^{2} b c d^{2} e^{5} + 3 \, a^{2} b^{2} d e^{6} + 2 \, a^{3} c d e^{6} - a^{3} b e^{7}\right )} e^{\left (-9\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {c^{4} d^{8} e^{7}}{x e + d} - \frac {4 \, b c^{3} d^{7} e^{8}}{x e + d} + \frac {6 \, b^{2} c^{2} d^{6} e^{9}}{x e + d} + \frac {4 \, a c^{3} d^{6} e^{9}}{x e + d} - \frac {4 \, b^{3} c d^{5} e^{10}}{x e + d} - \frac {12 \, a b c^{2} d^{5} e^{10}}{x e + d} + \frac {b^{4} d^{4} e^{11}}{x e + d} + \frac {12 \, a b^{2} c d^{4} e^{11}}{x e + d} + \frac {6 \, a^{2} c^{2} d^{4} e^{11}}{x e + d} - \frac {4 \, a b^{3} d^{3} e^{12}}{x e + d} - \frac {12 \, a^{2} b c d^{3} e^{12}}{x e + d} + \frac {6 \, a^{2} b^{2} d^{2} e^{13}}{x e + d} + \frac {4 \, a^{3} c d^{2} e^{13}}{x e + d} - \frac {4 \, a^{3} b d e^{14}}{x e + d} + \frac {a^{4} e^{15}}{x e + d}\right )} e^{\left (-16\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)^2,x, algorithm="giac")

[Out]

1/105*(15*c^4 - 70*(2*c^4*d*e - b*c^3*e^2)*e^(-1)/(x*e + d) + 42*(14*c^4*d^2*e^2 - 14*b*c^3*d*e^3 + 3*b^2*c^2*
e^4 + 2*a*c^3*e^4)*e^(-2)/(x*e + d)^2 - 105*(14*c^4*d^3*e^3 - 21*b*c^3*d^2*e^4 + 9*b^2*c^2*d*e^5 + 6*a*c^3*d*e
^5 - b^3*c*e^6 - 3*a*b*c^2*e^6)*e^(-3)/(x*e + d)^3 + 35*(70*c^4*d^4*e^4 - 140*b*c^3*d^3*e^5 + 90*b^2*c^2*d^2*e
^6 + 60*a*c^3*d^2*e^6 - 20*b^3*c*d*e^7 - 60*a*b*c^2*d*e^7 + b^4*e^8 + 12*a*b^2*c*e^8 + 6*a^2*c^2*e^8)*e^(-4)/(
x*e + d)^4 - 210*(14*c^4*d^5*e^5 - 35*b*c^3*d^4*e^6 + 30*b^2*c^2*d^3*e^7 + 20*a*c^3*d^3*e^7 - 10*b^3*c*d^2*e^8
- 30*a*b*c^2*d^2*e^8 + b^4*d*e^9 + 12*a*b^2*c*d*e^9 + 6*a^2*c^2*d*e^9 - a*b^3*e^10 - 3*a^2*b*c*e^10)*e^(-5)/(
x*e + d)^5 + 210*(14*c^4*d^6*e^6 - 42*b*c^3*d^5*e^7 + 45*b^2*c^2*d^4*e^8 + 30*a*c^3*d^4*e^8 - 20*b^3*c*d^3*e^9
- 60*a*b*c^2*d^3*e^9 + 3*b^4*d^2*e^10 + 36*a*b^2*c*d^2*e^10 + 18*a^2*c^2*d^2*e^10 - 6*a*b^3*d*e^11 - 18*a^2*b
*c*d*e^11 + 3*a^2*b^2*e^12 + 2*a^3*c*e^12)*e^(-6)/(x*e + d)^6)*(x*e + d)^7*e^(-9) + 4*(2*c^4*d^7 - 7*b*c^3*d^6
*e + 9*b^2*c^2*d^5*e^2 + 6*a*c^3*d^5*e^2 - 5*b^3*c*d^4*e^3 - 15*a*b*c^2*d^4*e^3 + b^4*d^3*e^4 + 12*a*b^2*c*d^3
*e^4 + 6*a^2*c^2*d^3*e^4 - 3*a*b^3*d^2*e^5 - 9*a^2*b*c*d^2*e^5 + 3*a^2*b^2*d*e^6 + 2*a^3*c*d*e^6 - a^3*b*e^7)*
e^(-9)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (c^4*d^8*e^7/(x*e + d) - 4*b*c^3*d^7*e^8/(x*e + d) + 6*b^2*c^2*d
^6*e^9/(x*e + d) + 4*a*c^3*d^6*e^9/(x*e + d) - 4*b^3*c*d^5*e^10/(x*e + d) - 12*a*b*c^2*d^5*e^10/(x*e + d) + b^
4*d^4*e^11/(x*e + d) + 12*a*b^2*c*d^4*e^11/(x*e + d) + 6*a^2*c^2*d^4*e^11/(x*e + d) - 4*a*b^3*d^3*e^12/(x*e +
d) - 12*a^2*b*c*d^3*e^12/(x*e + d) + 6*a^2*b^2*d^2*e^13/(x*e + d) + 4*a^3*c*d^2*e^13/(x*e + d) - 4*a^3*b*d*e^1
4/(x*e + d) + a^4*e^15/(x*e + d))*e^(-16)

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maple [B]  time = 0.06, size = 1159, normalized size = 2.72

result 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)^2,x)

[Out]

10/e^6*x^2*b*c^3*d^4+12/e^4*ln(e*x+d)*a*b^3*d^2+1/e^2*x^4*b^3*c+4/e^2*ln(e*x+d)*a^3*b-4/e^5*ln(e*x+d)*b^4*d^3-
8/e^9*ln(e*x+d)*c^4*d^7+2/e^2*x^3*a^2*c^2+2/3/e^2*x^6*b*c^3+4/5/e^2*x^5*a*c^3-1/e^5/(e*x+d)*b^4*d^4-1/e^9/(e*x
+d)*c^4*d^8+7/e^8*c^4*d^6*x-3/e^7*x^2*c^4*d^5+4/e^2*a^3*c*x+6/e^2*a^2*b^2*x+3/e^4*b^4*d^2*x+5/3/e^6*x^3*c^4*d^
4+2/e^2*x^2*a*b^3-1/e^3*x^2*b^4*d+6/5/e^2*x^5*b^2*c^2+3/5/e^4*x^5*c^4*d^2-1/e^5*x^4*c^4*d^3-24/e^7*ln(e*x+d)*a
*c^3*d^5+20/e^6*ln(e*x+d)*b^3*c*d^4+12/e^4/(e*x+d)*a^2*b*c*d^3-12/e^5/(e*x+d)*a*b^2*c*d^4+12/e^6/(e*x+d)*a*b*c
^2*d^5+36/e^4*ln(e*x+d)*a^2*b*c*d^2-48/e^5*ln(e*x+d)*a*b^2*c*d^3+60/e^6*ln(e*x+d)*a*b*c^2*d^4+1/3/e^2*x^3*b^4-
1/e/(e*x+d)*a^4-24/e^3*a^2*b*c*d*x+36/e^4*a*b^2*c*d^2*x-48/e^5*a*b*c^2*d^3*x-12/e^3*x^2*a*b^2*c*d+18/e^4*x^2*a
*b*c^2*d^2-8/e^3*x^3*a*b*c^2*d-12/e^5*x^2*b^2*c^2*d^3-16/3/e^5*x^3*b*c^3*d^3+4/e^2/(e*x+d)*d*a^3*b-4/e^3/(e*x+
d)*a^3*c*d^2-6/e^3/(e*x+d)*a^2*b^2*d^2-6/e^5/(e*x+d)*a^2*c^2*d^4+4/e^4/(e*x+d)*a*b^3*d^3-4/e^7/(e*x+d)*a*c^3*d
^6+4/e^6/(e*x+d)*b^3*c*d^5-6/e^7/(e*x+d)*b^2*c^2*d^6+4/e^8/(e*x+d)*b*c^3*d^7+1/7*c^4/e^2*x^7-36/e^7*ln(e*x+d)*
b^2*c^2*d^5+28/e^8*ln(e*x+d)*b*c^3*d^6-8/e^3*ln(e*x+d)*a^3*c*d-12/e^3*ln(e*x+d)*a^2*b^2*d-24/e^5*ln(e*x+d)*a^2
*c^2*d^3+6/e^4*x^3*b^2*c^2*d^2-8/5/e^3*x^5*b*c^3*d+3/e^2*x^4*a*b*c^2-2/e^3*x^4*a*c^3*d-3/e^3*x^4*b^2*c^2*d+4/e
^4*x^3*a*c^3*d^2+3/e^4*x^4*b*c^3*d^2-8/e^3*a*b^3*d*x+20/e^6*a*c^3*d^4*x-16/e^5*b^3*c*d^3*x+30/e^6*b^2*c^2*d^4*
x-24/e^7*b*c^3*d^5*x+18/e^4*a^2*c^2*d^2*x+4/e^2*x^3*a*b^2*c-8/3/e^3*x^3*b^3*c*d+6/e^2*x^2*a^2*b*c-6/e^3*x^2*a^
2*c^2*d-8/e^5*x^2*a*c^3*d^3+6/e^4*x^2*b^3*c*d^2-1/3*c^4*d/e^3*x^6

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maxima [A]  time = 1.12, size = 807, normalized size = 1.89 \begin {gather*} -\frac {c^{4} d^{8} - 4 \, b c^{3} d^{7} e - 4 \, a^{3} b d e^{7} + a^{4} e^{8} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 4 \, {\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} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6}}{e^{10} x + d e^{9}} + \frac {15 \, c^{4} e^{6} x^{7} - 35 \, {\left (c^{4} d e^{5} - 2 \, b c^{3} e^{6}\right )} x^{6} + 21 \, {\left (3 \, c^{4} d^{2} e^{4} - 8 \, b c^{3} d e^{5} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{6}\right )} x^{5} - 105 \, {\left (c^{4} d^{3} e^{3} - 3 \, b c^{3} d^{2} e^{4} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{5} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{6}\right )} x^{4} + 35 \, {\left (5 \, c^{4} d^{4} e^{2} - 16 \, b c^{3} d^{3} e^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{4} - 8 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{6}\right )} x^{3} - 105 \, {\left (3 \, c^{4} d^{5} e - 10 \, b c^{3} d^{4} e^{2} + 4 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{3} - 6 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{5} - 2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{6}\right )} x^{2} + 105 \, {\left (7 \, c^{4} d^{6} - 24 \, b c^{3} d^{5} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 16 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 8 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\right )} x}{105 \, e^{8}} - \frac {4 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} \log \left (e x + d\right )}{e^{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)^2,x, algorithm="maxima")

[Out]

-(c^4*d^8 - 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^8 + 2*(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 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2
*e^6)/(e^10*x + d*e^9) + 1/105*(15*c^4*e^6*x^7 - 35*(c^4*d*e^5 - 2*b*c^3*e^6)*x^6 + 21*(3*c^4*d^2*e^4 - 8*b*c^
3*d*e^5 + 2*(3*b^2*c^2 + 2*a*c^3)*e^6)*x^5 - 105*(c^4*d^3*e^3 - 3*b*c^3*d^2*e^4 + (3*b^2*c^2 + 2*a*c^3)*d*e^5
- (b^3*c + 3*a*b*c^2)*e^6)*x^4 + 35*(5*c^4*d^4*e^2 - 16*b*c^3*d^3*e^3 + 6*(3*b^2*c^2 + 2*a*c^3)*d^2*e^4 - 8*(b
^3*c + 3*a*b*c^2)*d*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^6)*x^3 - 105*(3*c^4*d^5*e - 10*b*c^3*d^4*e^2 + 4*(3
*b^2*c^2 + 2*a*c^3)*d^3*e^3 - 6*(b^3*c + 3*a*b*c^2)*d^2*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^5 - 2*(a*b^3
+ 3*a^2*b*c)*e^6)*x^2 + 105*(7*c^4*d^6 - 24*b*c^3*d^5*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 16*(b^3*c + 3*a*b
*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^4 - 8*(a*b^3 + 3*a^2*b*c)*d*e^5 + 2*(3*a^2*b^2 + 2*a^3*
c)*e^6)*x)/e^8 - 4*(2*c^4*d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b
*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)
*d*e^6)*log(e*x + d)/e^9

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mupad [B]  time = 0.75, size = 1679, normalized size = 3.94

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((4*a^3*c + 6*a^2*b^2)/e^2 + (2*d*((2*d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^2 + (d^2*((2*d*((4*b*c^3)/e^2 - (2
*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d
)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c
*(3*a*c + b^2))/e^2))/e))/e + (d^2*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2
+ (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))/e^2))/e^2 - (4*a*b*(3*
a*c + b^2))/e^2))/e - (d^2*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^2 + (d^2*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e
- (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*
a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))
/e^2))/e))/e^2) + x^6*((2*b*c^3)/(3*e^2) - (c^4*d)/(3*e^3)) - x^2*((d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^2 + (d
^2*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e^2 - (2*d*((2*d*((2
*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 -
(2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))/e^2))/e))/e + (d^2*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e
- (4*a*c^3 + 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c +
b^2))/e^2))/(2*e^2) - (2*a*b*(3*a*c + b^2))/e^2) + x^4*((d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3
+ 6*b^2*c^2)/e^2 + (c^4*d^2)/e^4))/(2*e) - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/(4*e^2) + (b*c*(3*a*c + b^2)
)/e^2) - x^5*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/(5*e) - (4*a*c^3 + 6*b^2*c^2)/(5*e^2) + (c^4*d^2)/(5*e^4))
+ x^3*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/(3*e^2) + (d^2*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*
b^2*c^2)/e^2 + (c^4*d^2)/e^4))/(3*e^2) - (2*d*((2*d*((2*d*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e - (4*a*c^3 + 6*b^
2*c^2)/e^2 + (c^4*d^2)/e^4))/e - (d^2*((4*b*c^3)/e^2 - (2*c^4*d)/e^3))/e^2 + (4*b*c*(3*a*c + b^2))/e^2))/(3*e)
) - (log(d + e*x)*(8*c^4*d^7 - 4*a^3*b*e^7 + 4*b^4*d^3*e^4 - 12*a*b^3*d^2*e^5 + 12*a^2*b^2*d*e^6 + 24*a*c^3*d^
5*e^2 - 20*b^3*c*d^4*e^3 + 24*a^2*c^2*d^3*e^4 + 36*b^2*c^2*d^5*e^2 + 8*a^3*c*d*e^6 - 28*b*c^3*d^6*e - 60*a*b*c
^2*d^4*e^3 + 48*a*b^2*c*d^3*e^4 - 36*a^2*b*c*d^2*e^5))/e^9 + (c^4*x^7)/(7*e^2) - (a^4*e^8 + c^4*d^8 + b^4*d^4*
e^4 - 4*a*b^3*d^3*e^5 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 - 4*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 6*a^2*c^2*d^
4*e^4 + 6*b^2*c^2*d^6*e^2 - 4*a^3*b*d*e^7 - 4*b*c^3*d^7*e - 12*a*b*c^2*d^5*e^3 + 12*a*b^2*c*d^4*e^4 - 12*a^2*b
*c*d^3*e^5)/(e*(d*e^8 + e^9*x))

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sympy [B]  time = 4.55, size = 847, normalized size = 1.99 \begin {gather*} \frac {c^{4} x^{7}}{7 e^{2}} + x^{6} \left (\frac {2 b c^{3}}{3 e^{2}} - \frac {c^{4} d}{3 e^{3}}\right ) + x^{5} \left (\frac {4 a c^{3}}{5 e^{2}} + \frac {6 b^{2} c^{2}}{5 e^{2}} - \frac {8 b c^{3} d}{5 e^{3}} + \frac {3 c^{4} d^{2}}{5 e^{4}}\right ) + x^{4} \left (\frac {3 a b c^{2}}{e^{2}} - \frac {2 a c^{3} d}{e^{3}} + \frac {b^{3} c}{e^{2}} - \frac {3 b^{2} c^{2} d}{e^{3}} + \frac {3 b c^{3} d^{2}}{e^{4}} - \frac {c^{4} d^{3}}{e^{5}}\right ) + x^{3} \left (\frac {2 a^{2} c^{2}}{e^{2}} + \frac {4 a b^{2} c}{e^{2}} - \frac {8 a b c^{2} d}{e^{3}} + \frac {4 a c^{3} d^{2}}{e^{4}} + \frac {b^{4}}{3 e^{2}} - \frac {8 b^{3} c d}{3 e^{3}} + \frac {6 b^{2} c^{2} d^{2}}{e^{4}} - \frac {16 b c^{3} d^{3}}{3 e^{5}} + \frac {5 c^{4} d^{4}}{3 e^{6}}\right ) + x^{2} \left (\frac {6 a^{2} b c}{e^{2}} - \frac {6 a^{2} c^{2} d}{e^{3}} + \frac {2 a b^{3}}{e^{2}} - \frac {12 a b^{2} c d}{e^{3}} + \frac {18 a b c^{2} d^{2}}{e^{4}} - \frac {8 a c^{3} d^{3}}{e^{5}} - \frac {b^{4} d}{e^{3}} + \frac {6 b^{3} c d^{2}}{e^{4}} - \frac {12 b^{2} c^{2} d^{3}}{e^{5}} + \frac {10 b c^{3} d^{4}}{e^{6}} - \frac {3 c^{4} d^{5}}{e^{7}}\right ) + x \left (\frac {4 a^{3} c}{e^{2}} + \frac {6 a^{2} b^{2}}{e^{2}} - \frac {24 a^{2} b c d}{e^{3}} + \frac {18 a^{2} c^{2} d^{2}}{e^{4}} - \frac {8 a b^{3} d}{e^{3}} + \frac {36 a b^{2} c d^{2}}{e^{4}} - \frac {48 a b c^{2} d^{3}}{e^{5}} + \frac {20 a c^{3} d^{4}}{e^{6}} + \frac {3 b^{4} d^{2}}{e^{4}} - \frac {16 b^{3} c d^{3}}{e^{5}} + \frac {30 b^{2} c^{2} d^{4}}{e^{6}} - \frac {24 b c^{3} d^{5}}{e^{7}} + \frac {7 c^{4} d^{6}}{e^{8}}\right ) + \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}}{d e^{9} + e^{10} x} + \frac {4 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{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)**2,x)

[Out]

c**4*x**7/(7*e**2) + x**6*(2*b*c**3/(3*e**2) - c**4*d/(3*e**3)) + x**5*(4*a*c**3/(5*e**2) + 6*b**2*c**2/(5*e**
2) - 8*b*c**3*d/(5*e**3) + 3*c**4*d**2/(5*e**4)) + x**4*(3*a*b*c**2/e**2 - 2*a*c**3*d/e**3 + b**3*c/e**2 - 3*b
**2*c**2*d/e**3 + 3*b*c**3*d**2/e**4 - c**4*d**3/e**5) + x**3*(2*a**2*c**2/e**2 + 4*a*b**2*c/e**2 - 8*a*b*c**2
*d/e**3 + 4*a*c**3*d**2/e**4 + b**4/(3*e**2) - 8*b**3*c*d/(3*e**3) + 6*b**2*c**2*d**2/e**4 - 16*b*c**3*d**3/(3
*e**5) + 5*c**4*d**4/(3*e**6)) + x**2*(6*a**2*b*c/e**2 - 6*a**2*c**2*d/e**3 + 2*a*b**3/e**2 - 12*a*b**2*c*d/e*
*3 + 18*a*b*c**2*d**2/e**4 - 8*a*c**3*d**3/e**5 - b**4*d/e**3 + 6*b**3*c*d**2/e**4 - 12*b**2*c**2*d**3/e**5 +
10*b*c**3*d**4/e**6 - 3*c**4*d**5/e**7) + x*(4*a**3*c/e**2 + 6*a**2*b**2/e**2 - 24*a**2*b*c*d/e**3 + 18*a**2*c
**2*d**2/e**4 - 8*a*b**3*d/e**3 + 36*a*b**2*c*d**2/e**4 - 48*a*b*c**2*d**3/e**5 + 20*a*c**3*d**4/e**6 + 3*b**4
*d**2/e**4 - 16*b**3*c*d**3/e**5 + 30*b**2*c**2*d**4/e**6 - 24*b*c**3*d**5/e**7 + 7*c**4*d**6/e**8) + (-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)/(d*e**9 + e**10*x) + 4*(b*e - 2*
c*d)*(a*e**2 - b*d*e + c*d**2)**3*log(d + e*x)/e**9

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