3.22.56 \(\int \frac {(a+b x+c x^2)^4}{(d+e x)^6} \, dx\) [2156]

Optimal. Leaf size=414 \[ \frac {c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right ) x}{e^8}-\frac {c^3 (3 c d-2 b e) x^2}{e^7}+\frac {c^4 x^3}{3 e^6}-\frac {\left (c d^2-b d e+a e^2\right )^4}{5 e^9 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^4}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{3 e^9 (d+e x)^3}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^2}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{e^9 (d+e x)}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \log (d+e x)}{e^9} \]

[Out]

c^2*(21*c^2*d^2+6*b^2*e^2-4*c*e*(-a*e+6*b*d))*x/e^8-c^3*(-2*b*e+3*c*d)*x^2/e^7+1/3*c^4*x^3/e^6-1/5*(a*e^2-b*d*
e+c*d^2)^4/e^9/(e*x+d)^5+(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^4-2/3*(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)^3+2*(-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)^2+(-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)-4*c*(-b*e+2*c*d)*(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 = 0.40, antiderivative size = 414, 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}{e^9 (d+e x)}+\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 )}{e^9 (d+e x)^2}-\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 )}{3 e^9 (d+e x)^3}-\frac {4 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {c^2 x \left (-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right )}{e^8}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^4}{5 e^9 (d+e x)^5}-\frac {c^3 x^2 (3 c d-2 b e)}{e^7}+\frac {c^4 x^3}{3 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^6,x]

[Out]

(c^2*(21*c^2*d^2 + 6*b^2*e^2 - 4*c*e*(6*b*d - a*e))*x)/e^8 - (c^3*(3*c*d - 2*b*e)*x^2)/e^7 + (c^4*x^3)/(3*e^6)
- (c*d^2 - b*d*e + a*e^2)^4/(5*e^9*(d + e*x)^5) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)^4)
- (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)))/(3*e^9*(d + e*x)^3) + (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)))/(e^9*(d + e*x)^2) - (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))/(e^9*(d + e*x)) - (4*c*(2*c*d - b*e)*(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)^6} \, dx &=\int \left (\frac {c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right )}{e^8}-\frac {2 c^3 (3 c d-2 b e) x}{e^7}+\frac {c^4 x^2}{e^6}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^6}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^5}+\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)^4}+\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)^3}+\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)^2}+\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)}\right ) \, dx\\ &=\frac {c^2 \left (21 c^2 d^2+6 b^2 e^2-4 c e (6 b d-a e)\right ) x}{e^8}-\frac {c^3 (3 c d-2 b e) x^2}{e^7}+\frac {c^4 x^3}{3 e^6}-\frac {\left (c d^2-b d e+a e^2\right )^4}{5 e^9 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^4}-\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 )}{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 ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{e^9 (d+e x)^2}-\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^9 (d+e x)}-\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 ) \log (d+e x)}{e^9}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 419, normalized size = 1.01 \begin {gather*} \frac {15 c^2 e \left (21 c^2 d^2+6 b^2 e^2+4 c e (-6 b d+a e)\right ) x+15 c^3 e^2 (-3 c d+2 b e) x^2+5 c^4 e^3 x^3-\frac {3 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^5}+\frac {15 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^4}-\frac {10 \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)^3}+\frac {30 (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 )}{(d+e x)^2}-\frac {15 \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}-60 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 ) \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)^6,x]

[Out]

(15*c^2*e*(21*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(-6*b*d + a*e))*x + 15*c^3*e^2*(-3*c*d + 2*b*e)*x^2 + 5*c^4*e^3*x^3
- (3*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^5 + (15*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^4 -
(10*(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)^3 + (30*(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)))/(d + e*x)^2 - (15*(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) - 60*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7
*b*d + 3*a*e))*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. \(901\) vs. \(2(408)=816\).
time = 0.79, size = 902, normalized size = 2.18 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)^6,x,method=_RETURNVERBOSE)

[Out]

c^2/e^8*(1/3*c^2*e^2*x^3+2*b*c*e^2*x^2-3*c^2*d*e*x^2+4*a*c*e^2*x+6*b^2*e^2*x-24*b*c*d*e*x+21*d^2*c^2*x)-1/5*(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)^5-1/3*(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)^3-1/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)+4*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)*ln(e*x+d)-1/2*(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)^2-1/4*(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^3-3
6*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)^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 865 vs. \(2 (413) = 826\).
time = 0.31, size = 865, normalized size = 2.09 \begin {gather*} -4 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e - b^{3} c e^{3} - 3 \, a b c^{2} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d\right )} e^{\left (-9\right )} \log \left (x e + d\right ) + \frac {1}{3} \, {\left (c^{4} x^{3} e^{2} - 3 \, {\left (3 \, c^{4} d e - 2 \, b c^{3} e^{2}\right )} x^{2} + 3 \, {\left (21 \, c^{4} d^{2} - 24 \, b c^{3} d e + 6 \, b^{2} c^{2} e^{2} + 4 \, a c^{3} e^{2}\right )} x\right )} e^{\left (-8\right )} - \frac {743 \, c^{4} d^{8} - 1377 \, b c^{3} d^{7} e + 261 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{6} - 137 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{5} + 3 \, a^{3} b d e^{7} + 3 \, {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{4} + 15 \, {\left (70 \, c^{4} d^{4} e^{4} - 140 \, 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} + 30 \, {\left (3 \, b^{2} c^{2} e^{6} + 2 \, a c^{3} e^{6}\right )} d^{2} - 20 \, {\left (b^{3} c e^{7} + 3 \, a b c^{2} e^{7}\right )} d\right )} x^{4} + 3 \, a^{4} e^{8} + 3 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d^{3} + 30 \, {\left (126 \, c^{4} d^{5} e^{3} - 245 \, b c^{3} d^{4} e^{4} + a b^{3} e^{8} + 3 \, a^{2} b c e^{8} + 50 \, {\left (3 \, b^{2} c^{2} e^{5} + 2 \, a c^{3} e^{5}\right )} d^{3} - 30 \, {\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} + {\left (3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} d^{2} + 10 \, {\left (518 \, c^{4} d^{6} e^{2} - 987 \, b c^{3} d^{5} e^{3} + 195 \, {\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} - 110 \, {\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} + 3 \, {\left (a b^{3} e^{7} + 3 \, a^{2} b c e^{7}\right )} d\right )} x^{2} + 5 \, {\left (638 \, c^{4} d^{7} e - 1197 \, b c^{3} d^{6} e^{2} + 231 \, {\left (3 \, b^{2} c^{2} e^{3} + 2 \, a c^{3} e^{3}\right )} d^{5} - 125 \, {\left (b^{3} c e^{4} + 3 \, a b c^{2} e^{4}\right )} d^{4} + 3 \, a^{3} b e^{8} + 3 \, {\left (b^{4} e^{5} + 12 \, a b^{2} c e^{5} + 6 \, a^{2} c^{2} e^{5}\right )} d^{3} + 3 \, {\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}{15 \, {\left (x^{5} e^{14} + 5 \, d x^{4} e^{13} + 10 \, d^{2} x^{3} e^{12} + 10 \, d^{3} x^{2} e^{11} + 5 \, d^{4} x e^{10} + d^{5} 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)^6,x, algorithm="maxima")

[Out]

-4*(14*c^4*d^3 - 21*b*c^3*d^2*e - b^3*c*e^3 - 3*a*b*c^2*e^3 + 3*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d)*e^(-9)*log(x*
e + d) + 1/3*(c^4*x^3*e^2 - 3*(3*c^4*d*e - 2*b*c^3*e^2)*x^2 + 3*(21*c^4*d^2 - 24*b*c^3*d*e + 6*b^2*c^2*e^2 + 4
*a*c^3*e^2)*x)*e^(-8) - 1/15*(743*c^4*d^8 - 1377*b*c^3*d^7*e + 261*(3*b^2*c^2*e^2 + 2*a*c^3*e^2)*d^6 - 137*(b^
3*c*e^3 + 3*a*b*c^2*e^3)*d^5 + 3*a^3*b*d*e^7 + 3*(b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*d^4 + 15*(70*c^4*d
^4*e^4 - 140*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 + 30*(3*b^2*c^2*e^6 + 2*a*c^3*e^6)*d^2 -
20*(b^3*c*e^7 + 3*a*b*c^2*e^7)*d)*x^4 + 3*a^4*e^8 + 3*(a*b^3*e^5 + 3*a^2*b*c*e^5)*d^3 + 30*(126*c^4*d^5*e^3 -
245*b*c^3*d^4*e^4 + a*b^3*e^8 + 3*a^2*b*c*e^8 + 50*(3*b^2*c^2*e^5 + 2*a*c^3*e^5)*d^3 - 30*(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 + (3*a^2*b^2*e^6 + 2*a^3*c*e^6)*d^2 + 10*(518
*c^4*d^6*e^2 - 987*b*c^3*d^5*e^3 + 195*(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 - 110*(
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 + 3*(a*b^3*e^7 + 3*a^2*b*c*e
^7)*d)*x^2 + 5*(638*c^4*d^7*e - 1197*b*c^3*d^6*e^2 + 231*(3*b^2*c^2*e^3 + 2*a*c^3*e^3)*d^5 - 125*(b^3*c*e^4 +
3*a*b*c^2*e^4)*d^4 + 3*a^3*b*e^8 + 3*(b^4*e^5 + 12*a*b^2*c*e^5 + 6*a^2*c^2*e^5)*d^3 + 3*(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^5*e^14 + 5*d*x^4*e^13 + 10*d^2*x^3*e^12 + 10*d^3*x^2*e^11 +
5*d^4*x*e^10 + d^5*e^9)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1209 vs. \(2 (413) = 826\).
time = 2.31, size = 1209, normalized size = 2.92 \begin {gather*} -\frac {743 \, c^{4} d^{8} - {\left (5 \, c^{4} x^{8} + 30 \, b c^{3} x^{7} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{6} - 15 \, a^{3} b x - 15 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{4} - 3 \, a^{4} - 30 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{3} - 10 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{2}\right )} e^{8} + {\left (20 \, c^{4} d x^{7} + 210 \, b c^{3} d x^{6} - 150 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{5} - 300 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{4} + 3 \, a^{3} b d + 30 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{3} + 30 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{2} + 5 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x\right )} e^{7} - {\left (140 \, c^{4} d^{2} x^{6} - 1500 \, b c^{3} d^{2} x^{5} - 150 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{4} + 900 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{3} - 30 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{2} - 15 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x - {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} e^{6} - {\left (1175 \, c^{4} d^{3} x^{5} - 1200 \, b c^{3} d^{3} x^{4} - 1200 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{3} + 1100 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{2} - 15 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3}\right )} e^{5} - {\left (1675 \, c^{4} d^{4} x^{4} + 3900 \, b c^{3} d^{4} x^{3} - 1800 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2} + 625 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} x - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4}\right )} e^{4} + {\left (850 \, c^{4} d^{5} x^{3} - 8100 \, b c^{3} d^{5} x^{2} + 1125 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x - 137 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5}\right )} e^{3} + {\left (3650 \, c^{4} d^{6} x^{2} - 5625 \, b c^{3} d^{6} x + 261 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6}\right )} e^{2} + {\left (2875 \, c^{4} d^{7} x - 1377 \, b c^{3} d^{7}\right )} e + 60 \, {\left (14 \, c^{4} d^{8} - {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{5} e^{8} + {\left (3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{5} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{4}\right )} e^{7} - {\left (21 \, b c^{3} d^{2} x^{5} - 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{4} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{3}\right )} e^{6} + {\left (14 \, c^{4} d^{3} x^{5} - 105 \, b c^{3} d^{3} x^{4} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{3} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{2}\right )} e^{5} + 5 \, {\left (14 \, c^{4} d^{4} x^{4} - 42 \, b c^{3} d^{4} x^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2} - {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} x\right )} e^{4} + {\left (140 \, c^{4} d^{5} x^{3} - 210 \, b c^{3} d^{5} x^{2} + 15 \, {\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} + {\left (140 \, c^{4} d^{6} x^{2} - 105 \, b c^{3} d^{6} x + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6}\right )} e^{2} + 7 \, {\left (10 \, c^{4} d^{7} x - 3 \, b c^{3} d^{7}\right )} e\right )} \log \left (x e + d\right )}{15 \, {\left (x^{5} e^{14} + 5 \, d x^{4} e^{13} + 10 \, d^{2} x^{3} e^{12} + 10 \, d^{3} x^{2} e^{11} + 5 \, d^{4} x e^{10} + d^{5} 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)^6,x, algorithm="fricas")

[Out]

-1/15*(743*c^4*d^8 - (5*c^4*x^8 + 30*b*c^3*x^7 + 30*(3*b^2*c^2 + 2*a*c^3)*x^6 - 15*a^3*b*x - 15*(b^4 + 12*a*b^
2*c + 6*a^2*c^2)*x^4 - 3*a^4 - 30*(a*b^3 + 3*a^2*b*c)*x^3 - 10*(3*a^2*b^2 + 2*a^3*c)*x^2)*e^8 + (20*c^4*d*x^7
+ 210*b*c^3*d*x^6 - 150*(3*b^2*c^2 + 2*a*c^3)*d*x^5 - 300*(b^3*c + 3*a*b*c^2)*d*x^4 + 3*a^3*b*d + 30*(b^4 + 12
*a*b^2*c + 6*a^2*c^2)*d*x^3 + 30*(a*b^3 + 3*a^2*b*c)*d*x^2 + 5*(3*a^2*b^2 + 2*a^3*c)*d*x)*e^7 - (140*c^4*d^2*x
^6 - 1500*b*c^3*d^2*x^5 - 150*(3*b^2*c^2 + 2*a*c^3)*d^2*x^4 + 900*(b^3*c + 3*a*b*c^2)*d^2*x^3 - 30*(b^4 + 12*a
*b^2*c + 6*a^2*c^2)*d^2*x^2 - 15*(a*b^3 + 3*a^2*b*c)*d^2*x - (3*a^2*b^2 + 2*a^3*c)*d^2)*e^6 - (1175*c^4*d^3*x^
5 - 1200*b*c^3*d^3*x^4 - 1200*(3*b^2*c^2 + 2*a*c^3)*d^3*x^3 + 1100*(b^3*c + 3*a*b*c^2)*d^3*x^2 - 15*(b^4 + 12*
a*b^2*c + 6*a^2*c^2)*d^3*x - 3*(a*b^3 + 3*a^2*b*c)*d^3)*e^5 - (1675*c^4*d^4*x^4 + 3900*b*c^3*d^4*x^3 - 1800*(3
*b^2*c^2 + 2*a*c^3)*d^4*x^2 + 625*(b^3*c + 3*a*b*c^2)*d^4*x - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4)*e^4 + (850
*c^4*d^5*x^3 - 8100*b*c^3*d^5*x^2 + 1125*(3*b^2*c^2 + 2*a*c^3)*d^5*x - 137*(b^3*c + 3*a*b*c^2)*d^5)*e^3 + (365
0*c^4*d^6*x^2 - 5625*b*c^3*d^6*x + 261*(3*b^2*c^2 + 2*a*c^3)*d^6)*e^2 + (2875*c^4*d^7*x - 1377*b*c^3*d^7)*e +
60*(14*c^4*d^8 - (b^3*c + 3*a*b*c^2)*x^5*e^8 + (3*(3*b^2*c^2 + 2*a*c^3)*d*x^5 - 5*(b^3*c + 3*a*b*c^2)*d*x^4)*e
^7 - (21*b*c^3*d^2*x^5 - 15*(3*b^2*c^2 + 2*a*c^3)*d^2*x^4 + 10*(b^3*c + 3*a*b*c^2)*d^2*x^3)*e^6 + (14*c^4*d^3*
x^5 - 105*b*c^3*d^3*x^4 + 30*(3*b^2*c^2 + 2*a*c^3)*d^3*x^3 - 10*(b^3*c + 3*a*b*c^2)*d^3*x^2)*e^5 + 5*(14*c^4*d
^4*x^4 - 42*b*c^3*d^4*x^3 + 6*(3*b^2*c^2 + 2*a*c^3)*d^4*x^2 - (b^3*c + 3*a*b*c^2)*d^4*x)*e^4 + (140*c^4*d^5*x^
3 - 210*b*c^3*d^5*x^2 + 15*(3*b^2*c^2 + 2*a*c^3)*d^5*x - (b^3*c + 3*a*b*c^2)*d^5)*e^3 + (140*c^4*d^6*x^2 - 105
*b*c^3*d^6*x + 3*(3*b^2*c^2 + 2*a*c^3)*d^6)*e^2 + 7*(10*c^4*d^7*x - 3*b*c^3*d^7)*e)*log(x*e + d))/(x^5*e^14 +
5*d*x^4*e^13 + 10*d^2*x^3*e^12 + 10*d^3*x^2*e^11 + 5*d^4*x*e^10 + d^5*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)**6,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 841 vs. \(2 (413) = 826\).
time = 1.45, size = 841, normalized size = 2.03 \begin {gather*} -4 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 9 \, b^{2} c^{2} d e^{2} + 6 \, a c^{3} d e^{2} - b^{3} c e^{3} - 3 \, a b c^{2} e^{3}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{3} \, {\left (c^{4} x^{3} e^{12} - 9 \, c^{4} d x^{2} e^{11} + 63 \, c^{4} d^{2} x e^{10} + 6 \, b c^{3} x^{2} e^{12} - 72 \, b c^{3} d x e^{11} + 18 \, b^{2} c^{2} x e^{12} + 12 \, a c^{3} x e^{12}\right )} e^{\left (-18\right )} - \frac {{\left (743 \, c^{4} d^{8} - 1377 \, b c^{3} d^{7} e + 783 \, b^{2} c^{2} d^{6} e^{2} + 522 \, a c^{3} d^{6} e^{2} - 137 \, b^{3} c d^{5} e^{3} - 411 \, a b c^{2} d^{5} e^{3} + 3 \, b^{4} d^{4} e^{4} + 36 \, a b^{2} c d^{4} e^{4} + 18 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a b^{3} d^{3} e^{5} + 9 \, a^{2} b c d^{3} e^{5} + 3 \, a^{2} b^{2} d^{2} e^{6} + 2 \, a^{3} c d^{2} e^{6} + 3 \, a^{3} b d e^{7} + 15 \, {\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 )} x^{4} + 3 \, a^{4} e^{8} + 30 \, {\left (126 \, c^{4} d^{5} e^{3} - 245 \, b c^{3} d^{4} e^{4} + 150 \, b^{2} c^{2} d^{3} e^{5} + 100 \, a c^{3} d^{3} e^{5} - 30 \, b^{3} c d^{2} e^{6} - 90 \, a b c^{2} d^{2} e^{6} + b^{4} d e^{7} + 12 \, a b^{2} c d e^{7} + 6 \, a^{2} c^{2} d e^{7} + a b^{3} e^{8} + 3 \, a^{2} b c e^{8}\right )} x^{3} + 10 \, {\left (518 \, c^{4} d^{6} e^{2} - 987 \, b c^{3} d^{5} e^{3} + 585 \, b^{2} c^{2} d^{4} e^{4} + 390 \, a c^{3} d^{4} e^{4} - 110 \, b^{3} c d^{3} e^{5} - 330 \, 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} + 3 \, a b^{3} d e^{7} + 9 \, a^{2} b c d e^{7} + 3 \, a^{2} b^{2} e^{8} + 2 \, a^{3} c e^{8}\right )} x^{2} + 5 \, {\left (638 \, c^{4} d^{7} e - 1197 \, b c^{3} d^{6} e^{2} + 693 \, b^{2} c^{2} d^{5} e^{3} + 462 \, a c^{3} d^{5} e^{3} - 125 \, b^{3} c d^{4} e^{4} - 375 \, a b c^{2} d^{4} e^{4} + 3 \, b^{4} d^{3} e^{5} + 36 \, a b^{2} c d^{3} e^{5} + 18 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a b^{3} d^{2} e^{6} + 9 \, a^{2} b c d^{2} e^{6} + 3 \, a^{2} b^{2} d e^{7} + 2 \, a^{3} c d e^{7} + 3 \, a^{3} b e^{8}\right )} x\right )} e^{\left (-9\right )}}{15 \, {\left (x e + d\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^6,x, algorithm="giac")

[Out]

-4*(14*c^4*d^3 - 21*b*c^3*d^2*e + 9*b^2*c^2*d*e^2 + 6*a*c^3*d*e^2 - b^3*c*e^3 - 3*a*b*c^2*e^3)*e^(-9)*log(abs(
x*e + d)) + 1/3*(c^4*x^3*e^12 - 9*c^4*d*x^2*e^11 + 63*c^4*d^2*x*e^10 + 6*b*c^3*x^2*e^12 - 72*b*c^3*d*x*e^11 +
18*b^2*c^2*x*e^12 + 12*a*c^3*x*e^12)*e^(-18) - 1/15*(743*c^4*d^8 - 1377*b*c^3*d^7*e + 783*b^2*c^2*d^6*e^2 + 52
2*a*c^3*d^6*e^2 - 137*b^3*c*d^5*e^3 - 411*a*b*c^2*d^5*e^3 + 3*b^4*d^4*e^4 + 36*a*b^2*c*d^4*e^4 + 18*a^2*c^2*d^
4*e^4 + 3*a*b^3*d^3*e^5 + 9*a^2*b*c*d^3*e^5 + 3*a^2*b^2*d^2*e^6 + 2*a^3*c*d^2*e^6 + 3*a^3*b*d*e^7 + 15*(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)*x^4 + 3*a^4*e^8 + 30*(126*c^4*d^5*e^3 - 245*b*c^3*d^4*e^4 + 150*b^2*c^2
*d^3*e^5 + 100*a*c^3*d^3*e^5 - 30*b^3*c*d^2*e^6 - 90*a*b*c^2*d^2*e^6 + b^4*d*e^7 + 12*a*b^2*c*d*e^7 + 6*a^2*c^
2*d*e^7 + a*b^3*e^8 + 3*a^2*b*c*e^8)*x^3 + 10*(518*c^4*d^6*e^2 - 987*b*c^3*d^5*e^3 + 585*b^2*c^2*d^4*e^4 + 390
*a*c^3*d^4*e^4 - 110*b^3*c*d^3*e^5 - 330*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 + 3*a*b^3*d*e^7 + 9*a^2*b*c*d*e^7 + 3*a^2*b^2*e^8 + 2*a^3*c*e^8)*x^2 + 5*(638*c^4*d^7*e - 1197*b*c^3*d^6*
e^2 + 693*b^2*c^2*d^5*e^3 + 462*a*c^3*d^5*e^3 - 125*b^3*c*d^4*e^4 - 375*a*b*c^2*d^4*e^4 + 3*b^4*d^3*e^5 + 36*a
*b^2*c*d^3*e^5 + 18*a^2*c^2*d^3*e^5 + 3*a*b^3*d^2*e^6 + 9*a^2*b*c*d^2*e^6 + 3*a^2*b^2*d*e^7 + 2*a^3*c*d*e^7 +
3*a^3*b*e^8)*x)*e^(-9)/(x*e + d)^5

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Mupad [B]
time = 0.21, size = 959, normalized size = 2.32 \begin {gather*} x^2\,\left (\frac {2\,b\,c^3}{e^6}-\frac {3\,c^4\,d}{e^7}\right )-x\,\left (\frac {6\,d\,\left (\frac {4\,b\,c^3}{e^6}-\frac {6\,c^4\,d}{e^7}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^6}+\frac {15\,c^4\,d^2}{e^8}\right )-\frac {x^3\,\left (6\,a^2\,b\,c\,e^7+12\,a^2\,c^2\,d\,e^6+2\,a\,b^3\,e^7+24\,a\,b^2\,c\,d\,e^6-180\,a\,b\,c^2\,d^2\,e^5+200\,a\,c^3\,d^3\,e^4+2\,b^4\,d\,e^6-60\,b^3\,c\,d^2\,e^5+300\,b^2\,c^2\,d^3\,e^4-490\,b\,c^3\,d^4\,e^3+252\,c^4\,d^5\,e^2\right )+x\,\left (a^3\,b\,e^7+\frac {2\,a^3\,c\,d\,e^6}{3}+a^2\,b^2\,d\,e^6+3\,a^2\,b\,c\,d^2\,e^5+6\,a^2\,c^2\,d^3\,e^4+a\,b^3\,d^2\,e^5+12\,a\,b^2\,c\,d^3\,e^4-125\,a\,b\,c^2\,d^4\,e^3+154\,a\,c^3\,d^5\,e^2+b^4\,d^3\,e^4-\frac {125\,b^3\,c\,d^4\,e^3}{3}+231\,b^2\,c^2\,d^5\,e^2-399\,b\,c^3\,d^6\,e+\frac {638\,c^4\,d^7}{3}\right )+x^4\,\left (6\,a^2\,c^2\,e^7+12\,a\,b^2\,c\,e^7-60\,a\,b\,c^2\,d\,e^6+60\,a\,c^3\,d^2\,e^5+b^4\,e^7-20\,b^3\,c\,d\,e^6+90\,b^2\,c^2\,d^2\,e^5-140\,b\,c^3\,d^3\,e^4+70\,c^4\,d^4\,e^3\right )+\frac {3\,a^4\,e^8+3\,a^3\,b\,d\,e^7+2\,a^3\,c\,d^2\,e^6+3\,a^2\,b^2\,d^2\,e^6+9\,a^2\,b\,c\,d^3\,e^5+18\,a^2\,c^2\,d^4\,e^4+3\,a\,b^3\,d^3\,e^5+36\,a\,b^2\,c\,d^4\,e^4-411\,a\,b\,c^2\,d^5\,e^3+522\,a\,c^3\,d^6\,e^2+3\,b^4\,d^4\,e^4-137\,b^3\,c\,d^5\,e^3+783\,b^2\,c^2\,d^6\,e^2-1377\,b\,c^3\,d^7\,e+743\,c^4\,d^8}{15\,e}+x^2\,\left (\frac {4\,a^3\,c\,e^7}{3}+2\,a^2\,b^2\,e^7+6\,a^2\,b\,c\,d\,e^6+12\,a^2\,c^2\,d^2\,e^5+2\,a\,b^3\,d\,e^6+24\,a\,b^2\,c\,d^2\,e^5-220\,a\,b\,c^2\,d^3\,e^4+260\,a\,c^3\,d^4\,e^3+2\,b^4\,d^2\,e^5-\frac {220\,b^3\,c\,d^3\,e^4}{3}+390\,b^2\,c^2\,d^4\,e^3-658\,b\,c^3\,d^5\,e^2+\frac {1036\,c^4\,d^6\,e}{3}\right )}{d^5\,e^8+5\,d^4\,e^9\,x+10\,d^3\,e^{10}\,x^2+10\,d^2\,e^{11}\,x^3+5\,d\,e^{12}\,x^4+e^{13}\,x^5}-\frac {\ln \left (d+e\,x\right )\,\left (-4\,b^3\,c\,e^3+36\,b^2\,c^2\,d\,e^2-84\,b\,c^3\,d^2\,e-12\,a\,b\,c^2\,e^3+56\,c^4\,d^3+24\,a\,c^3\,d\,e^2\right )}{e^9}+\frac {c^4\,x^3}{3\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^4/(d + e*x)^6,x)

[Out]

x^2*((2*b*c^3)/e^6 - (3*c^4*d)/e^7) - x*((6*d*((4*b*c^3)/e^6 - (6*c^4*d)/e^7))/e - (4*a*c^3 + 6*b^2*c^2)/e^6 +
(15*c^4*d^2)/e^8) - (x^3*(2*a*b^3*e^7 + 2*b^4*d*e^6 + 252*c^4*d^5*e^2 + 200*a*c^3*d^3*e^4 + 12*a^2*c^2*d*e^6
- 490*b*c^3*d^4*e^3 - 60*b^3*c*d^2*e^5 + 300*b^2*c^2*d^3*e^4 + 6*a^2*b*c*e^7 + 24*a*b^2*c*d*e^6 - 180*a*b*c^2*
d^2*e^5) + x*((638*c^4*d^7)/3 + a^3*b*e^7 + b^4*d^3*e^4 + a*b^3*d^2*e^5 + a^2*b^2*d*e^6 + 154*a*c^3*d^5*e^2 -
(125*b^3*c*d^4*e^3)/3 + 6*a^2*c^2*d^3*e^4 + 231*b^2*c^2*d^5*e^2 + (2*a^3*c*d*e^6)/3 - 399*b*c^3*d^6*e - 125*a*
b*c^2*d^4*e^3 + 12*a*b^2*c*d^3*e^4 + 3*a^2*b*c*d^2*e^5) + x^4*(b^4*e^7 + 6*a^2*c^2*e^7 + 70*c^4*d^4*e^3 + 60*a
*c^3*d^2*e^5 - 140*b*c^3*d^3*e^4 + 90*b^2*c^2*d^2*e^5 + 12*a*b^2*c*e^7 - 20*b^3*c*d*e^6 - 60*a*b*c^2*d*e^6) +
(3*a^4*e^8 + 743*c^4*d^8 + 3*b^4*d^4*e^4 + 3*a*b^3*d^3*e^5 + 522*a*c^3*d^6*e^2 + 2*a^3*c*d^2*e^6 - 137*b^3*c*d
^5*e^3 + 3*a^2*b^2*d^2*e^6 + 18*a^2*c^2*d^4*e^4 + 783*b^2*c^2*d^6*e^2 + 3*a^3*b*d*e^7 - 1377*b*c^3*d^7*e - 411
*a*b*c^2*d^5*e^3 + 36*a*b^2*c*d^4*e^4 + 9*a^2*b*c*d^3*e^5)/(15*e) + x^2*((4*a^3*c*e^7)/3 + (1036*c^4*d^6*e)/3
+ 2*a^2*b^2*e^7 + 2*b^4*d^2*e^5 + 260*a*c^3*d^4*e^3 - 658*b*c^3*d^5*e^2 - (220*b^3*c*d^3*e^4)/3 + 12*a^2*c^2*d
^2*e^5 + 390*b^2*c^2*d^4*e^3 + 2*a*b^3*d*e^6 + 6*a^2*b*c*d*e^6 - 220*a*b*c^2*d^3*e^4 + 24*a*b^2*c*d^2*e^5))/(d
^5*e^8 + e^13*x^5 + 5*d^4*e^9*x + 5*d*e^12*x^4 + 10*d^3*e^10*x^2 + 10*d^2*e^11*x^3) - (log(d + e*x)*(56*c^4*d^
3 - 4*b^3*c*e^3 + 36*b^2*c^2*d*e^2 - 12*a*b*c^2*e^3 + 24*a*c^3*d*e^2 - 84*b*c^3*d^2*e))/e^9 + (c^4*x^3)/(3*e^6
)

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