3.12.33 \(\int \frac {(A+B x) (a+c x^2)^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=189 \[ -\frac {2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6 (d+e x)^4}+\frac {c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (d+e x)^2}-\frac {c^2 (5 B d-A e) \log (d+e x)}{e^6}+\frac {B c^2 x}{e^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.17, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \begin {gather*} -\frac {2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)}+\frac {c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (d+e x)^2}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6 (d+e x)^4}-\frac {c^2 (5 B d-A e) \log (d+e x)}{e^6}+\frac {B c^2 x}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^5,x]

[Out]

(B*c^2*x)/e^5 + ((B*d - A*e)*(c*d^2 + a*e^2)^2)/(4*e^6*(d + e*x)^4) - ((c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e
+ a*B*e^2))/(3*e^6*(d + e*x)^3) + (c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(e^6*(d + e*x)^2) - (2
*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(e^6*(d + e*x)) - (c^2*(5*B*d - A*e)*Log[d + e*x])/e^6

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^5} \, dx &=\int \left (\frac {B c^2}{e^5}+\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^5}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^4}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^3}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^2}+\frac {c^2 (-5 B d+A e)}{e^5 (d+e x)}\right ) \, dx\\ &=\frac {B c^2 x}{e^5}+\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{4 e^6 (d+e x)^4}-\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{3 e^6 (d+e x)^3}+\frac {c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^6 (d+e x)^2}-\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right )}{e^6 (d+e x)}-\frac {c^2 (5 B d-A e) \log (d+e x)}{e^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 221, normalized size = 1.17 \begin {gather*} \frac {A e \left (-3 a^2 e^4-2 a c e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-B \left (a^2 e^4 (d+4 e x)+6 a c e^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+c^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )-12 c^2 (d+e x)^4 (5 B d-A e) \log (d+e x)}{12 e^6 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^5,x]

[Out]

(A*e*(-3*a^2*e^4 - 2*a*c*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + c^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3
*x^3)) - B*(a^2*e^4*(d + 4*e*x) + 6*a*c*e^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + c^2*(77*d^5 + 248*d^
4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) - 12*c^2*(5*B*d - A*e)*(d + e*x)^4*Log[
d + e*x])/(12*e^6*(d + e*x)^4)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^5,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^5, x]

________________________________________________________________________________________

fricas [B]  time = 0.39, size = 405, normalized size = 2.14 \begin {gather*} \frac {12 \, B c^{2} e^{5} x^{5} + 48 \, B c^{2} d e^{4} x^{4} - 77 \, B c^{2} d^{5} + 25 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} - B a^{2} d e^{4} - 3 \, A a^{2} e^{5} - 24 \, {\left (2 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} - 12 \, {\left (21 \, B c^{2} d^{3} e^{2} - 9 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} - 4 \, {\left (62 \, B c^{2} d^{4} e - 22 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 2 \, A a c d e^{4} + B a^{2} e^{5}\right )} x - 12 \, {\left (5 \, B c^{2} d^{5} - A c^{2} d^{4} e + {\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 4 \, {\left (5 \, B c^{2} d^{2} e^{3} - A c^{2} d e^{4}\right )} x^{3} + 6 \, {\left (5 \, B c^{2} d^{3} e^{2} - A c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (5 \, B c^{2} d^{4} e - A c^{2} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/12*(12*B*c^2*e^5*x^5 + 48*B*c^2*d*e^4*x^4 - 77*B*c^2*d^5 + 25*A*c^2*d^4*e - 6*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^
3 - B*a^2*d*e^4 - 3*A*a^2*e^5 - 24*(2*B*c^2*d^2*e^3 - 2*A*c^2*d*e^4 + B*a*c*e^5)*x^3 - 12*(21*B*c^2*d^3*e^2 -
9*A*c^2*d^2*e^3 + 3*B*a*c*d*e^4 + A*a*c*e^5)*x^2 - 4*(62*B*c^2*d^4*e - 22*A*c^2*d^3*e^2 + 6*B*a*c*d^2*e^3 + 2*
A*a*c*d*e^4 + B*a^2*e^5)*x - 12*(5*B*c^2*d^5 - A*c^2*d^4*e + (5*B*c^2*d*e^4 - A*c^2*e^5)*x^4 + 4*(5*B*c^2*d^2*
e^3 - A*c^2*d*e^4)*x^3 + 6*(5*B*c^2*d^3*e^2 - A*c^2*d^2*e^3)*x^2 + 4*(5*B*c^2*d^4*e - A*c^2*d^3*e^2)*x)*log(e*
x + d))/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)

________________________________________________________________________________________

giac [B]  time = 0.17, size = 372, normalized size = 1.97 \begin {gather*} {\left (x e + d\right )} B c^{2} e^{\left (-6\right )} + {\left (5 \, B c^{2} d - A c^{2} e\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {1}{12} \, {\left (\frac {120 \, B c^{2} d^{2} e^{22}}{x e + d} - \frac {60 \, B c^{2} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac {20 \, B c^{2} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B c^{2} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac {48 \, A c^{2} d e^{23}}{x e + d} + \frac {36 \, A c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac {16 \, A c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac {3 \, A c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {24 \, B a c e^{24}}{x e + d} - \frac {36 \, B a c d e^{24}}{{\left (x e + d\right )}^{2}} + \frac {24 \, B a c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac {6 \, B a c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac {12 \, A a c e^{25}}{{\left (x e + d\right )}^{2}} - \frac {16 \, A a c d e^{25}}{{\left (x e + d\right )}^{3}} + \frac {6 \, A a c d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a^{2} e^{26}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a^{2} d e^{26}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a^{2} e^{27}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^5,x, algorithm="giac")

[Out]

(x*e + d)*B*c^2*e^(-6) + (5*B*c^2*d - A*c^2*e)*e^(-6)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*(120*B*c^2*d
^2*e^22/(x*e + d) - 60*B*c^2*d^3*e^22/(x*e + d)^2 + 20*B*c^2*d^4*e^22/(x*e + d)^3 - 3*B*c^2*d^5*e^22/(x*e + d)
^4 - 48*A*c^2*d*e^23/(x*e + d) + 36*A*c^2*d^2*e^23/(x*e + d)^2 - 16*A*c^2*d^3*e^23/(x*e + d)^3 + 3*A*c^2*d^4*e
^23/(x*e + d)^4 + 24*B*a*c*e^24/(x*e + d) - 36*B*a*c*d*e^24/(x*e + d)^2 + 24*B*a*c*d^2*e^24/(x*e + d)^3 - 6*B*
a*c*d^3*e^24/(x*e + d)^4 + 12*A*a*c*e^25/(x*e + d)^2 - 16*A*a*c*d*e^25/(x*e + d)^3 + 6*A*a*c*d^2*e^25/(x*e + d
)^4 + 4*B*a^2*e^26/(x*e + d)^3 - 3*B*a^2*d*e^26/(x*e + d)^4 + 3*A*a^2*e^27/(x*e + d)^4)*e^(-28)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 356, normalized size = 1.88 \begin {gather*} -\frac {A \,a^{2}}{4 \left (e x +d \right )^{4} e}-\frac {A a c \,d^{2}}{2 \left (e x +d \right )^{4} e^{3}}-\frac {A \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}+\frac {B \,a^{2} d}{4 \left (e x +d \right )^{4} e^{2}}+\frac {B a c \,d^{3}}{2 \left (e x +d \right )^{4} e^{4}}+\frac {B \,c^{2} d^{5}}{4 \left (e x +d \right )^{4} e^{6}}+\frac {4 A a c d}{3 \left (e x +d \right )^{3} e^{3}}+\frac {4 A \,c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {B \,a^{2}}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 B a c \,d^{2}}{\left (e x +d \right )^{3} e^{4}}-\frac {5 B \,c^{2} d^{4}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {A a c}{\left (e x +d \right )^{2} e^{3}}-\frac {3 A \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{5}}+\frac {3 B a c d}{\left (e x +d \right )^{2} e^{4}}+\frac {5 B \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{6}}+\frac {4 A \,c^{2} d}{\left (e x +d \right ) e^{5}}+\frac {A \,c^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {2 B a c}{\left (e x +d \right ) e^{4}}-\frac {10 B \,c^{2} d^{2}}{\left (e x +d \right ) e^{6}}-\frac {5 B \,c^{2} d \ln \left (e x +d \right )}{e^{6}}+\frac {B \,c^{2} x}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2/(e*x+d)^5,x)

[Out]

B*c^2/e^5*x+4/(e*x+d)*A*c^2*d/e^5-2*c/e^4/(e*x+d)*B*a-10/(e*x+d)*B*c^2*d^2/e^6-1/4/e/(e*x+d)^4*A*a^2-1/2/e^3/(
e*x+d)^4*A*d^2*a*c-1/4/(e*x+d)^4*A*c^2*d^4/e^5+1/4/e^2/(e*x+d)^4*B*d*a^2+1/2/e^4/(e*x+d)^4*B*d^3*a*c+1/4/(e*x+
d)^4*B*c^2*d^5/e^6-c/e^3/(e*x+d)^2*a*A-3/(e*x+d)^2*A*c^2*d^2/e^5+3*c/e^4/(e*x+d)^2*a*B*d+5/(e*x+d)^2*B*c^2*d^3
/e^6+4/3/e^3/(e*x+d)^3*A*a*c*d+4/3/(e*x+d)^3*A*c^2*d^3/e^5-1/3/e^2/(e*x+d)^3*B*a^2-2/e^4/(e*x+d)^3*B*a*c*d^2-5
/3/(e*x+d)^3*B*c^2*d^4/e^6+A*c^2/e^5*ln(e*x+d)-5*B*c^2*d/e^6*ln(e*x+d)

________________________________________________________________________________________

maxima [A]  time = 0.58, size = 279, normalized size = 1.48 \begin {gather*} -\frac {77 \, B c^{2} d^{5} - 25 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} + 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + 3 \, A a^{2} e^{5} + 24 \, {\left (5 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 12 \, {\left (25 \, B c^{2} d^{3} e^{2} - 9 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} + 4 \, {\left (65 \, B c^{2} d^{4} e - 22 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 2 \, A a c d e^{4} + B a^{2} e^{5}\right )} x}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac {B c^{2} x}{e^{5}} - \frac {{\left (5 \, B c^{2} d - A c^{2} e\right )} \log \left (e x + d\right )}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/12*(77*B*c^2*d^5 - 25*A*c^2*d^4*e + 6*B*a*c*d^3*e^2 + 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 + 3*A*a^2*e^5 + 24*(5*B
*c^2*d^2*e^3 - 2*A*c^2*d*e^4 + B*a*c*e^5)*x^3 + 12*(25*B*c^2*d^3*e^2 - 9*A*c^2*d^2*e^3 + 3*B*a*c*d*e^4 + A*a*c
*e^5)*x^2 + 4*(65*B*c^2*d^4*e - 22*A*c^2*d^3*e^2 + 6*B*a*c*d^2*e^3 + 2*A*a*c*d*e^4 + B*a^2*e^5)*x)/(e^10*x^4 +
4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6) + B*c^2*x/e^5 - (5*B*c^2*d - A*c^2*e)*log(e*x + d)/e^6

________________________________________________________________________________________

mupad [B]  time = 1.84, size = 277, normalized size = 1.47 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,c^2\,e-5\,B\,c^2\,d\right )}{e^6}-\frac {x^3\,\left (10\,B\,c^2\,d^2\,e^2-4\,A\,c^2\,d\,e^3+2\,B\,a\,c\,e^4\right )+x\,\left (\frac {B\,a^2\,e^4}{3}+2\,B\,a\,c\,d^2\,e^2+\frac {2\,A\,a\,c\,d\,e^3}{3}+\frac {65\,B\,c^2\,d^4}{3}-\frac {22\,A\,c^2\,d^3\,e}{3}\right )+\frac {B\,a^2\,d\,e^4+3\,A\,a^2\,e^5+6\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3+77\,B\,c^2\,d^5-25\,A\,c^2\,d^4\,e}{12\,e}+x^2\,\left (25\,B\,c^2\,d^3\,e-9\,A\,c^2\,d^2\,e^2+3\,B\,a\,c\,d\,e^3+A\,a\,c\,e^4\right )}{d^4\,e^5+4\,d^3\,e^6\,x+6\,d^2\,e^7\,x^2+4\,d\,e^8\,x^3+e^9\,x^4}+\frac {B\,c^2\,x}{e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^2*(A + B*x))/(d + e*x)^5,x)

[Out]

(log(d + e*x)*(A*c^2*e - 5*B*c^2*d))/e^6 - (x^3*(2*B*a*c*e^4 - 4*A*c^2*d*e^3 + 10*B*c^2*d^2*e^2) + x*((B*a^2*e
^4)/3 + (65*B*c^2*d^4)/3 - (22*A*c^2*d^3*e)/3 + (2*A*a*c*d*e^3)/3 + 2*B*a*c*d^2*e^2) + (3*A*a^2*e^5 + 77*B*c^2
*d^5 + B*a^2*d*e^4 - 25*A*c^2*d^4*e + 2*A*a*c*d^2*e^3 + 6*B*a*c*d^3*e^2)/(12*e) + x^2*(A*a*c*e^4 + 25*B*c^2*d^
3*e - 9*A*c^2*d^2*e^2 + 3*B*a*c*d*e^3))/(d^4*e^5 + e^9*x^4 + 4*d^3*e^6*x + 4*d*e^8*x^3 + 6*d^2*e^7*x^2) + (B*c
^2*x)/e^5

________________________________________________________________________________________

sympy [A]  time = 21.57, size = 304, normalized size = 1.61 \begin {gather*} \frac {B c^{2} x}{e^{5}} - \frac {c^{2} \left (- A e + 5 B d\right ) \log {\left (d + e x \right )}}{e^{6}} + \frac {- 3 A a^{2} e^{5} - 2 A a c d^{2} e^{3} + 25 A c^{2} d^{4} e - B a^{2} d e^{4} - 6 B a c d^{3} e^{2} - 77 B c^{2} d^{5} + x^{3} \left (48 A c^{2} d e^{4} - 24 B a c e^{5} - 120 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (- 12 A a c e^{5} + 108 A c^{2} d^{2} e^{3} - 36 B a c d e^{4} - 300 B c^{2} d^{3} e^{2}\right ) + x \left (- 8 A a c d e^{4} + 88 A c^{2} d^{3} e^{2} - 4 B a^{2} e^{5} - 24 B a c d^{2} e^{3} - 260 B c^{2} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**5,x)

[Out]

B*c**2*x/e**5 - c**2*(-A*e + 5*B*d)*log(d + e*x)/e**6 + (-3*A*a**2*e**5 - 2*A*a*c*d**2*e**3 + 25*A*c**2*d**4*e
- B*a**2*d*e**4 - 6*B*a*c*d**3*e**2 - 77*B*c**2*d**5 + x**3*(48*A*c**2*d*e**4 - 24*B*a*c*e**5 - 120*B*c**2*d*
*2*e**3) + x**2*(-12*A*a*c*e**5 + 108*A*c**2*d**2*e**3 - 36*B*a*c*d*e**4 - 300*B*c**2*d**3*e**2) + x*(-8*A*a*c
*d*e**4 + 88*A*c**2*d**3*e**2 - 4*B*a**2*e**5 - 24*B*a*c*d**2*e**3 - 260*B*c**2*d**4*e))/(12*d**4*e**6 + 48*d*
*3*e**7*x + 72*d**2*e**8*x**2 + 48*d*e**9*x**3 + 12*e**10*x**4)

________________________________________________________________________________________