3.331 \(\int \frac{(f+g x^2)^2 \log (c (d+e x^2)^p)}{x^{11}} \, dx\)

Optimal. Leaf size=253 \[ -\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac{f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac{g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}+\frac{e^2 p \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right )}{60 d^4 x^2}-\frac{e p \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right )}{120 d^3 x^4}-\frac{e^3 p \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{60 d^5}+\frac{e^3 p \log (x) \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right )}{30 d^5}+\frac{e f p (2 e f-5 d g)}{60 d^2 x^6}-\frac{e f^2 p}{40 d x^8} \]

[Out]

-(e*f^2*p)/(40*d*x^8) + (e*f*(2*e*f - 5*d*g)*p)/(60*d^2*x^6) - (e*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p)/(12
0*d^3*x^4) + (e^2*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p)/(60*d^4*x^2) + (e^3*(6*e^2*f^2 - 15*d*e*f*g + 10*d^
2*g^2)*p*Log[x])/(30*d^5) - (e^3*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p*Log[d + e*x^2])/(60*d^5) - (f^2*Log[c
*(d + e*x^2)^p])/(10*x^10) - (f*g*Log[c*(d + e*x^2)^p])/(4*x^8) - (g^2*Log[c*(d + e*x^2)^p])/(6*x^6)

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Rubi [A]  time = 0.333418, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2475, 43, 2414, 12, 893} \[ -\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac{f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac{g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}+\frac{e^2 p \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right )}{60 d^4 x^2}-\frac{e p \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right )}{120 d^3 x^4}-\frac{e^3 p \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{60 d^5}+\frac{e^3 p \log (x) \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right )}{30 d^5}+\frac{e f p (2 e f-5 d g)}{60 d^2 x^6}-\frac{e f^2 p}{40 d x^8} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^11,x]

[Out]

-(e*f^2*p)/(40*d*x^8) + (e*f*(2*e*f - 5*d*g)*p)/(60*d^2*x^6) - (e*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p)/(12
0*d^3*x^4) + (e^2*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p)/(60*d^4*x^2) + (e^3*(6*e^2*f^2 - 15*d*e*f*g + 10*d^
2*g^2)*p*Log[x])/(30*d^5) - (e^3*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p*Log[d + e*x^2])/(60*d^5) - (f^2*Log[c
*(d + e*x^2)^p])/(10*x^10) - (f*g*Log[c*(d + e*x^2)^p])/(4*x^8) - (g^2*Log[c*(d + e*x^2)^p])/(6*x^6)

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2414

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(x_)^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = IntHide[x^m*(f + g*x^r)^q, x]}, Dist[a + b*Log[c*(d + e*x)^n], u, x] - Dist[b*e*n, Int[SimplifyI
ntegrand[u/(d + e*x), x], x], x] /; InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x]
&& IntegerQ[m] && IntegerQ[q] && IntegerQ[r]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(f+g x)^2 \log \left (c (d+e x)^p\right )}{x^6} \, dx,x,x^2\right )\\ &=-\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac{f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac{g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac{1}{2} (e p) \operatorname{Subst}\left (\int \frac{-6 f^2-15 f g x-10 g^2 x^2}{30 x^5 (d+e x)} \, dx,x,x^2\right )\\ &=-\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac{f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac{g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac{1}{60} (e p) \operatorname{Subst}\left (\int \frac{-6 f^2-15 f g x-10 g^2 x^2}{x^5 (d+e x)} \, dx,x,x^2\right )\\ &=-\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac{f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac{g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac{1}{60} (e p) \operatorname{Subst}\left (\int \left (-\frac{6 f^2}{d x^5}-\frac{3 f (-2 e f+5 d g)}{d^2 x^4}+\frac{-6 e^2 f^2+15 d e f g-10 d^2 g^2}{d^3 x^3}+\frac{e \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right )}{d^4 x^2}-\frac{e^2 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right )}{d^5 x}+\frac{e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right )}{d^5 (d+e x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{e f^2 p}{40 d x^8}+\frac{e f (2 e f-5 d g) p}{60 d^2 x^6}-\frac{e \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p}{120 d^3 x^4}+\frac{e^2 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p}{60 d^4 x^2}+\frac{e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p \log (x)}{30 d^5}-\frac{e^3 \left (6 e^2 f^2-15 d e f g+10 d^2 g^2\right ) p \log \left (d+e x^2\right )}{60 d^5}-\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{10 x^{10}}-\frac{f g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^8}-\frac{g^2 \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.219371, size = 215, normalized size = 0.85 \[ -\frac{2 d^5 \left (6 f^2+15 f g x^2+10 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )+d e p x^2 \left (-d^2 e x^2 \left (4 f^2+15 f g x^2+20 g^2 x^4\right )+d^3 \left (3 f^2+10 f g x^2+10 g^2 x^4\right )+6 d e^2 f x^4 \left (f+5 g x^2\right )-12 e^3 f^2 x^6\right )-4 e^3 p x^{10} \log (x) \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right )+2 e^3 p x^{10} \left (10 d^2 g^2-15 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{120 d^5 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^11,x]

[Out]

-(d*e*p*x^2*(-12*e^3*f^2*x^6 + 6*d*e^2*f*x^4*(f + 5*g*x^2) + d^3*(3*f^2 + 10*f*g*x^2 + 10*g^2*x^4) - d^2*e*x^2
*(4*f^2 + 15*f*g*x^2 + 20*g^2*x^4)) - 4*e^3*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p*x^10*Log[x] + 2*e^3*(6*e^2
*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p*x^10*Log[d + e*x^2] + 2*d^5*(6*f^2 + 15*f*g*x^2 + 10*g^2*x^4)*Log[c*(d + e*x
^2)^p])/(120*d^5*x^10)

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Maple [C]  time = 0.415, size = 748, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^2+f)^2*ln(c*(e*x^2+d)^p)/x^11,x)

[Out]

-1/60*(10*g^2*x^4+15*f*g*x^2+6*f^2)/x^10*ln((e*x^2+d)^p)+1/120*(-15*I*Pi*d^5*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(
I*c*(e*x^2+d)^p)^2-12*ln(c)*d^5*f^2-15*I*Pi*d^5*f*g*x^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-20*ln(c)*d^5*g^2*x^4
+10*I*Pi*d^5*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-60*ln(x)*d*e^4*f*g*p*x^10+30*ln(e*x^2
+d)*d*e^4*f*g*p*x^10-10*I*Pi*d^5*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-10*I*Pi*d^5*g^2*x^4*csgn(I*(e*x^2+d
)^p)*csgn(I*c*(e*x^2+d)^p)^2+24*ln(x)*e^5*f^2*p*x^10-12*ln(e*x^2+d)*e^5*f^2*p*x^10-30*ln(c)*d^5*f*g*x^2+15*I*P
i*d^5*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-6*I*Pi*d^5*f^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(
I*c)+15*I*Pi*d^5*f*g*x^2*csgn(I*c*(e*x^2+d)^p)^3+20*d^3*e^2*g^2*p*x^8+12*d*e^4*f^2*p*x^8-10*d^4*e*g^2*p*x^6-6*
d^2*e^3*f^2*p*x^6+4*d^3*e^2*f^2*p*x^4-3*d^4*e*f^2*p*x^2+6*I*Pi*d^5*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^
p)*csgn(I*c)+40*ln(x)*d^2*e^3*g^2*p*x^10-20*ln(e*x^2+d)*d^2*e^3*g^2*p*x^10-30*d^2*e^3*f*g*p*x^8+15*d^3*e^2*f*g
*p*x^6-10*d^4*e*f*g*p*x^4+10*I*Pi*d^5*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^3-6*I*Pi*d^5*f^2*csgn(I*(e*x^2+d)^p)*csgn(
I*c*(e*x^2+d)^p)^2+6*I*Pi*d^5*f^2*csgn(I*c*(e*x^2+d)^p)^3)/d^5/x^10

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Maxima [A]  time = 1.03691, size = 301, normalized size = 1.19 \begin{align*} -\frac{1}{120} \, e p{\left (\frac{2 \,{\left (6 \, e^{4} f^{2} - 15 \, d e^{3} f g + 10 \, d^{2} e^{2} g^{2}\right )} \log \left (e x^{2} + d\right )}{d^{5}} - \frac{2 \,{\left (6 \, e^{4} f^{2} - 15 \, d e^{3} f g + 10 \, d^{2} e^{2} g^{2}\right )} \log \left (x^{2}\right )}{d^{5}} - \frac{2 \,{\left (6 \, e^{3} f^{2} - 15 \, d e^{2} f g + 10 \, d^{2} e g^{2}\right )} x^{6} - 3 \, d^{3} f^{2} -{\left (6 \, d e^{2} f^{2} - 15 \, d^{2} e f g + 10 \, d^{3} g^{2}\right )} x^{4} + 2 \,{\left (2 \, d^{2} e f^{2} - 5 \, d^{3} f g\right )} x^{2}}{d^{4} x^{8}}\right )} - \frac{{\left (10 \, g^{2} x^{4} + 15 \, f g x^{2} + 6 \, f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{60 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^11,x, algorithm="maxima")

[Out]

-1/120*e*p*(2*(6*e^4*f^2 - 15*d*e^3*f*g + 10*d^2*e^2*g^2)*log(e*x^2 + d)/d^5 - 2*(6*e^4*f^2 - 15*d*e^3*f*g + 1
0*d^2*e^2*g^2)*log(x^2)/d^5 - (2*(6*e^3*f^2 - 15*d*e^2*f*g + 10*d^2*e*g^2)*x^6 - 3*d^3*f^2 - (6*d*e^2*f^2 - 15
*d^2*e*f*g + 10*d^3*g^2)*x^4 + 2*(2*d^2*e*f^2 - 5*d^3*f*g)*x^2)/(d^4*x^8)) - 1/60*(10*g^2*x^4 + 15*f*g*x^2 + 6
*f^2)*log((e*x^2 + d)^p*c)/x^10

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Fricas [A]  time = 2.07507, size = 587, normalized size = 2.32 \begin{align*} \frac{4 \,{\left (6 \, e^{5} f^{2} - 15 \, d e^{4} f g + 10 \, d^{2} e^{3} g^{2}\right )} p x^{10} \log \left (x\right ) - 3 \, d^{4} e f^{2} p x^{2} + 2 \,{\left (6 \, d e^{4} f^{2} - 15 \, d^{2} e^{3} f g + 10 \, d^{3} e^{2} g^{2}\right )} p x^{8} -{\left (6 \, d^{2} e^{3} f^{2} - 15 \, d^{3} e^{2} f g + 10 \, d^{4} e g^{2}\right )} p x^{6} + 2 \,{\left (2 \, d^{3} e^{2} f^{2} - 5 \, d^{4} e f g\right )} p x^{4} - 2 \,{\left (10 \, d^{5} g^{2} p x^{4} +{\left (6 \, e^{5} f^{2} - 15 \, d e^{4} f g + 10 \, d^{2} e^{3} g^{2}\right )} p x^{10} + 15 \, d^{5} f g p x^{2} + 6 \, d^{5} f^{2} p\right )} \log \left (e x^{2} + d\right ) - 2 \,{\left (10 \, d^{5} g^{2} x^{4} + 15 \, d^{5} f g x^{2} + 6 \, d^{5} f^{2}\right )} \log \left (c\right )}{120 \, d^{5} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^11,x, algorithm="fricas")

[Out]

1/120*(4*(6*e^5*f^2 - 15*d*e^4*f*g + 10*d^2*e^3*g^2)*p*x^10*log(x) - 3*d^4*e*f^2*p*x^2 + 2*(6*d*e^4*f^2 - 15*d
^2*e^3*f*g + 10*d^3*e^2*g^2)*p*x^8 - (6*d^2*e^3*f^2 - 15*d^3*e^2*f*g + 10*d^4*e*g^2)*p*x^6 + 2*(2*d^3*e^2*f^2
- 5*d^4*e*f*g)*p*x^4 - 2*(10*d^5*g^2*p*x^4 + (6*e^5*f^2 - 15*d*e^4*f*g + 10*d^2*e^3*g^2)*p*x^10 + 15*d^5*f*g*p
*x^2 + 6*d^5*f^2*p)*log(e*x^2 + d) - 2*(10*d^5*g^2*x^4 + 15*d^5*f*g*x^2 + 6*d^5*f^2)*log(c))/(d^5*x^10)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**2+f)**2*ln(c*(e*x**2+d)**p)/x**11,x)

[Out]

Timed out

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Giac [B]  time = 1.20026, size = 1809, normalized size = 7.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^11,x, algorithm="giac")

[Out]

-1/120*(20*(x^2*e + d)^5*d^2*g^2*p*e^4*log(x^2*e + d) - 100*(x^2*e + d)^4*d^3*g^2*p*e^4*log(x^2*e + d) + 200*(
x^2*e + d)^3*d^4*g^2*p*e^4*log(x^2*e + d) - 180*(x^2*e + d)^2*d^5*g^2*p*e^4*log(x^2*e + d) + 60*(x^2*e + d)*d^
6*g^2*p*e^4*log(x^2*e + d) - 20*(x^2*e + d)^5*d^2*g^2*p*e^4*log(x^2*e) + 100*(x^2*e + d)^4*d^3*g^2*p*e^4*log(x
^2*e) - 200*(x^2*e + d)^3*d^4*g^2*p*e^4*log(x^2*e) + 200*(x^2*e + d)^2*d^5*g^2*p*e^4*log(x^2*e) - 100*(x^2*e +
d)*d^6*g^2*p*e^4*log(x^2*e) + 20*d^7*g^2*p*e^4*log(x^2*e) - 20*(x^2*e + d)^4*d^3*g^2*p*e^4 + 90*(x^2*e + d)^3
*d^4*g^2*p*e^4 - 150*(x^2*e + d)^2*d^5*g^2*p*e^4 + 110*(x^2*e + d)*d^6*g^2*p*e^4 - 30*d^7*g^2*p*e^4 - 30*(x^2*
e + d)^5*d*f*g*p*e^5*log(x^2*e + d) + 150*(x^2*e + d)^4*d^2*f*g*p*e^5*log(x^2*e + d) - 300*(x^2*e + d)^3*d^3*f
*g*p*e^5*log(x^2*e + d) + 300*(x^2*e + d)^2*d^4*f*g*p*e^5*log(x^2*e + d) - 120*(x^2*e + d)*d^5*f*g*p*e^5*log(x
^2*e + d) + 30*(x^2*e + d)^5*d*f*g*p*e^5*log(x^2*e) - 150*(x^2*e + d)^4*d^2*f*g*p*e^5*log(x^2*e) + 300*(x^2*e
+ d)^3*d^3*f*g*p*e^5*log(x^2*e) - 300*(x^2*e + d)^2*d^4*f*g*p*e^5*log(x^2*e) + 150*(x^2*e + d)*d^5*f*g*p*e^5*l
og(x^2*e) - 30*d^6*f*g*p*e^5*log(x^2*e) + 20*(x^2*e + d)^2*d^5*g^2*e^4*log(c) - 40*(x^2*e + d)*d^6*g^2*e^4*log
(c) + 20*d^7*g^2*e^4*log(c) + 30*(x^2*e + d)^4*d^2*f*g*p*e^5 - 135*(x^2*e + d)^3*d^3*f*g*p*e^5 + 235*(x^2*e +
d)^2*d^4*f*g*p*e^5 - 185*(x^2*e + d)*d^5*f*g*p*e^5 + 55*d^6*f*g*p*e^5 + 12*(x^2*e + d)^5*f^2*p*e^6*log(x^2*e +
d) - 60*(x^2*e + d)^4*d*f^2*p*e^6*log(x^2*e + d) + 120*(x^2*e + d)^3*d^2*f^2*p*e^6*log(x^2*e + d) - 120*(x^2*
e + d)^2*d^3*f^2*p*e^6*log(x^2*e + d) + 60*(x^2*e + d)*d^4*f^2*p*e^6*log(x^2*e + d) - 12*(x^2*e + d)^5*f^2*p*e
^6*log(x^2*e) + 60*(x^2*e + d)^4*d*f^2*p*e^6*log(x^2*e) - 120*(x^2*e + d)^3*d^2*f^2*p*e^6*log(x^2*e) + 120*(x^
2*e + d)^2*d^3*f^2*p*e^6*log(x^2*e) - 60*(x^2*e + d)*d^4*f^2*p*e^6*log(x^2*e) + 12*d^5*f^2*p*e^6*log(x^2*e) +
30*(x^2*e + d)*d^5*f*g*e^5*log(c) - 30*d^6*f*g*e^5*log(c) - 12*(x^2*e + d)^4*d*f^2*p*e^6 + 54*(x^2*e + d)^3*d^
2*f^2*p*e^6 - 94*(x^2*e + d)^2*d^3*f^2*p*e^6 + 77*(x^2*e + d)*d^4*f^2*p*e^6 - 25*d^5*f^2*p*e^6 + 12*d^5*f^2*e^
6*log(c))*e^(-1)/((x^2*e + d)^5*d^5 - 5*(x^2*e + d)^4*d^6 + 10*(x^2*e + d)^3*d^7 - 10*(x^2*e + d)^2*d^8 + 5*(x
^2*e + d)*d^9 - d^10)