3.332 \(\int x^2 (f+g x^2)^2 \log (c (d+e x^2)^p) \, dx\)

Optimal. Leaf size=278 \[ \frac{1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac{2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 d^{3/2} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{4 d^2 f g p x}{5 e^2}+\frac{4 d^{5/2} f g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{5 e^{5/2}}-\frac{2 d^2 g^2 p x^3}{21 e^2}+\frac{2 d^3 g^2 p x}{7 e^3}-\frac{2 d^{7/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+\frac{2 d f^2 p x}{3 e}+\frac{4 d f g p x^3}{15 e}+\frac{2 d g^2 p x^5}{35 e}-\frac{2}{9} f^2 p x^3-\frac{4}{25} f g p x^5-\frac{2}{49} g^2 p x^7 \]

[Out]

(2*d*f^2*p*x)/(3*e) - (4*d^2*f*g*p*x)/(5*e^2) + (2*d^3*g^2*p*x)/(7*e^3) - (2*f^2*p*x^3)/9 + (4*d*f*g*p*x^3)/(1
5*e) - (2*d^2*g^2*p*x^3)/(21*e^2) - (4*f*g*p*x^5)/25 + (2*d*g^2*p*x^5)/(35*e) - (2*g^2*p*x^7)/49 - (2*d^(3/2)*
f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + (4*d^(5/2)*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*e^(5/2)) - (
2*d^(7/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) + (f^2*x^3*Log[c*(d + e*x^2)^p])/3 + (2*f*g*x^5*Log[c
*(d + e*x^2)^p])/5 + (g^2*x^7*Log[c*(d + e*x^2)^p])/7

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Rubi [A]  time = 0.236472, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2476, 2455, 302, 205} \[ \frac{1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac{2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 d^{3/2} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{4 d^2 f g p x}{5 e^2}+\frac{4 d^{5/2} f g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{5 e^{5/2}}-\frac{2 d^2 g^2 p x^3}{21 e^2}+\frac{2 d^3 g^2 p x}{7 e^3}-\frac{2 d^{7/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+\frac{2 d f^2 p x}{3 e}+\frac{4 d f g p x^3}{15 e}+\frac{2 d g^2 p x^5}{35 e}-\frac{2}{9} f^2 p x^3-\frac{4}{25} f g p x^5-\frac{2}{49} g^2 p x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*f^2*p*x)/(3*e) - (4*d^2*f*g*p*x)/(5*e^2) + (2*d^3*g^2*p*x)/(7*e^3) - (2*f^2*p*x^3)/9 + (4*d*f*g*p*x^3)/(1
5*e) - (2*d^2*g^2*p*x^3)/(21*e^2) - (4*f*g*p*x^5)/25 + (2*d*g^2*p*x^5)/(35*e) - (2*g^2*p*x^7)/49 - (2*d^(3/2)*
f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + (4*d^(5/2)*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*e^(5/2)) - (
2*d^(7/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) + (f^2*x^3*Log[c*(d + e*x^2)^p])/3 + (2*f*g*x^5*Log[c
*(d + e*x^2)^p])/5 + (g^2*x^7*Log[c*(d + e*x^2)^p])/7

Rule 2476

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

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int x^2 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^2 x^2 \log \left (c \left (d+e x^2\right )^p\right )+2 f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^2 \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=\frac{1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac{2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{1}{3} \left (2 e f^2 p\right ) \int \frac{x^4}{d+e x^2} \, dx-\frac{1}{5} (4 e f g p) \int \frac{x^6}{d+e x^2} \, dx-\frac{1}{7} \left (2 e g^2 p\right ) \int \frac{x^8}{d+e x^2} \, dx\\ &=\frac{1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac{2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{1}{3} \left (2 e f^2 p\right ) \int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx-\frac{1}{5} (4 e f g p) \int \left (\frac{d^2}{e^3}-\frac{d x^2}{e^2}+\frac{x^4}{e}-\frac{d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx-\frac{1}{7} \left (2 e g^2 p\right ) \int \left (-\frac{d^3}{e^4}+\frac{d^2 x^2}{e^3}-\frac{d x^4}{e^2}+\frac{x^6}{e}+\frac{d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{2 d f^2 p x}{3 e}-\frac{4 d^2 f g p x}{5 e^2}+\frac{2 d^3 g^2 p x}{7 e^3}-\frac{2}{9} f^2 p x^3+\frac{4 d f g p x^3}{15 e}-\frac{2 d^2 g^2 p x^3}{21 e^2}-\frac{4}{25} f g p x^5+\frac{2 d g^2 p x^5}{35 e}-\frac{2}{49} g^2 p x^7+\frac{1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac{2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{\left (2 d^2 f^2 p\right ) \int \frac{1}{d+e x^2} \, dx}{3 e}+\frac{\left (4 d^3 f g p\right ) \int \frac{1}{d+e x^2} \, dx}{5 e^2}-\frac{\left (2 d^4 g^2 p\right ) \int \frac{1}{d+e x^2} \, dx}{7 e^3}\\ &=\frac{2 d f^2 p x}{3 e}-\frac{4 d^2 f g p x}{5 e^2}+\frac{2 d^3 g^2 p x}{7 e^3}-\frac{2}{9} f^2 p x^3+\frac{4 d f g p x^3}{15 e}-\frac{2 d^2 g^2 p x^3}{21 e^2}-\frac{4}{25} f g p x^5+\frac{2 d g^2 p x^5}{35 e}-\frac{2}{49} g^2 p x^7-\frac{2 d^{3/2} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}+\frac{4 d^{5/2} f g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{5 e^{5/2}}-\frac{2 d^{7/2} g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+\frac{1}{3} f^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac{2}{5} f g x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )\\ \end{align*}

Mathematica [A]  time = 0.165823, size = 188, normalized size = 0.68 \[ \frac{\sqrt{e} x \left (105 e^3 x^2 \left (35 f^2+42 f g x^2+15 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )+2 p \left (-105 d^2 e g \left (42 f+5 g x^2\right )+1575 d^3 g^2+105 d e^2 \left (35 f^2+14 f g x^2+3 g^2 x^4\right )-e^3 x^2 \left (1225 f^2+882 f g x^2+225 g^2 x^4\right )\right )\right )-210 d^{3/2} p \left (15 d^2 g^2-42 d e f g+35 e^2 f^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{11025 e^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-210*d^(3/2)*(35*e^2*f^2 - 42*d*e*f*g + 15*d^2*g^2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + Sqrt[e]*x*(2*p*(1575*d^3*
g^2 - 105*d^2*e*g*(42*f + 5*g*x^2) + 105*d*e^2*(35*f^2 + 14*f*g*x^2 + 3*g^2*x^4) - e^3*x^2*(1225*f^2 + 882*f*g
*x^2 + 225*g^2*x^4)) + 105*e^3*x^2*(35*f^2 + 42*f*g*x^2 + 15*g^2*x^4)*Log[c*(d + e*x^2)^p]))/(11025*e^(7/2))

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Maple [C]  time = 0.602, size = 761, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*f^2*p*x^3+1/7*ln(c)*g^2*x^7+1/3*ln(c)*f^2*x^3+(1/7*g^2*x^7+2/5*f*g*x^5+1/3*f^2*x^3)*ln((e*x^2+d)^p)+2/7*d
^3*g^2*p*x/e^3-2/21*d^2*g^2*p*x^3/e^2+2/35*d*g^2*p*x^5/e+2/3*d*f^2*p*x/e+2/5*ln(c)*f*g*x^5-1/14*I*Pi*g^2*x^7*c
sgn(I*c*(e*x^2+d)^p)^3-2/49*g^2*p*x^7-4/25*f*g*p*x^5-1/5*I*Pi*f*g*x^5*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p
)*csgn(I*c)-1/14*I*Pi*g^2*x^7*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-1/6*I*Pi*f^2*x^3*csgn(I*c*(e
*x^2+d)^p)^3-2/5/e^3*(-d*e)^(1/2)*p*d^2*ln(-(-d*e)^(1/2)*x-d)*f*g+2/5/e^3*(-d*e)^(1/2)*p*d^2*ln((-d*e)^(1/2)*x
-d)*f*g+1/5*I*Pi*f*g*x^5*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+1/5*I*Pi*f*g*x^5*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^
2+d)^p)^2+1/14*I*Pi*g^2*x^7*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+1/14*I*Pi*g^2*x^7*csgn(I*c*(e*x^2+d)^p
)^2*csgn(I*c)-4/5*d^2*f*g*p*x/e^2-1/6*I*Pi*f^2*x^3*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+4/15*d*
f*g*p*x^3/e+1/7/e^4*(-d*e)^(1/2)*p*d^3*ln(-(-d*e)^(1/2)*x-d)*g^2-1/7/e^4*(-d*e)^(1/2)*p*d^3*ln((-d*e)^(1/2)*x-
d)*g^2+1/3/e^2*(-d*e)^(1/2)*p*d*ln(-(-d*e)^(1/2)*x-d)*f^2-1/3/e^2*(-d*e)^(1/2)*p*d*ln((-d*e)^(1/2)*x-d)*f^2-1/
5*I*Pi*f*g*x^5*csgn(I*c*(e*x^2+d)^p)^3+1/6*I*Pi*f^2*x^3*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+1/6*I*Pi*f
^2*x^3*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.68206, size = 1138, normalized size = 4.09 \begin{align*} \left [-\frac{450 \, e^{3} g^{2} p x^{7} + 126 \,{\left (14 \, e^{3} f g - 5 \, d e^{2} g^{2}\right )} p x^{5} + 70 \,{\left (35 \, e^{3} f^{2} - 42 \, d e^{2} f g + 15 \, d^{2} e g^{2}\right )} p x^{3} - 105 \,{\left (35 \, d e^{2} f^{2} - 42 \, d^{2} e f g + 15 \, d^{3} g^{2}\right )} p \sqrt{-\frac{d}{e}} \log \left (\frac{e x^{2} - 2 \, e x \sqrt{-\frac{d}{e}} - d}{e x^{2} + d}\right ) - 210 \,{\left (35 \, d e^{2} f^{2} - 42 \, d^{2} e f g + 15 \, d^{3} g^{2}\right )} p x - 105 \,{\left (15 \, e^{3} g^{2} p x^{7} + 42 \, e^{3} f g p x^{5} + 35 \, e^{3} f^{2} p x^{3}\right )} \log \left (e x^{2} + d\right ) - 105 \,{\left (15 \, e^{3} g^{2} x^{7} + 42 \, e^{3} f g x^{5} + 35 \, e^{3} f^{2} x^{3}\right )} \log \left (c\right )}{11025 \, e^{3}}, -\frac{450 \, e^{3} g^{2} p x^{7} + 126 \,{\left (14 \, e^{3} f g - 5 \, d e^{2} g^{2}\right )} p x^{5} + 70 \,{\left (35 \, e^{3} f^{2} - 42 \, d e^{2} f g + 15 \, d^{2} e g^{2}\right )} p x^{3} + 210 \,{\left (35 \, d e^{2} f^{2} - 42 \, d^{2} e f g + 15 \, d^{3} g^{2}\right )} p \sqrt{\frac{d}{e}} \arctan \left (\frac{e x \sqrt{\frac{d}{e}}}{d}\right ) - 210 \,{\left (35 \, d e^{2} f^{2} - 42 \, d^{2} e f g + 15 \, d^{3} g^{2}\right )} p x - 105 \,{\left (15 \, e^{3} g^{2} p x^{7} + 42 \, e^{3} f g p x^{5} + 35 \, e^{3} f^{2} p x^{3}\right )} \log \left (e x^{2} + d\right ) - 105 \,{\left (15 \, e^{3} g^{2} x^{7} + 42 \, e^{3} f g x^{5} + 35 \, e^{3} f^{2} x^{3}\right )} \log \left (c\right )}{11025 \, e^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/11025*(450*e^3*g^2*p*x^7 + 126*(14*e^3*f*g - 5*d*e^2*g^2)*p*x^5 + 70*(35*e^3*f^2 - 42*d*e^2*f*g + 15*d^2*e
*g^2)*p*x^3 - 105*(35*d*e^2*f^2 - 42*d^2*e*f*g + 15*d^3*g^2)*p*sqrt(-d/e)*log((e*x^2 - 2*e*x*sqrt(-d/e) - d)/(
e*x^2 + d)) - 210*(35*d*e^2*f^2 - 42*d^2*e*f*g + 15*d^3*g^2)*p*x - 105*(15*e^3*g^2*p*x^7 + 42*e^3*f*g*p*x^5 +
35*e^3*f^2*p*x^3)*log(e*x^2 + d) - 105*(15*e^3*g^2*x^7 + 42*e^3*f*g*x^5 + 35*e^3*f^2*x^3)*log(c))/e^3, -1/1102
5*(450*e^3*g^2*p*x^7 + 126*(14*e^3*f*g - 5*d*e^2*g^2)*p*x^5 + 70*(35*e^3*f^2 - 42*d*e^2*f*g + 15*d^2*e*g^2)*p*
x^3 + 210*(35*d*e^2*f^2 - 42*d^2*e*f*g + 15*d^3*g^2)*p*sqrt(d/e)*arctan(e*x*sqrt(d/e)/d) - 210*(35*d*e^2*f^2 -
42*d^2*e*f*g + 15*d^3*g^2)*p*x - 105*(15*e^3*g^2*p*x^7 + 42*e^3*f*g*p*x^5 + 35*e^3*f^2*p*x^3)*log(e*x^2 + d)
- 105*(15*e^3*g^2*x^7 + 42*e^3*f*g*x^5 + 35*e^3*f^2*x^3)*log(c))/e^3]

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

[Out]

Timed out

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Giac [A]  time = 1.27028, size = 332, normalized size = 1.19 \begin{align*} -\frac{2 \,{\left (15 \, d^{4} g^{2} p - 42 \, d^{3} f g p e + 35 \, d^{2} f^{2} p e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{7}{2}\right )}}{105 \, \sqrt{d}} + \frac{1}{11025} \,{\left (1575 \, g^{2} p x^{7} e^{3} \log \left (x^{2} e + d\right ) - 450 \, g^{2} p x^{7} e^{3} + 1575 \, g^{2} x^{7} e^{3} \log \left (c\right ) + 630 \, d g^{2} p x^{5} e^{2} + 4410 \, f g p x^{5} e^{3} \log \left (x^{2} e + d\right ) - 1764 \, f g p x^{5} e^{3} - 1050 \, d^{2} g^{2} p x^{3} e + 4410 \, f g x^{5} e^{3} \log \left (c\right ) + 2940 \, d f g p x^{3} e^{2} + 3675 \, f^{2} p x^{3} e^{3} \log \left (x^{2} e + d\right ) + 3150 \, d^{3} g^{2} p x - 2450 \, f^{2} p x^{3} e^{3} - 8820 \, d^{2} f g p x e + 3675 \, f^{2} x^{3} e^{3} \log \left (c\right ) + 7350 \, d f^{2} p x e^{2}\right )} e^{\left (-3\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(15*d^4*g^2*p - 42*d^3*f*g*p*e + 35*d^2*f^2*p*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-7/2)/sqrt(d) + 1/11025
*(1575*g^2*p*x^7*e^3*log(x^2*e + d) - 450*g^2*p*x^7*e^3 + 1575*g^2*x^7*e^3*log(c) + 630*d*g^2*p*x^5*e^2 + 4410
*f*g*p*x^5*e^3*log(x^2*e + d) - 1764*f*g*p*x^5*e^3 - 1050*d^2*g^2*p*x^3*e + 4410*f*g*x^5*e^3*log(c) + 2940*d*f
*g*p*x^3*e^2 + 3675*f^2*p*x^3*e^3*log(x^2*e + d) + 3150*d^3*g^2*p*x - 2450*f^2*p*x^3*e^3 - 8820*d^2*f*g*p*x*e
+ 3675*f^2*x^3*e^3*log(c) + 7350*d*f^2*p*x*e^2)*e^(-3)