3.4.12 \(\int x (f+g x^2) \log (c (d+e x^2)^p) \, dx\) [312]

Optimal. Leaf size=94 \[ -\frac {(e f-d g) p x^2}{4 e}-\frac {p \left (f+g x^2\right )^2}{8 g}-\frac {(e f-d g)^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g} \]

[Out]

-1/4*(-d*g+e*f)*p*x^2/e-1/8*p*(g*x^2+f)^2/g-1/4*(-d*g+e*f)^2*p*ln(e*x^2+d)/e^2/g+1/4*(g*x^2+f)^2*ln(c*(e*x^2+d
)^p)/g

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Rubi [A]
time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2525, 2442, 45} \begin {gather*} \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {p (e f-d g)^2 \log \left (d+e x^2\right )}{4 e^2 g}-\frac {p x^2 (e f-d g)}{4 e}-\frac {p \left (f+g x^2\right )^2}{8 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((e*f - d*g)*p*x^2)/e - (p*(f + g*x^2)^2)/(8*g) - ((e*f - d*g)^2*p*Log[d + e*x^2])/(4*e^2*g) + ((f + g*x^
2)^2*Log[c*(d + e*x^2)^p])/(4*g)

Rule 45

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 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2525

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])

Rubi steps

\begin {align*} \int x \left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int (f+g x) \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )\\ &=\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {(e p) \text {Subst}\left (\int \frac {(f+g x)^2}{d+e x} \, dx,x,x^2\right )}{4 g}\\ &=\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {(e p) \text {Subst}\left (\int \left (\frac {g (e f-d g)}{e^2}+\frac {(e f-d g)^2}{e^2 (d+e x)}+\frac {g (f+g x)}{e}\right ) \, dx,x,x^2\right )}{4 g}\\ &=-\frac {(e f-d g) p x^2}{4 e}-\frac {p \left (f+g x^2\right )^2}{8 g}-\frac {(e f-d g)^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 98, normalized size = 1.04 \begin {gather*} \frac {d g p x^2}{4 e}-\frac {1}{8} g p x^4-\frac {d^2 g p \log \left (d+e x^2\right )}{4 e^2}+\frac {1}{4} g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f \left (-p x^2+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*g*p*x^2)/(4*e) - (g*p*x^4)/8 - (d^2*g*p*Log[d + e*x^2])/(4*e^2) + (g*x^4*Log[c*(d + e*x^2)^p])/4 + (f*(-(p*
x^2) + ((d + e*x^2)*Log[c*(d + e*x^2)^p])/e))/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.44, size = 3275, normalized size = 34.84

method	result	size
risch	\(\text {Expression too large to display}\)	\(3275\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(g*x^2+f)^2/g*ln((e*x^2+d)^p)+1/8*(-4*I*ln(e*x^2+d)*Pi*d*e*f*g*p*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)
*csgn(I*c)+4*e^2*f^2*p^2-Pi^2*e^2*f^2*csgn(I*c*(e*x^2+d)^p)^6+4*I*ln(e*x^2+d)*Pi*d*e*f*g*p*csgn(I*c*(e*x^2+d)^
p)^2*csgn(I*c)-2*I*Pi*d*e*g^2*p*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+6*I*Pi*e^2*f*g*p*x^2*c
sgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-2*I*Pi*d*e*f*g*p*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*
csgn(I*c)+2*Pi^2*e^2*f^2*csgn(I*c*(e*x^2+d)^p)^5*csgn(I*c)-Pi^2*e^2*f^2*csgn(I*c*(e*x^2+d)^p)^4*csgn(I*c)^2-Pi
^2*e^2*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^6-Pi^2*e^2*f^2*csgn(I*(e*x^2+d)^p)^2*csgn(I*c*(e*x^2+d)^p)^4+2*Pi^2*e^2*f
^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^5+4*I*ln(e*x^2+d)*Pi*d*e*f*g*p*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^
2+d)^p)^2-8*I*Pi*ln(c)*e^2*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-Pi^2*e^2*g^2*x^4*csgn(I
*(e*x^2+d)^p)^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)^2-4*Pi^2*e^2*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^
p)^4*csgn(I*c)+2*Pi^2*e^2*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^3*csgn(I*c)^2-2*Pi^2*e^2*f*g*x^2*c
sgn(I*(e*x^2+d)^p)^2*csgn(I*c*(e*x^2+d)^p)^4+4*Pi^2*e^2*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^5-4*
I*Pi*e^2*f^2*p*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-4*I*Pi*e^2*f^2*p*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+
2*Pi^2*e^2*g^2*x^4*csgn(I*(e*x^2+d)^p)^2*csgn(I*c*(e*x^2+d)^p)^3*csgn(I*c)-4*I*Pi*ln(c)*e^2*g^2*x^4*csgn(I*c*(
e*x^2+d)^p)^3+2*I*Pi*e^2*g^2*p*x^4*csgn(I*c*(e*x^2+d)^p)^3+4*I*Pi*ln(c)*e^2*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(
e*x^2+d)^p)^2+4*I*Pi*ln(c)*e^2*f^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+4*Pi^2*e^2*f*g*x^2*csgn(I*c*(e*x^2+d)^p)^
5*csgn(I*c)-2*Pi^2*e^2*f*g*x^2*csgn(I*c*(e*x^2+d)^p)^4*csgn(I*c)^2+2*I*ln(e*x^2+d)*Pi*d^2*g^2*p*csgn(I*c*(e*x^
2+d)^p)^3+2*I*ln(e*x^2+d)*Pi*e^2*f^2*p*csgn(I*c*(e*x^2+d)^p)^3+4*ln(c)^2*e^2*f^2-2*Pi^2*e^2*f*g*x^2*csgn(I*c*(
e*x^2+d)^p)^6+2*Pi^2*e^2*f^2*csgn(I*(e*x^2+d)^p)^2*csgn(I*c*(e*x^2+d)^p)^3*csgn(I*c)-Pi^2*e^2*f^2*csgn(I*(e*x^
2+d)^p)^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)^2-Pi^2*e^2*g^2*x^4*csgn(I*(e*x^2+d)^p)^2*csgn(I*c*(e*x^2+d)^p)^4+2
*Pi^2*e^2*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^5+2*Pi^2*e^2*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^5*csgn(
I*c)-4*Pi^2*e^2*f^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^4*csgn(I*c)+2*Pi^2*e^2*f^2*csgn(I*(e*x^2+d)^p)*c
sgn(I*c*(e*x^2+d)^p)^3*csgn(I*c)^2-4*I*Pi*ln(c)*e^2*f^2*csgn(I*c*(e*x^2+d)^p)^3+4*I*Pi*e^2*f^2*p*csgn(I*c*(e*x
^2+d)^p)^3-Pi^2*e^2*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^4*csgn(I*c)^2-4*ln(e*x^2+d)*d*e*f*g*p^2+4*ln(c)*d*e*g^2*p*x^
2-12*ln(c)*e^2*f*g*p*x^2+4*ln(c)*d*e*f*g*p-8*I*Pi*ln(c)*e^2*f*g*x^2*csgn(I*c*(e*x^2+d)^p)^3-2*I*Pi*d*e*g^2*p*x
^2*csgn(I*c*(e*x^2+d)^p)^3+6*I*Pi*e^2*f*g*p*x^2*csgn(I*c*(e*x^2+d)^p)^3-2*I*ln(e*x^2+d)*Pi*e^2*f^2*p*csgn(I*c*
(e*x^2+d)^p)^2*csgn(I*c)+4*I*Pi*ln(c)*e^2*g^2*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+4*I*Pi*ln(c)*e^2
*g^2*x^4*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-2*I*Pi*e^2*g^2*p*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-2*
I*Pi*e^2*g^2*p*x^4*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-2*Pi^2*e^2*f*g*x^2*csgn(I*(e*x^2+d)^p)^2*csgn(I*c*(e*x^2+
d)^p)^2*csgn(I*c)^2-8*Pi^2*e^2*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^4*csgn(I*c)+4*Pi^2*e^2*f*g*x^
2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^3*csgn(I*c)^2+4*Pi^2*e^2*f*g*x^2*csgn(I*(e*x^2+d)^p)^2*csgn(I*c*(e
*x^2+d)^p)^3*csgn(I*c)+4*I*Pi*e^2*f^2*p*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-2*I*ln(e*x^2+d)*Pi
*d^2*g^2*p*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-2*I*ln(e*x^2+d)*Pi*d^2*g^2*p*csgn(I*c*(e*x^2+d)^p)^2*cs
gn(I*c)-2*I*ln(e*x^2+d)*Pi*e^2*f^2*p*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-4*I*Pi*ln(c)*e^2*f^2*csgn(I*(
e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-2*I*Pi*d*e*f*g*p*csgn(I*c*(e*x^2+d)^p)^3+8*I*Pi*ln(c)*e^2*f*g*x^2*
csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+2*I*Pi*d*e*g^2*p*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-4*I*ln(e*x^
2+d)*Pi*d*e*f*g*p*csgn(I*c*(e*x^2+d)^p)^3+2*I*ln(e*x^2+d)*Pi*e^2*f^2*p*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^
p)*csgn(I*c)+2*I*Pi*d*e*f*g*p*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-4*I*Pi*ln(c)*e^2*g^2*x^4*csgn(I*(e*x^2+d)^p)*c
sgn(I*c*(e*x^2+d)^p)*csgn(I*c)+2*I*Pi*e^2*g^2*p*x^4*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+8*I*Pi
*ln(c)*e^2*f*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+2*I*Pi*d*e*g^2*p*x^2*csgn(I*c*(e*x^2+d)^p)^2*cs
gn(I*c)-6*I*Pi*e^2*f*g*p*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-6*I*Pi*e^2*f*g*p*x^2*csgn(I*c*(e*x^2+
d)^p)^2*csgn(I*c)+2*I*Pi*d*e*f*g*p*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+2*I*ln(e*x^2+d)*Pi*d^2*g^2*p*cs
gn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+8*ln(e*x^2+d)*ln(c)*d*e*f*g*p-4*ln(e*x^2+d)*ln(c)*d^2*g^2*p-
4*ln(e*x^2+d)*ln(c)*e^2*f^2*p-4*ln(c)*e^2*g^2*p*x^4+8*ln(c)^2*e^2*f*g*x^2+2*ln(e*x^2+d)*d^2*g^2*p^2+2*ln(e*x^2
+d)*e^2*f^2*p^2+4*ln(c)^2*e^2*g^2*x^4-8*ln(c)*e^2*f^2*p+e^2*g^2*p^2*x^4+d^2*g^2*p^2-2*d*e*g^2*p^2*x^2+4*e^2*f*
g*p^2*x^2-4*d*e*f*g*p^2)/e^2/g/(I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-I*Pi*csgn(I*(e*x^2+d)^p)*csgn
(I*c*(e*x^2+d)^p)*csgn(I*c)-I*Pi*csgn(I*c*(e*x^...

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Maxima [A]
time = 0.27, size = 101, normalized size = 1.07 \begin {gather*} -\frac {{\left (2 \, {\left (d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x^{2} e + d\right ) + {\left (g^{2} x^{4} e - 2 \, {\left (d g^{2} - 2 \, f g e\right )} x^{2}\right )} e^{\left (-2\right )}\right )} p e}{8 \, g} + \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{4 \, g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 94, normalized size = 1.00 \begin {gather*} \frac {1}{8} \, {\left (2 \, d g p x^{2} e + 2 \, {\left (g x^{4} + 2 \, f x^{2}\right )} e^{2} \log \left (c\right ) - {\left (g p x^{4} + 4 \, f p x^{2}\right )} e^{2} - 2 \, {\left (d^{2} g p - 2 \, d f p e - {\left (g p x^{4} + 2 \, f p x^{2}\right )} e^{2}\right )} \log \left (x^{2} e + d\right )\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*d*g*p*x^2*e + 2*(g*x^4 + 2*f*x^2)*e^2*log(c) - (g*p*x^4 + 4*f*p*x^2)*e^2 - 2*(d^2*g*p - 2*d*f*p*e - (g*
p*x^4 + 2*f*p*x^2)*e^2)*log(x^2*e + d))*e^(-2)

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Sympy [A]
time = 17.02, size = 126, normalized size = 1.34 \begin {gather*} \begin {cases} - \frac {d^{2} g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{4 e^{2}} + \frac {d f \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2 e} + \frac {d g p x^{2}}{4 e} - \frac {f p x^{2}}{2} + \frac {f x^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2} - \frac {g p x^{4}}{8} + \frac {g x^{4} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{4} & \text {for}\: e \neq 0 \\\left (\frac {f x^{2}}{2} + \frac {g x^{4}}{4}\right ) \log {\left (c d^{p} \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-d**2*g*log(c*(d + e*x**2)**p)/(4*e**2) + d*f*log(c*(d + e*x**2)**p)/(2*e) + d*g*p*x**2/(4*e) - f*p
*x**2/2 + f*x**2*log(c*(d + e*x**2)**p)/2 - g*p*x**4/8 + g*x**4*log(c*(d + e*x**2)**p)/4, Ne(e, 0)), ((f*x**2/
2 + g*x**4/4)*log(c*d**p), True))

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Giac [A]
time = 4.26, size = 155, normalized size = 1.65 \begin {gather*} \frac {1}{8} \, {\left (2 \, {\left (x^{2} e + d\right )}^{2} g p \log \left (x^{2} e + d\right ) - {\left (x^{2} e + d\right )}^{2} g p + 2 \, {\left (x^{2} e + d\right )}^{2} g \log \left (c\right )\right )} e^{\left (-2\right )} + \frac {1}{2} \, {\left ({\left (x^{2} e - {\left (x^{2} e + d\right )} \log \left (x^{2} e + d\right ) + d\right )} d g p - {\left (x^{2} e - {\left (x^{2} e + d\right )} \log \left (x^{2} e + d\right ) + d\right )} f p e - {\left (x^{2} e + d\right )} d g \log \left (c\right ) + {\left (x^{2} e + d\right )} f e \log \left (c\right )\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(x^2*e + d)^2*g*p*log(x^2*e + d) - (x^2*e + d)^2*g*p + 2*(x^2*e + d)^2*g*log(c))*e^(-2) + 1/2*((x^2*e -
(x^2*e + d)*log(x^2*e + d) + d)*d*g*p - (x^2*e - (x^2*e + d)*log(x^2*e + d) + d)*f*p*e - (x^2*e + d)*d*g*log(
c) + (x^2*e + d)*f*e*log(c))*e^(-2)

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Mupad [B]
time = 0.31, size = 78, normalized size = 0.83 \begin {gather*} \ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {g\,x^4}{4}+\frac {f\,x^2}{2}\right )-x^2\,\left (\frac {f\,p}{2}-\frac {d\,g\,p}{4\,e}\right )-\frac {g\,p\,x^4}{8}-\frac {\ln \left (e\,x^2+d\right )\,\left (d^2\,g\,p-2\,d\,e\,f\,p\right )}{4\,e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*(d + e*x^2)^p)*((f*x^2)/2 + (g*x^4)/4) - x^2*((f*p)/2 - (d*g*p)/(4*e)) - (g*p*x^4)/8 - (log(d + e*x^2)*(
d^2*g*p - 2*d*e*f*p))/(4*e^2)

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