Integrand size = 25, antiderivative size = 178 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=-4 f g p x+\frac {2 d g^2 p x}{3 e}-\frac {2}{9} g^2 p x^3+\frac {2 \sqrt {e} f^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {4 \sqrt {d} f g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {2 d^{3/2} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+2 f g x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g^2 x^3 \log \left (c \left (d+e x^2\right )^p\right ) \] Output:
-4*f*g*p*x+2/3*d*g^2*p*x/e-2/9*g^2*p*x^3+2*e^(1/2)*f^2*p*arctan(e^(1/2)*x/ d^(1/2))/d^(1/2)+4*d^(1/2)*f*g*p*arctan(e^(1/2)*x/d^(1/2))/e^(1/2)-2/3*d^( 3/2)*g^2*p*arctan(e^(1/2)*x/d^(1/2))/e^(3/2)-f^2*ln(c*(e*x^2+d)^p)/x+2*f*g *x*ln(c*(e*x^2+d)^p)+1/3*g^2*x^3*ln(c*(e*x^2+d)^p)
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.63 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\frac {1}{9} \left (-\frac {2 g p x \left (18 e f-3 d g+e g x^2\right )}{e}+\frac {6 \left (3 e^2 f^2+6 d e f g-d^2 g^2\right ) p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}+\left (-\frac {9 f^2}{x}+18 f g x+3 g^2 x^3\right ) \log \left (c \left (d+e x^2\right )^p\right )\right ) \] Input:
Integrate[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^2,x]
Output:
((-2*g*p*x*(18*e*f - 3*d*g + e*g*x^2))/e + (6*(3*e^2*f^2 + 6*d*e*f*g - d^2 *g^2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(3/2)) + ((-9*f^2)/x + 18* f*g*x + 3*g^2*x^3)*Log[c*(d + e*x^2)^p])/9
Time = 0.59 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2926, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle \int \left (\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+2 f g \log \left (c \left (d+e x^2\right )^p\right )+g^2 x^2 \log \left (c \left (d+e x^2\right )^p\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 d^{3/2} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}+\frac {2 \sqrt {e} f^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {4 \sqrt {d} f g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+2 f g x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d g^2 p x}{3 e}-4 f g p x-\frac {2}{9} g^2 p x^3\) |
Input:
Int[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^2,x]
Output:
-4*f*g*p*x + (2*d*g^2*p*x)/(3*e) - (2*g^2*p*x^3)/9 + (2*Sqrt[e]*f^2*p*ArcT an[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (4*Sqrt[d]*f*g*p*ArcTan[(Sqrt[e]*x)/Sqr t[d]])/Sqrt[e] - (2*d^(3/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) - (f^2*Log[c*(d + e*x^2)^p])/x + 2*f*g*x*Log[c*(d + e*x^2)^p] + (g^2*x^3* Log[c*(d + e*x^2)^p])/3
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]
Time = 3.43 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.71
| method | result | size |
| parts | \(\frac {g^{2} x^{3} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{3}+2 f g x \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )-\frac {f^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{x}-\frac {2 e p \left (-\frac {g \left (-\frac {1}{3} e g \,x^{3}+x d g -6 e f x \right )}{e^{2}}+\frac {\left (d^{2} g^{2}-6 f g e d -3 f^{2} e^{2}\right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}\right )}{3}\) | \(127\) |
| risch | \(-\frac {\left (-g^{2} x^{4}-6 f g \,x^{2}+3 f^{2}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{3 x}+\frac {9 i \pi e \,f^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+18 i \pi e f g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+3 i \pi e \,g^{2} x^{4} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+18 i \pi e f g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-9 i \pi e \,f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-18 i \pi e f g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-9 i \pi e \,f^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+9 i \pi e \,f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-3 i \pi e \,g^{2} x^{4} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+3 i \pi e \,g^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-3 i \pi e \,g^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-18 i \pi e f g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+6 \ln \left (c \right ) e \,g^{2} x^{4}-4 e p \,g^{2} x^{4}+36 \ln \left (c \right ) e f g \,x^{2}+12 d \,g^{2} p \,x^{2}-72 e p f g \,x^{2}-18 \ln \left (c \right ) e \,f^{2}+6 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d^{4} g^{4} p^{2}-12 d^{3} e f \,g^{3} p^{2}+30 d^{2} e^{2} f^{2} g^{2} p^{2}+36 d \,e^{3} f^{3} g \,p^{2}+9 e^{4} f^{4} p^{2}+d \,\textit {\_Z}^{2} e \right )}{\sum }\textit {\_R} \ln \left (\left (2 d^{4} g^{4} p^{2}-24 d^{3} e f \,g^{3} p^{2}+60 d^{2} e^{2} f^{2} g^{2} p^{2}+72 d \,e^{3} f^{3} g \,p^{2}+18 e^{4} f^{4} p^{2}+3 \textit {\_R}^{2} d e \right ) x +\left (d^{3} g^{2} p -6 d^{2} e f g p -3 d \,e^{2} f^{2} p \right ) \textit {\_R} \right )\right ) x}{18 e x}\) | \(702\) |
Input:
int((g*x^2+f)^2*ln(c*(e*x^2+d)^p)/x^2,x,method=_RETURNVERBOSE)
Output:
1/3*g^2*x^3*ln(c*(e*x^2+d)^p)+2*f*g*x*ln(c*(e*x^2+d)^p)-f^2*ln(c*(e*x^2+d) ^p)/x-2/3*e*p*(-g/e^2*(-1/3*e*g*x^3+x*d*g-6*e*f*x)+(d^2*g^2-6*d*e*f*g-3*e^ 2*f^2)/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2)))
Time = 0.09 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.06 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\left [-\frac {2 \, d e^{2} g^{2} p x^{4} - 3 \, {\left (3 \, e^{2} f^{2} + 6 \, d e f g - d^{2} g^{2}\right )} \sqrt {-d e} p x \log \left (\frac {e x^{2} + 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) + 6 \, {\left (6 \, d e^{2} f g - d^{2} e g^{2}\right )} p x^{2} - 3 \, {\left (d e^{2} g^{2} p x^{4} + 6 \, d e^{2} f g p x^{2} - 3 \, d e^{2} f^{2} p\right )} \log \left (e x^{2} + d\right ) - 3 \, {\left (d e^{2} g^{2} x^{4} + 6 \, d e^{2} f g x^{2} - 3 \, d e^{2} f^{2}\right )} \log \left (c\right )}{9 \, d e^{2} x}, -\frac {2 \, d e^{2} g^{2} p x^{4} - 6 \, {\left (3 \, e^{2} f^{2} + 6 \, d e f g - d^{2} g^{2}\right )} \sqrt {d e} p x \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + 6 \, {\left (6 \, d e^{2} f g - d^{2} e g^{2}\right )} p x^{2} - 3 \, {\left (d e^{2} g^{2} p x^{4} + 6 \, d e^{2} f g p x^{2} - 3 \, d e^{2} f^{2} p\right )} \log \left (e x^{2} + d\right ) - 3 \, {\left (d e^{2} g^{2} x^{4} + 6 \, d e^{2} f g x^{2} - 3 \, d e^{2} f^{2}\right )} \log \left (c\right )}{9 \, d e^{2} x}\right ] \] Input:
integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^2,x, algorithm="fricas")
Output:
[-1/9*(2*d*e^2*g^2*p*x^4 - 3*(3*e^2*f^2 + 6*d*e*f*g - d^2*g^2)*sqrt(-d*e)* p*x*log((e*x^2 + 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) + 6*(6*d*e^2*f*g - d^2*e *g^2)*p*x^2 - 3*(d*e^2*g^2*p*x^4 + 6*d*e^2*f*g*p*x^2 - 3*d*e^2*f^2*p)*log( e*x^2 + d) - 3*(d*e^2*g^2*x^4 + 6*d*e^2*f*g*x^2 - 3*d*e^2*f^2)*log(c))/(d* e^2*x), -1/9*(2*d*e^2*g^2*p*x^4 - 6*(3*e^2*f^2 + 6*d*e*f*g - d^2*g^2)*sqrt (d*e)*p*x*arctan(sqrt(d*e)*x/d) + 6*(6*d*e^2*f*g - d^2*e*g^2)*p*x^2 - 3*(d *e^2*g^2*p*x^4 + 6*d*e^2*f*g*p*x^2 - 3*d*e^2*f^2*p)*log(e*x^2 + d) - 3*(d* e^2*g^2*x^4 + 6*d*e^2*f*g*x^2 - 3*d*e^2*f^2)*log(c))/(d*e^2*x)]
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (182) = 364\).
Time = 61.97 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.25 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\begin {cases} \left (- \frac {f^{2}}{x} + 2 f g x + \frac {g^{2} x^{3}}{3}\right ) \log {\left (0^{p} c \right )} & \text {for}\: d = 0 \wedge e = 0 \\\left (- \frac {f^{2}}{x} + 2 f g x + \frac {g^{2} x^{3}}{3}\right ) \log {\left (c d^{p} \right )} & \text {for}\: e = 0 \\- \frac {2 f^{2} p}{x} - \frac {f^{2} \log {\left (c \left (e x^{2}\right )^{p} \right )}}{x} - 4 f g p x + 2 f g x \log {\left (c \left (e x^{2}\right )^{p} \right )} - \frac {2 g^{2} p x^{3}}{9} + \frac {g^{2} x^{3} \log {\left (c \left (e x^{2}\right )^{p} \right )}}{3} & \text {for}\: d = 0 \\- \frac {2 d^{2} g^{2} p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{3 e^{2} \sqrt {- \frac {d}{e}}} + \frac {d^{2} g^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{3 e^{2} \sqrt {- \frac {d}{e}}} + \frac {4 d f g p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{e \sqrt {- \frac {d}{e}}} - \frac {2 d f g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{e \sqrt {- \frac {d}{e}}} + \frac {2 d g^{2} p x}{3 e} + \frac {2 f^{2} p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{\sqrt {- \frac {d}{e}}} - \frac {f^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\sqrt {- \frac {d}{e}}} - \frac {f^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x} - 4 f g p x + 2 f g x \log {\left (c \left (d + e x^{2}\right )^{p} \right )} - \frac {2 g^{2} p x^{3}}{9} + \frac {g^{2} x^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{3} & \text {otherwise} \end {cases} \] Input:
integrate((g*x**2+f)**2*ln(c*(e*x**2+d)**p)/x**2,x)
Output:
Piecewise(((-f**2/x + 2*f*g*x + g**2*x**3/3)*log(0**p*c), Eq(d, 0) & Eq(e, 0)), ((-f**2/x + 2*f*g*x + g**2*x**3/3)*log(c*d**p), Eq(e, 0)), (-2*f**2* p/x - f**2*log(c*(e*x**2)**p)/x - 4*f*g*p*x + 2*f*g*x*log(c*(e*x**2)**p) - 2*g**2*p*x**3/9 + g**2*x**3*log(c*(e*x**2)**p)/3, Eq(d, 0)), (-2*d**2*g** 2*p*log(x - sqrt(-d/e))/(3*e**2*sqrt(-d/e)) + d**2*g**2*log(c*(d + e*x**2) **p)/(3*e**2*sqrt(-d/e)) + 4*d*f*g*p*log(x - sqrt(-d/e))/(e*sqrt(-d/e)) - 2*d*f*g*log(c*(d + e*x**2)**p)/(e*sqrt(-d/e)) + 2*d*g**2*p*x/(3*e) + 2*f** 2*p*log(x - sqrt(-d/e))/sqrt(-d/e) - f**2*log(c*(d + e*x**2)**p)/sqrt(-d/e ) - f**2*log(c*(d + e*x**2)**p)/x - 4*f*g*p*x + 2*f*g*x*log(c*(d + e*x**2) **p) - 2*g**2*p*x**3/9 + g**2*x**3*log(c*(d + e*x**2)**p)/3, True))
Exception generated. \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.76 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=-\frac {1}{9} \, {\left (2 \, g^{2} p - 3 \, g^{2} \log \left (c\right )\right )} x^{3} + \frac {1}{3} \, {\left (g^{2} p x^{3} + 6 \, f g p x - \frac {3 \, f^{2} p}{x}\right )} \log \left (e x^{2} + d\right ) - \frac {f^{2} \log \left (c\right )}{x} - \frac {2 \, {\left (6 \, e f g p - d g^{2} p - 3 \, e f g \log \left (c\right )\right )} x}{3 \, e} + \frac {2 \, {\left (3 \, e^{2} f^{2} p + 6 \, d e f g p - d^{2} g^{2} p\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{3 \, \sqrt {d e} e} \] Input:
integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^2,x, algorithm="giac")
Output:
-1/9*(2*g^2*p - 3*g^2*log(c))*x^3 + 1/3*(g^2*p*x^3 + 6*f*g*p*x - 3*f^2*p/x )*log(e*x^2 + d) - f^2*log(c)/x - 2/3*(6*e*f*g*p - d*g^2*p - 3*e*f*g*log(c ))*x/e + 2/3*(3*e^2*f^2*p + 6*d*e*f*g*p - d^2*g^2*p)*arctan(e*x/sqrt(d*e)) /(sqrt(d*e)*e)
Time = 25.71 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.01 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\frac {2\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,p\,x\,\left (-d^2\,g^2+6\,d\,e\,f\,g+3\,e^2\,f^2\right )}{\sqrt {d}\,\left (-p\,d^2\,g^2+6\,p\,d\,e\,f\,g+3\,p\,e^2\,f^2\right )}\right )\,\left (-d^2\,g^2+6\,d\,e\,f\,g+3\,e^2\,f^2\right )}{3\,\sqrt {d}\,e^{3/2}}-x\,\left (4\,f\,g\,p-\frac {2\,d\,g^2\,p}{3\,e}\right )-\frac {2\,g^2\,p\,x^3}{9}-\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2+2\,f\,g\,x^2+g^2\,x^4}{x}-\frac {\frac {4\,g^2\,x^4}{3}+4\,f\,g\,x^2}{x}\right ) \] Input:
int((log(c*(d + e*x^2)^p)*(f + g*x^2)^2)/x^2,x)
Output:
(2*p*atan((e^(1/2)*p*x*(3*e^2*f^2 - d^2*g^2 + 6*d*e*f*g))/(d^(1/2)*(3*e^2* f^2*p - d^2*g^2*p + 6*d*e*f*g*p)))*(3*e^2*f^2 - d^2*g^2 + 6*d*e*f*g))/(3*d ^(1/2)*e^(3/2)) - x*(4*f*g*p - (2*d*g^2*p)/(3*e)) - (2*g^2*p*x^3)/9 - log( c*(d + e*x^2)^p)*((f^2 + g^2*x^4 + 2*f*g*x^2)/x - ((4*g^2*x^4)/3 + 4*f*g*x ^2)/x)
Time = 0.23 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.09 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx=\frac {-6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d^{2} g^{2} p x +36 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d e f g p x +18 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) e^{2} f^{2} p x -9 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d \,e^{2} f^{2}+18 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d \,e^{2} f g \,x^{2}+3 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d \,e^{2} g^{2} x^{4}+6 d^{2} e \,g^{2} p \,x^{2}-36 d \,e^{2} f g p \,x^{2}-2 d \,e^{2} g^{2} p \,x^{4}}{9 d \,e^{2} x} \] Input:
int((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^2,x)
Output:
( - 6*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d**2*g**2*p*x + 36*sqr t(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d*e*f*g*p*x + 18*sqrt(e)*sqrt(d )*atan((e*x)/(sqrt(e)*sqrt(d)))*e**2*f**2*p*x - 9*log((d + e*x**2)**p*c)*d *e**2*f**2 + 18*log((d + e*x**2)**p*c)*d*e**2*f*g*x**2 + 3*log((d + e*x**2 )**p*c)*d*e**2*g**2*x**4 + 6*d**2*e*g**2*p*x**2 - 36*d*e**2*f*g*p*x**2 - 2 *d*e**2*g**2*p*x**4)/(9*d*e**2*x)