\(\int \frac {(f+g x^2)^2 \log (c (d+e x^2)^p)}{x^3} \, dx\) [327]

Optimal result

Integrand size = 25, antiderivative size = 135 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=-\frac {1}{2} g^2 p x^2+\frac {e f^2 p \log (x)}{d}-\frac {e f^2 p \log \left (d+e x^2\right )}{2 d}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {g^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}+f g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f g p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\frac {e f^2 p \log (x)}{d}-\frac {e f^2 p \log \left (d+e x^2\right )}{2 d}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}-\frac {1}{2} g^2 \left (p x^2-\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}\right )+f g \left (\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {d+e x^2}{d}\right )\right ) \] Input:

Integrate[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2925, 2863, 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.

\(\displaystyle \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx\)

\(\Big \downarrow \) 2925

\(\displaystyle \frac {1}{2} \int \frac {\left (g x^2+f\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{x^4}dx^2\)

\(\Big \downarrow \) 2863

\(\displaystyle \frac {1}{2} \int \left (\frac {\log \left (c \left (e x^2+d\right )^p\right ) f^2}{x^4}+\frac {2 g \log \left (c \left (e x^2+d\right )^p\right ) f}{x^2}+g^2 \log \left (c \left (e x^2+d\right )^p\right )\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+2 f g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {g^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {e f^2 p \log \left (x^2\right )}{d}-\frac {e f^2 p \log \left (d+e x^2\right )}{d}+2 f g p \operatorname {PolyLog}\left (2,\frac {e x^2}{d}+1\right )-g^2 p x^2\right )\)

Input:

Int[((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^3,x]
 

Output:

(-(g^2*p*x^2) + (e*f^2*p*Log[x^2])/d - (e*f^2*p*Log[d + e*x^2])/d - (f^2*L 
og[c*(d + e*x^2)^p])/x^2 + (g^2*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e + 2*f* 
g*Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p] + 2*f*g*p*PolyLog[2, 1 + (e*x^2)/ 
d])/2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2863

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

rule 2925

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Si 
mplify[(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] && Integer 
Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 
] || IGtQ[q, 0])
 

Maple [A] (verified)

Time = 2.74 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.56

method	result	size
parts	\(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) g^{2} x^{2}}{2}-\frac {f^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2 x^{2}}+2 \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) g f \ln \left (x \right )-e p \left (\frac {g^{2} x^{2}}{2 e}-\frac {\left (d^{2} g^{2}-f^{2} e^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 d \,e^{2}}-\frac {f^{2} \ln \left (x \right )}{d}+4 g f \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}\right )\right )\)	\(211\)
risch	\(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) g^{2} x^{2}}{2}+2 \ln \left (\left (e \,x^{2}+d \right )^{p}\right ) g f \ln \left (x \right )-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f^{2}}{2 x^{2}}-\frac {g^{2} p \,x^{2}}{2}+\frac {p d \ln \left (e \,x^{2}+d \right ) g^{2}}{2 e}-\frac {e \,f^{2} p \ln \left (e \,x^{2}+d \right )}{2 d}+\frac {e \,f^{2} p \ln \left (x \right )}{d}-2 p g f \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-2 p g f \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-2 p g f \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-2 p g f \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {g^{2} x^{2}}{2}-\frac {f^{2}}{2 x^{2}}+2 g f \ln \left (x \right )\right )\)	\(349\)

Input:

int((g*x^2+f)^2*ln(c*(e*x^2+d)^p)/x^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}} \,d x } \] Input:

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^3,x, algorithm="fricas")
 

Output:

integral((g^2*x^4 + 2*f*g*x^2 + f^2)*log((e*x^2 + d)^p*c)/x^3, x)
 

Sympy [F]

\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\int \frac {\left (f + g x^{2}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x^{3}}\, dx \] Input:

integrate((g*x**2+f)**2*ln(c*(e*x**2+d)**p)/x**3,x)
 

Output:

Integral((f + g*x**2)**2*log(c*(d + e*x**2)**p)/x**3, x)
 

Maxima [F]

\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}} \,d x } \] Input:

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^3,x, algorithm="maxima")
 

Output:

integrate((g*x^2 + f)^2*log((e*x^2 + d)^p*c)/x^3, x)
 

Giac [F]

\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{3}} \,d x } \] Input:

integrate((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^3,x, algorithm="giac")
 

Output:

integrate((g*x^2 + f)^2*log((e*x^2 + d)^p*c)/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,{\left (g\,x^2+f\right )}^2}{x^3} \,d x \] Input:

int((log(c*(d + e*x^2)^p)*(f + g*x^2)^2)/x^3,x)
 

Output:

int((log(c*(d + e*x^2)^p)*(f + g*x^2)^2)/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\frac {-4 \left (\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}{e \,x^{5}+d \,x^{3}}d x \right ) d^{3} f g p \,x^{2}+{\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}^{2} d e f g \,x^{2}-2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{2} f g p +\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{2} g^{2} p \,x^{2}-\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d e \,f^{2} p -2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d e f g p \,x^{2}+\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d e \,g^{2} p \,x^{4}-\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{2} f^{2} p \,x^{2}+4 \,\mathrm {log}\left (x \right ) d e f g \,p^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) e^{2} f^{2} p^{2} x^{2}-d e \,g^{2} p^{2} x^{4}}{2 d e p \,x^{2}} \] Input:

int((g*x^2+f)^2*log(c*(e*x^2+d)^p)/x^3,x)
 

Output:

( - 4*int(log((d + e*x**2)**p*c)/(d*x**3 + e*x**5),x)*d**3*f*g*p*x**2 + lo 
g((d + e*x**2)**p*c)**2*d*e*f*g*x**2 - 2*log((d + e*x**2)**p*c)*d**2*f*g*p 
 + log((d + e*x**2)**p*c)*d**2*g**2*p*x**2 - log((d + e*x**2)**p*c)*d*e*f* 
*2*p - 2*log((d + e*x**2)**p*c)*d*e*f*g*p*x**2 + log((d + e*x**2)**p*c)*d* 
e*g**2*p*x**4 - log((d + e*x**2)**p*c)*e**2*f**2*p*x**2 + 4*log(x)*d*e*f*g 
*p**2*x**2 + 2*log(x)*e**2*f**2*p**2*x**2 - d*e*g**2*p**2*x**4)/(2*d*e*p*x 
**2)