3.4.14 \(\int \frac {(f+g x^2) \log (c (d+e x^2)^p)}{x^3} \, dx\) [314]

3.4.14.1 Optimal result

Integrand size = 23, antiderivative size = 93 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\frac {e f p \log (x)}{d}-\frac {e f p \log \left (d+e x^2\right )}{2 d}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {1}{2} g \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} g p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right ) \]

output

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

3.4.14.2 Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^3} \, dx=\frac {e f p \log (x)}{d}-\frac {e f p \log \left (d+e x^2\right )}{2 d}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2}+\frac {1}{2} g \left (\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {d+e x^2}{d}\right )\right ) \]

input

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

output

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

3.4.14.3 Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.

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

\(\Big \downarrow \) 2925

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.4.14.3.1 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])
 

3.4.14.4 Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.67

method	result	size
parts	\(-\frac {f \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2 x^{2}}+\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) g \ln \left (x \right )-p e \left (\frac {f \ln \left (e \,x^{2}+d \right )}{2 d}-\frac {f \ln \left (x \right )}{d}+2 g \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}\right )\right )\)	\(155\)
risch	\(\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) g \ln \left (x \right )-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f}{2 x^{2}}-\frac {e f p \ln \left (e \,x^{2}+d \right )}{2 d}+\frac {e f p \ln \left (x \right )}{d}-p g \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p g \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p g \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p g \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (-\frac {f}{2 x^{2}}+g \ln \left (x \right )\right )\)	\(280\)

input

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

output

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

3.4.14.5 Fricas [F]

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

input

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

output

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

3.4.14.6 Sympy [F]

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

input

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

output

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

3.4.14.7 Maxima [F]

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

input

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

output

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

3.4.14.8 Giac [F]

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

input

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

output

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

3.4.14.9 Mupad [F(-1)]

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

input

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

output

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