\(\int \frac {(f+g x^2) \log (c (d+e x^2)^p)}{x} \, dx\) [313]

Optimal result

Integrand size = 23, antiderivative size = 82 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=-\frac {1}{2} g p x^2+\frac {g \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e}+\frac {1}{2} f \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x} \, dx=\frac {1}{2} g \left (-p x^2+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}\right )+\frac {1}{2} f \left (\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {d+e x^2}{d}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91, 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.

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

\(\Big \downarrow \) 2925

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(74)=148\).

Time = 0.81 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.91

method	result	size
parts	\(\frac {x^{2} g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2}+\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (x \right )-e p \left (\frac {g \,x^{2}}{2 e}-\frac {g d \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+2 f \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}\right )\right )\)	\(157\)
risch	\(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) g \,x^{2}}{2}+\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (x \right )-\frac {g p \,x^{2}}{2}+\frac {p g d \ln \left (e \,x^{2}+d \right )}{2 e}-p f \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p f \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p f \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p f \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {g \,x^{2}}{2}+f \ln \left (x \right )\right )\)	\(278\)

Input:

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

Output:

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

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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