\(\int \frac {\log (c (a+\frac {b}{x^2})^p)}{d+e x} \, dx\) [250]

Optimal result

Integrand size = 20, antiderivative size = 241 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx=\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e}+\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e}-\frac {p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e}+\frac {2 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx=\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e}+\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e}-\frac {p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{e}+\frac {2 p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e} \] Input:

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

Output:

(Log[c*(a + b/x^2)^p]*Log[d + e*x])/e + (2*p*Log[-((e*x)/d)]*Log[d + e*x]) 
/e - (p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e 
*x])/e - (p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Lo 
g[d + e*x])/e + (2*p*PolyLog[2, (d + e*x)/d])/e - (p*PolyLog[2, (Sqrt[-a]* 
(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/e - (p*PolyLog[2, (Sqrt[-a]*(d + e*x 
))/(Sqrt[-a]*d + Sqrt[b]*e)])/e
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2912, 2005, 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.

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

\(\Big \downarrow \) 2912

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 2863

\(\displaystyle \frac {2 b p \int \left (\frac {\log (d+e x)}{b x}-\frac {a x \log (d+e x)}{b \left (a x^2+b\right )}\right )dx}{e}+\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Log[c*(a + b/x^2)^p]*Log[d + e*x])/e + (2*b*p*((Log[-((e*x)/d)]*Log[d + e 
*x])/b - (Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + 
 e*x])/(2*b) - (Log[-((e*(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e)) 
]*Log[d + e*x])/(2*b) - PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt 
[b]*e)]/(2*b) - PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)]/ 
(2*b) + PolyLog[2, 1 + (e*x)/d]/b))/e
 

Defintions of rubi rules used

rule 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

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 2912

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

Maple [A] (verified)

Time = 2.02 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.97

method	result	size
parts	\(\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \ln \left (e x +d \right )}{e}+2 p b e \left (\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b \,e^{2}}+\frac {\left (-\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}+d a -a \left (e x +d \right )}{e \sqrt {-a b}+d a}\right )+\ln \left (\frac {e \sqrt {-a b}-d a +a \left (e x +d \right )}{e \sqrt {-a b}-d a}\right )\right )}{2 a}-\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}+d a -a \left (e x +d \right )}{e \sqrt {-a b}+d a}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}-d a +a \left (e x +d \right )}{e \sqrt {-a b}-d a}\right )}{2 a}\right ) a}{b \,e^{2}}\right )\)	\(234\)

Input:

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

Output:

ln(c*(a+b/x^2)^p)*ln(e*x+d)/e+2*p*b*e*((dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d) 
)/b/e^2+(-1/2*ln(e*x+d)*(ln((e*(-a*b)^(1/2)+d*a-a*(e*x+d))/(e*(-a*b)^(1/2) 
+d*a))+ln((e*(-a*b)^(1/2)-d*a+a*(e*x+d))/(e*(-a*b)^(1/2)-d*a)))/a-1/2*(dil 
og((e*(-a*b)^(1/2)+d*a-a*(e*x+d))/(e*(-a*b)^(1/2)+d*a))+dilog((e*(-a*b)^(1 
/2)-d*a+a*(e*x+d))/(e*(-a*b)^(1/2)-d*a)))/a)/b*a/e^2)
 

Fricas [F]

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

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

Output:

integral(log(c*((a*x^2 + b)/x^2)^p)/(e*x + d), x)
 

Sympy [F]

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

integrate(ln(c*(a+b/x**2)**p)/(e*x+d),x)
 

Output:

Integral(log(c*(a + b/x**2)**p)/(d + e*x), x)
 

Maxima [F]

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

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

Output:

integrate(log((a + b/x^2)^p*c)/(e*x + d), x)
 

Giac [F]

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

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

Output:

integrate(log((a + b/x^2)^p*c)/(e*x + d), x)
 

Mupad [F(-1)]

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

int(log(c*(a + b/x^2)^p)/(d + e*x),x)
 

Output:

int(log(c*(a + b/x^2)^p)/(d + e*x), x)
 

Reduce [F]

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

int(log(c*(a+b/x^2)^p)/(e*x+d),x)
 

Output:

int(log(((a*x**2 + b)**p*c)/x**(2*p))/(d + e*x),x)