\(\int \frac {\log (c (a+\frac {b}{x^2})^p)}{x^2 (d+e x)} \, dx\) [252]

Optimal result

Integrand size = 23, antiderivative size = 357 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {2 p}{d x}+\frac {2 \sqrt {a} p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^2}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^2}+\frac {2 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^2}+\frac {e p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^2}\right )}{2 d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^2}+\frac {2 e p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.92 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {\frac {4 d p}{x}-\frac {4 \sqrt {a} d p \arctan \left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {b}}-\frac {2 d \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}+e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )+2 e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)+4 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)-2 e p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)-2 e p \log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)+e p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^2}\right )-2 e p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )-2 e p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )+4 e p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 d^2} \] Input:

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

Output:

((4*d*p)/x - (4*Sqrt[a]*d*p*ArcTan[Sqrt[b]/(Sqrt[a]*x)])/Sqrt[b] - (2*d*Lo 
g[c*(a + b/x^2)^p])/x + e*Log[c*(a + b/x^2)^p]*Log[-(b/(a*x^2))] + 2*e*Log 
[c*(a + b/x^2)^p]*Log[d + e*x] + 4*e*p*Log[-((e*x)/d)]*Log[d + e*x] - 2*e* 
p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x] - 
2*e*p*Log[(e*(Sqrt[b] + Sqrt[-a]*x))/(-(Sqrt[-a]*d) + Sqrt[b]*e)]*Log[d + 
e*x] + e*p*PolyLog[2, 1 + b/(a*x^2)] - 2*e*p*PolyLog[2, (Sqrt[-a]*(d + e*x 
))/(Sqrt[-a]*d - Sqrt[b]*e)] - 2*e*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt 
[-a]*d + Sqrt[b]*e)] + 4*e*p*PolyLog[2, 1 + (e*x)/d])/(2*d^2)
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2916, 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.

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

\(\Big \downarrow \) 2916

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2916

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

Maple [A] (verified)

Time = 1.65 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.17

method	result	size
parts	\(\frac {e \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{d x}-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) e \ln \left (x \right )}{d^{2}}+2 p b \left (\frac {e \left (\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b}-\frac {\left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}+d a -a \left (e x +d \right )}{e \sqrt {-a b}+d a}\right )+\ln \left (\frac {e \sqrt {-a b}-d a +a \left (e x +d \right )}{e \sqrt {-a b}-d a}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}+d a -a \left (e x +d \right )}{e \sqrt {-a b}+d a}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}-d a +a \left (e x +d \right )}{e \sqrt {-a b}-d a}\right )}{2 a}\right ) a}{b}\right )}{d^{2}}+\frac {1}{d b x}+\frac {a \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{d b \sqrt {a b}}-\frac {e \left (\frac {\ln \left (x \right )^{2}}{2 b}-\frac {\left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-a x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {a x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {-a x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {a x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 a}\right ) a}{b}\right )}{d^{2}}\right )\)	\(418\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(log(((a*x**2 + b)**p*c)/x**(2*p))/(d*x**2 + e*x**3),x)