\(\int \frac {\log (c (a+\frac {b}{x^2})^p)}{x^3 (d+e x)} \, dx\) [253]

Optimal result

Integrand size = 23, antiderivative size = 414 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3 (d+e x)} \, dx=\frac {p}{2 d x^2}-\frac {2 e p}{d^2 x}-\frac {2 \sqrt {a} e p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^2}\right )}{2 d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^3}-\frac {2 e^2 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d^3} \] Output:

1/2*p/d/x^2-2*e*p/d^2/x-2*a^(1/2)*e*p*arctan(a^(1/2)*x/b^(1/2))/b^(1/2)/d^ 
2-1/2*(a+b/x^2)*ln(c*(a+b/x^2)^p)/b/d+e*ln(c*(a+b/x^2)^p)/d^2/x-1/2*e^2*ln 
(c*(a+b/x^2)^p)*ln(-b/a/x^2)/d^3-e^2*ln(c*(a+b/x^2)^p)*ln(e*x+d)/d^3-2*e^2 
*p*ln(-e*x/d)*ln(e*x+d)/d^3+e^2*p*ln(e*(b^(1/2)-(-a)^(1/2)*x)/((-a)^(1/2)* 
d+b^(1/2)*e))*ln(e*x+d)/d^3+e^2*p*ln(-e*(b^(1/2)+(-a)^(1/2)*x)/((-a)^(1/2) 
*d-b^(1/2)*e))*ln(e*x+d)/d^3-1/2*e^2*p*polylog(2,1+b/a/x^2)/d^3+e^2*p*poly 
log(2,(-a)^(1/2)*(e*x+d)/((-a)^(1/2)*d-b^(1/2)*e))/d^3+e^2*p*polylog(2,(-a 
)^(1/2)*(e*x+d)/((-a)^(1/2)*d+b^(1/2)*e))/d^3-2*e^2*p*polylog(2,1+e*x/d)/d 
^3
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.93 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3 (d+e x)} \, dx=\frac {-\frac {4 d e p}{x}+\frac {4 \sqrt {a} d e p \arctan \left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {b}}+\frac {2 d e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}+d^2 \left (\frac {p}{x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{b}\right )-e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )-2 e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)-4 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)+2 e^2 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)+2 e^2 p \log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)-e^2 p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^2}\right )+2 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )+2 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )-4 e^2 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 d^3} \] Input:

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

Output:

((-4*d*e*p)/x + (4*Sqrt[a]*d*e*p*ArcTan[Sqrt[b]/(Sqrt[a]*x)])/Sqrt[b] + (2 
*d*e*Log[c*(a + b/x^2)^p])/x + d^2*(p/x^2 - ((a + b/x^2)*Log[c*(a + b/x^2) 
^p])/b) - e^2*Log[c*(a + b/x^2)^p]*Log[-(b/(a*x^2))] - 2*e^2*Log[c*(a + b/ 
x^2)^p]*Log[d + e*x] - 4*e^2*p*Log[-((e*x)/d)]*Log[d + e*x] + 2*e^2*p*Log[ 
(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x] + 2*e^2* 
p*Log[(e*(Sqrt[b] + Sqrt[-a]*x))/(-(Sqrt[-a]*d) + Sqrt[b]*e)]*Log[d + e*x] 
 - e^2*p*PolyLog[2, 1 + b/(a*x^2)] + 2*e^2*p*PolyLog[2, (Sqrt[-a]*(d + e*x 
))/(Sqrt[-a]*d - Sqrt[b]*e)] + 2*e^2*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sq 
rt[-a]*d + Sqrt[b]*e)] - 4*e^2*p*PolyLog[2, 1 + (e*x)/d])/(2*d^3)
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 414, 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.

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

\(\Big \downarrow \) 2916

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {a} e p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {e^2 \log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^3}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}-\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {b}{a x^2}+1\right )}{2 d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^3}-\frac {2 e^2 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}-\frac {2 e p}{d^2 x}+\frac {p}{2 d x^2}\)

Input:

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

Output:

p/(2*d*x^2) - (2*e*p)/(d^2*x) - (2*Sqrt[a]*e*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]] 
)/(Sqrt[b]*d^2) - ((a + b/x^2)*Log[c*(a + b/x^2)^p])/(2*b*d) + (e*Log[c*(a 
 + b/x^2)^p])/(d^2*x) - (e^2*Log[c*(a + b/x^2)^p]*Log[-(b/(a*x^2))])/(2*d^ 
3) - (e^2*Log[c*(a + b/x^2)^p]*Log[d + e*x])/d^3 - (2*e^2*p*Log[-((e*x)/d) 
]*Log[d + e*x])/d^3 + (e^2*p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + 
Sqrt[b]*e)]*Log[d + e*x])/d^3 + (e^2*p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x))/(S 
qrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/d^3 - (e^2*p*PolyLog[2, 1 + b/(a*x^ 
2)])/(2*d^3) + (e^2*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b 
]*e)])/d^3 + (e^2*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]* 
e)])/d^3 - (2*e^2*p*PolyLog[2, 1 + (e*x)/d])/d^3
 

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 = 2.08 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.18

method	result	size
parts	\(-\frac {e^{2} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \ln \left (e x +d \right )}{d^{3}}-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {e \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{d^{2} x}+p b \left (\frac {1}{2 d b \,x^{2}}-\frac {2 e}{d^{2} b x}+\frac {a \ln \left (x \right )}{d \,b^{2}}-\frac {a \ln \left (a \,x^{2}+b \right )}{2 d \,b^{2}}-\frac {2 a e \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{d^{2} b \sqrt {a b}}+\frac {2 e^{2} \left (\frac {\ln \left (x \right )^{2}}{2 b}-\frac {\left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-a x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {a x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {-a x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {a x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 a}\right ) a}{b}\right )}{d^{3}}-\frac {2 e^{2} \left (\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b}-\frac {\left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}+d a -a \left (e x +d \right )}{e \sqrt {-a b}+d a}\right )+\ln \left (\frac {e \sqrt {-a b}-d a +a \left (e x +d \right )}{e \sqrt {-a b}-d a}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}+d a -a \left (e x +d \right )}{e \sqrt {-a b}+d a}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}-d a +a \left (e x +d \right )}{e \sqrt {-a b}-d a}\right )}{2 a}\right ) a}{b}\right )}{d^{3}}\right )\)	\(488\)

Input:

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

Output:

-e^2*ln(c*(a+b/x^2)^p)*ln(e*x+d)/d^3-1/2*ln(c*(a+b/x^2)^p)/d/x^2+ln(c*(a+b 
/x^2)^p)*e^2/d^3*ln(x)+e*ln(c*(a+b/x^2)^p)/d^2/x+p*b*(1/2/d/b/x^2-2/d^2*e/ 
b/x+1/d/b^2*a*ln(x)-1/2/d/b^2*a*ln(a*x^2+b)-2/d^2/b*a*e/(a*b)^(1/2)*arctan 
(a*x/(a*b)^(1/2))+2*e^2/d^3*(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 
)-2*e^2/d^3*((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))
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

integrate(ln(c*(a+b/x**2)**p)/x**3/(e*x+d),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 4*int(log(((a*x**2 + b)**p*c)/x**(2*p))/(a*d*x**4 + a*e*x**5 + b*d*x** 
2 + b*e*x**3),x)*b**2*e*p*x**2 + 4*int(log(((a*x**2 + b)**p*c)/x**(2*p))/( 
a*d*x**3 + a*e*x**4 + b*d*x + b*e*x**2),x)*a*b*d*p*x**2 + log(((a*x**2 + b 
)**p*c)/x**(2*p))**2*a*x**2 - 2*log(((a*x**2 + b)**p*c)/x**(2*p))*a*p*x**2 
 - 2*log(((a*x**2 + b)**p*c)/x**(2*p))*b*p + 2*b*p**2)/(4*b*d*p*x**2)