\(\int \frac {\log (c (a+b x^2)^p)}{x (d+e x)} \, dx\) [230]

Optimal result

Integrand size = 23, antiderivative size = 247 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x (d+e x)} \, dx=\frac {p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d}+\frac {\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d}-\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.93 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x (d+e x)} \, dx=\frac {2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+2 p \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )-2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )+p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 247, 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.

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

\(\Big \downarrow \) 2916

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 1.10 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.26

method	result	size
parts	\(-\frac {\ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d}+\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) \ln \left (x \right )}{d}-2 p b \left (\frac {\frac {\ln \left (x \right ) \left (\ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 b}}{d}-\frac {\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}}{d}\right )\)	\(311\)
risch	\(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (e x +d \right )}{d}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (x \right )}{d}-\frac {p \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}-\frac {p \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}-\frac {p \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}-\frac {p \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}+\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d}+\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (-\frac {\ln \left (e x +d \right )}{d}+\frac {\ln \left (x \right )}{d}\right )\)	\(455\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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