\(\int \frac {\log (c (a+b x^2)^p)}{x^2 (d+e x)} \, dx\) [231]

Optimal result

Integrand size = 23, antiderivative size = 306 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {2 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d^2} \] Output:

2*b^(1/2)*p*arctan(b^(1/2)*x/a^(1/2))/a^(1/2)/d-e*p*ln(e*((-a)^(1/2)-b^(1/ 
2)*x)/(b^(1/2)*d+(-a)^(1/2)*e))*ln(e*x+d)/d^2-e*p*ln(-e*((-a)^(1/2)+b^(1/2 
)*x)/(b^(1/2)*d-(-a)^(1/2)*e))*ln(e*x+d)/d^2-ln(c*(b*x^2+a)^p)/d/x-1/2*e*l 
n(-b*x^2/a)*ln(c*(b*x^2+a)^p)/d^2+e*ln(e*x+d)*ln(c*(b*x^2+a)^p)/d^2-e*p*po 
lylog(2,b^(1/2)*(e*x+d)/(b^(1/2)*d-(-a)^(1/2)*e))/d^2-e*p*polylog(2,b^(1/2 
)*(e*x+d)/(b^(1/2)*d+(-a)^(1/2)*e))/d^2-1/2*e*p*polylog(2,1+b*x^2/a)/d^2
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.92 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=-\frac {-\frac {4 \sqrt {b} d p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+2 e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+2 e p \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+\frac {2 d \log \left (c \left (a+b x^2\right )^p\right )}{x}+e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )-2 e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+2 e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+2 e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )+e p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 306, 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.

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

\(\Big \downarrow \) 2916

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*Sqrt[b]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*d) - (e*p*Log[(e*(Sqrt[ 
-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/d^2 - (e*p*Log[- 
((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/d^2 - 
 Log[c*(a + b*x^2)^p]/(d*x) - (e*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p])/( 
2*d^2) + (e*Log[d + e*x]*Log[c*(a + b*x^2)^p])/d^2 - (e*p*PolyLog[2, (Sqrt 
[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/d^2 - (e*p*PolyLog[2, (Sqrt[b]*( 
d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/d^2 - (e*p*PolyLog[2, 1 + (b*x^2)/a]) 
/(2*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 = 1.18 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.16

method	result	size
parts	\(\frac {e \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d^{2}}-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d x}-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-2 p b \left (\frac {e \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}\right )}{d^{2}}-\frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{d \sqrt {a b}}-\frac {e \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 b}\right )}{d^{2}}\right )\)	\(354\)
risch	\(\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{d x}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-\frac {p e \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d^{2}}-\frac {p e \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d^{2}}-\frac {p e \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d^{2}}-\frac {p e \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d^{2}}+\frac {2 p b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{d \sqrt {a b}}+\frac {p e \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\frac {p e \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\frac {p e \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\frac {p e \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )\)	\(514\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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