Integrand size = 23, antiderivative size = 371 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=-\frac {2 \sqrt {b} e p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {b p \log (x)}{a d}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d^3} \]
b*p*ln(x)/a/d-1/2*b*p*ln(b*x^2+a)/a/d-1/2*ln(c*(b*x^2+a)^p)/d/x^2+e*ln(c*( b*x^2+a)^p)/d^2/x+1/2*e^2*ln(-b*x^2/a)*ln(c*(b*x^2+a)^p)/d^3-e^2*ln(e*x+d) *ln(c*(b*x^2+a)^p)/d^3+e^2*p*ln(e*x+d)*ln(e*((-a)^(1/2)-x*b^(1/2))/(e*(-a) ^(1/2)+d*b^(1/2)))/d^3+e^2*p*ln(e*x+d)*ln(-e*((-a)^(1/2)+x*b^(1/2))/(-e*(- a)^(1/2)+d*b^(1/2)))/d^3+1/2*e^2*p*polylog(2,1+b*x^2/a)/d^3+e^2*p*polylog( 2,(e*x+d)*b^(1/2)/(-e*(-a)^(1/2)+d*b^(1/2)))/d^3+e^2*p*polylog(2,(e*x+d)*b ^(1/2)/(e*(-a)^(1/2)+d*b^(1/2)))/d^3-2*e*p*arctan(x*b^(1/2)/a^(1/2))*b^(1/ 2)/d^2/a^(1/2)
Time = 0.13 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.94 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\frac {-\frac {4 \sqrt {b} d e p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {2 b d^2 p \log (x)}{a}+2 e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+2 e^2 p \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)-\frac {b d^2 p \log \left (a+b x^2\right )}{a}-\frac {d^2 \log \left (c \left (a+b x^2\right )^p\right )}{x^2}+\frac {2 d e \log \left (c \left (a+b x^2\right )^p\right )}{x}-2 e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+2 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+2 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )+e^2 \left (\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )+p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )\right )}{2 d^3} \]
((-4*Sqrt[b]*d*e*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (2*b*d^2*p*Log[x ])/a + 2*e^2*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Lo g[d + e*x] + 2*e^2*p*Log[(e*(Sqrt[-a] + Sqrt[b]*x))/(-(Sqrt[b]*d) + Sqrt[- a]*e)]*Log[d + e*x] - (b*d^2*p*Log[a + b*x^2])/a - (d^2*Log[c*(a + b*x^2)^ p])/x^2 + (2*d*e*Log[c*(a + b*x^2)^p])/x - 2*e^2*Log[d + e*x]*Log[c*(a + b *x^2)^p] + 2*e^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e) ] + 2*e^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)] + e^2 *(Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p] + p*PolyLog[2, 1 + (b*x^2)/a]))/( 2*d^3)
Time = 0.65 (sec) , antiderivative size = 371, 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.
| \(\displaystyle \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 2916 |
| \(\displaystyle \int \left (-\frac {e^3 \log \left (c \left (a+b x^2\right )^p\right )}{d^3 (d+e x)}+\frac {e^2 \log \left (c \left (a+b x^2\right )^p\right )}{d^3 x}-\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x^2}+\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {b} e p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{2 d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}+\frac {b p \log (x)}{a d}\) |
(-2*Sqrt[b]*e*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*d^2) + (b*p*Log[x])/ (a*d) + (e^2*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Lo g[d + e*x])/d^3 + (e^2*p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqr t[-a]*e))]*Log[d + e*x])/d^3 - (b*p*Log[a + b*x^2])/(2*a*d) - Log[c*(a + b *x^2)^p]/(2*d*x^2) + (e*Log[c*(a + b*x^2)^p])/(d^2*x) + (e^2*Log[-((b*x^2) /a)]*Log[c*(a + b*x^2)^p])/(2*d^3) - (e^2*Log[d + e*x]*Log[c*(a + b*x^2)^p ])/d^3 + (e^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/ d^3 + (e^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/d^3 + (e^2*p*PolyLog[2, 1 + (b*x^2)/a])/(2*d^3)
3.3.32.3.1 Defintions of rubi rules used
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]
Time = 1.40 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.11
| method | result | size |
| parts | \(-\frac {e^{2} \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d^{3}}-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {e \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d^{2} x}-p b \left (-\frac {\ln \left (x \right )}{d a}+\frac {\ln \left (b \,x^{2}+a \right )}{2 d a}+\frac {2 e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{d^{2} \sqrt {a b}}+\frac {2 e^{2} \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 b}\right )}{d^{3}}-\frac {2 e^{2} \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}\right )}{d^{3}}\right )\) | \(410\) |
| risch | \(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e}{d^{2} x}+\frac {b p \ln \left (x \right )}{a d}-\frac {b p \ln \left (b \,x^{2}+a \right )}{2 a d}-\frac {2 p b e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{d^{2} \sqrt {a b}}-\frac {p \,e^{2} \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{3}}-\frac {p \,e^{2} \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{3}}-\frac {p \,e^{2} \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{3}}-\frac {p \,e^{2} \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{3}}+\frac {p \,e^{2} \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^{3}}+\frac {p \,e^{2} \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^{3}}+\frac {p \,e^{2} \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d^{3}}+\frac {p \,e^{2} \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d^{3}}+\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^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {1}{2 d \,x^{2}}+\frac {e^{2} \ln \left (x \right )}{d^{3}}+\frac {e}{d^{2} x}\right )\) | \(594\) |
-e^2*ln(e*x+d)*ln(c*(b*x^2+a)^p)/d^3-1/2*ln(c*(b*x^2+a)^p)/d/x^2+ln(c*(b*x ^2+a)^p)*e^2/d^3*ln(x)+e*ln(c*(b*x^2+a)^p)/d^2/x-p*b*(-1/d/a*ln(x)+1/2/d/a *ln(b*x^2+a)+2/d^2*e/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+2*e^2/d^3*(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)-2*e^2/d^3*(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))+dilog((e*(-a*b)^(1/2)+(e*x+d)*b-b*d)/(e*(-a*b)^(1/2)-b*d)))/b))
\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{x^3\,\left (d+e\,x\right )} \,d x \]