Integrand size = 22, antiderivative size = 477 \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {3 n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}} \]
1/2*ln(c*(b*x+a)^n)^3*ln(b*((-d)^(1/2)-x*e^(1/2))/(b*(-d)^(1/2)+a*e^(1/2)) )/(-d)^(1/2)/e^(1/2)-1/2*ln(c*(b*x+a)^n)^3*ln(b*((-d)^(1/2)+x*e^(1/2))/(b* (-d)^(1/2)-a*e^(1/2)))/(-d)^(1/2)/e^(1/2)-3/2*n*ln(c*(b*x+a)^n)^2*polylog( 2,-(b*x+a)*e^(1/2)/(b*(-d)^(1/2)-a*e^(1/2)))/(-d)^(1/2)/e^(1/2)+3/2*n*ln(c *(b*x+a)^n)^2*polylog(2,(b*x+a)*e^(1/2)/(b*(-d)^(1/2)+a*e^(1/2)))/(-d)^(1/ 2)/e^(1/2)+3*n^2*ln(c*(b*x+a)^n)*polylog(3,-(b*x+a)*e^(1/2)/(b*(-d)^(1/2)- a*e^(1/2)))/(-d)^(1/2)/e^(1/2)-3*n^2*ln(c*(b*x+a)^n)*polylog(3,(b*x+a)*e^( 1/2)/(b*(-d)^(1/2)+a*e^(1/2)))/(-d)^(1/2)/e^(1/2)-3*n^3*polylog(4,-(b*x+a) *e^(1/2)/(b*(-d)^(1/2)-a*e^(1/2)))/(-d)^(1/2)/e^(1/2)+3*n^3*polylog(4,(b*x +a)*e^(1/2)/(b*(-d)^(1/2)+a*e^(1/2)))/(-d)^(1/2)/e^(1/2)
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.58 \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\frac {-2 n^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log ^3(a+b x)+6 n^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log ^2(a+b x) \log \left (c (a+b x)^n\right )-6 n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (a+b x) \log ^2\left (c (a+b x)^n\right )+2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log ^3\left (c (a+b x)^n\right )+i n^3 \log ^3(a+b x) \log \left (1-\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )-3 i n^2 \log ^2(a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )+3 i n \log (a+b x) \log ^2\left (c (a+b x)^n\right ) \log \left (1-\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )-i n^3 \log ^3(a+b x) \log \left (1-\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )+3 i n^2 \log ^2(a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )-3 i n \log (a+b x) \log ^2\left (c (a+b x)^n\right ) \log \left (1-\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )+3 i n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )-3 i n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )-6 i n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )+6 i n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )+6 i n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )-6 i n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )}{2 \sqrt {d} \sqrt {e}} \]
(-2*n^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[a + b*x]^3 + 6*n^2*ArcTan[(Sqrt[e] *x)/Sqrt[d]]*Log[a + b*x]^2*Log[c*(a + b*x)^n] - 6*n*ArcTan[(Sqrt[e]*x)/Sq rt[d]]*Log[a + b*x]*Log[c*(a + b*x)^n]^2 + 2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*L og[c*(a + b*x)^n]^3 + I*n^3*Log[a + b*x]^3*Log[1 - (Sqrt[e]*(a + b*x))/((- I)*b*Sqrt[d] + a*Sqrt[e])] - (3*I)*n^2*Log[a + b*x]^2*Log[c*(a + b*x)^n]*L og[1 - (Sqrt[e]*(a + b*x))/((-I)*b*Sqrt[d] + a*Sqrt[e])] + (3*I)*n*Log[a + b*x]*Log[c*(a + b*x)^n]^2*Log[1 - (Sqrt[e]*(a + b*x))/((-I)*b*Sqrt[d] + a *Sqrt[e])] - I*n^3*Log[a + b*x]^3*Log[1 - (Sqrt[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])] + (3*I)*n^2*Log[a + b*x]^2*Log[c*(a + b*x)^n]*Log[1 - (Sqrt [e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])] - (3*I)*n*Log[a + b*x]*Log[c*(a + b*x)^n]^2*Log[1 - (Sqrt[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])] + (3*I) *n*Log[c*(a + b*x)^n]^2*PolyLog[2, (Sqrt[e]*(a + b*x))/((-I)*b*Sqrt[d] + a *Sqrt[e])] - (3*I)*n*Log[c*(a + b*x)^n]^2*PolyLog[2, (Sqrt[e]*(a + b*x))/( I*b*Sqrt[d] + a*Sqrt[e])] - (6*I)*n^2*Log[c*(a + b*x)^n]*PolyLog[3, (Sqrt[ e]*(a + b*x))/((-I)*b*Sqrt[d] + a*Sqrt[e])] + (6*I)*n^2*Log[c*(a + b*x)^n] *PolyLog[3, (Sqrt[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])] + (6*I)*n^3*Pol yLog[4, (Sqrt[e]*(a + b*x))/((-I)*b*Sqrt[d] + a*Sqrt[e])] - (6*I)*n^3*Poly Log[4, (Sqrt[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])])/(2*Sqrt[d]*Sqrt[e])
Time = 0.80 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2856, 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.
| \(\displaystyle \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx\) |
| \(\Big \downarrow \) 2856 |
| \(\displaystyle \int \left (\frac {\sqrt {-d} \log ^3\left (c (a+b x)^n\right )}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log ^3\left (c (a+b x)^n\right )}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {e} (a+b x)}{\sqrt {e} a+b \sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (a+b x)}{\sqrt {e} a+b \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{a \sqrt {e}+b \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {3 n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {3 n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} (a+b x)}{\sqrt {e} a+b \sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}\) |
(Log[c*(a + b*x)^n]^3*Log[(b*(Sqrt[-d] - Sqrt[e]*x))/(b*Sqrt[-d] + a*Sqrt[ e])])/(2*Sqrt[-d]*Sqrt[e]) - (Log[c*(a + b*x)^n]^3*Log[(b*(Sqrt[-d] + Sqrt [e]*x))/(b*Sqrt[-d] - a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(2* Sqrt[-d]*Sqrt[e]) + (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, (Sqrt[e]*(a + b*x ))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) + (3*n^2*Log[c*(a + b*x )^n]*PolyLog[3, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d ]*Sqrt[e]) - (3*n^2*Log[c*(a + b*x)^n]*PolyLog[3, (Sqrt[e]*(a + b*x))/(b*S qrt[-d] + a*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) - (3*n^3*PolyLog[4, -((Sqrt[e]*( a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) + (3*n^3*PolyLog[ 4, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(Sqrt[-d]*Sqrt[e])
3.4.29.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. )*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) ^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
\[\int \frac {\ln \left (c \left (b x +a \right )^{n}\right )^{3}}{e \,x^{2}+d}d x\]
\[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d} \,d x } \]
\[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{3}}{d + e x^{2}}\, dx \]
Exception generated. \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d} \,d x } \]
Timed out. \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^3}{e\,x^2+d} \,d x \]