Integrand size = 24, antiderivative size = 541 \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
1/2*ln(c*(d+e*x^(1/2))^p)*ln(e*((-f)^(1/4)-g^(1/4)*x^(1/2))/(e*(-f)^(1/4)+ d*g^(1/4)))/(-f)^(1/2)/g^(1/2)+1/2*ln(c*(d+e*x^(1/2))^p)*ln(e*((-f)^(1/4)+ g^(1/4)*x^(1/2))/(e*(-f)^(1/4)-d*g^(1/4)))/(-f)^(1/2)/g^(1/2)-1/2*ln(c*(d+ e*x^(1/2))^p)*ln(e*(g^(1/4)*x^(1/2)+(-(-f)^(1/2))^(1/2))/(-d*g^(1/4)+e*(-( -f)^(1/2))^(1/2)))/(-f)^(1/2)/g^(1/2)-1/2*ln(c*(d+e*x^(1/2))^p)*ln(e*(-g^( 1/4)*x^(1/2)+(-(-f)^(1/2))^(1/2))/(d*g^(1/4)+e*(-(-f)^(1/2))^(1/2)))/(-f)^ (1/2)/g^(1/2)+1/2*p*polylog(2,-g^(1/4)*(d+e*x^(1/2))/(e*(-f)^(1/4)-d*g^(1/ 4)))/(-f)^(1/2)/g^(1/2)+1/2*p*polylog(2,g^(1/4)*(d+e*x^(1/2))/(e*(-f)^(1/4 )+d*g^(1/4)))/(-f)^(1/2)/g^(1/2)-1/2*p*polylog(2,-g^(1/4)*(d+e*x^(1/2))/(- d*g^(1/4)+e*(-(-f)^(1/2))^(1/2)))/(-f)^(1/2)/g^(1/2)-1/2*p*polylog(2,g^(1/ 4)*(d+e*x^(1/2))/(d*g^(1/4)+e*(-(-f)^(1/2))^(1/2)))/(-f)^(1/2)/g^(1/2)
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.78 \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )-\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-i \sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+i \sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-i d \sqrt [4]{g}}\right )+\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )+p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )-p \operatorname {PolyLog}\left (2,\frac {i \sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{i e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
(Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e*(-f)^( 1/4) + d*g^(1/4))] - Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) - I*g^(1/ 4)*Sqrt[x]))/(e*(-f)^(1/4) + I*d*g^(1/4))] - Log[c*(d + e*Sqrt[x])^p]*Log[ (e*((-f)^(1/4) + I*g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) - I*d*g^(1/4))] + Log[c *(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) - d*g^(1/4))] + p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) - d* g^(1/4)))] - p*PolyLog[2, (I*g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + I*d* g^(1/4))] - p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(I*e*(-f)^(1/4) + d*g^( 1/4))] + p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4)) ])/(2*Sqrt[-f]*Sqrt[g])
Time = 0.94 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2922, 2863, 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.
| \(\displaystyle \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx\) |
| \(\Big \downarrow \) 2922 |
| \(\displaystyle 2 \int \frac {\sqrt {x} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{g x^2+f}d\sqrt {x}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle 2 \int \left (-\frac {\sqrt {g} \sqrt {x} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{2 \sqrt {-f} \left (\sqrt {-f} \sqrt {g}-g x\right )}-\frac {\sqrt {g} \sqrt {x} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{2 \sqrt {-f} \left (g x+\sqrt {-f} \sqrt {g}\right )}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt {-\sqrt {-f}}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{4 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{\sqrt [4]{g} d+e \sqrt {-\sqrt {-f}}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{\sqrt [4]{g} d+e \sqrt [4]{-f}}\right )}{4 \sqrt {-f} \sqrt {g}}\right )\) |
2*(-1/4*(Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x ]))/(e*Sqrt[-Sqrt[-f]] + d*g^(1/4))])/(Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e*S qrt[x])^p]*Log[(e*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4 ))])/(4*Sqrt[-f]*Sqrt[g]) - (Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[- f]] + g^(1/4)*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] - d*g^(1/4))])/(4*Sqrt[-f]*Sqrt [g]) + (Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(e *(-f)^(1/4) - d*g^(1/4))])/(4*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, -((g^(1/4) *(d + e*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] - d*g^(1/4)))])/(4*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) - d*g^(1/4)))])/ (4*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*Sqrt[-Sq rt[-f]] + d*g^(1/4))])/(4*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))])/(4*Sqrt[-f]*Sqrt[g]))
3.3.66.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> With[{k = Denominator[n]}, Simp[k S ubst[Int[x^(k - 1)*(f + g*x^(k*s))^r*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x] , x, x^(1/k)], x] /; IntegerQ[k*s]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && FractionQ[n]
\[\int \frac {\ln \left (c \left (d +e \sqrt {x}\right )^{p}\right )}{g \,x^{2}+f}d x\]
\[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
\[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^p\right )}{g\,x^2+f} \,d x \]