Integrand size = 38, antiderivative size = 519 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \operatorname {PolyLog}\left (2,-\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \operatorname {PolyLog}\left (2,-\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}} \]
1/2*I*b*g*p*q*arcsin(h*x/g)^2*(1-h^2*x^2/g^2)^(1/2)/h/(-h*x+g)^(1/2)/(h*x+ g)^(1/2)+g*arcsin(h*x/g)*(a+b*ln(c*(d*(f*x+e)^p)^q))*(1-h^2*x^2/g^2)^(1/2) /h/(-h*x+g)^(1/2)/(h*x+g)^(1/2)-b*g*p*q*arcsin(h*x/g)*ln(1+(I*h*x/g+(1-h^2 *x^2/g^2)^(1/2))*f*g/(I*e*h-(-e^2*h^2+f^2*g^2)^(1/2)))*(1-h^2*x^2/g^2)^(1/ 2)/h/(-h*x+g)^(1/2)/(h*x+g)^(1/2)-b*g*p*q*arcsin(h*x/g)*ln(1+(I*h*x/g+(1-h ^2*x^2/g^2)^(1/2))*f*g/(I*e*h+(-e^2*h^2+f^2*g^2)^(1/2)))*(1-h^2*x^2/g^2)^( 1/2)/h/(-h*x+g)^(1/2)/(h*x+g)^(1/2)+I*b*g*p*q*polylog(2,-(I*h*x/g+(1-h^2*x ^2/g^2)^(1/2))*f*g/(I*e*h-(-e^2*h^2+f^2*g^2)^(1/2)))*(1-h^2*x^2/g^2)^(1/2) /h/(-h*x+g)^(1/2)/(h*x+g)^(1/2)+I*b*g*p*q*polylog(2,-(I*h*x/g+(1-h^2*x^2/g ^2)^(1/2))*f*g/(I*e*h+(-e^2*h^2+f^2*g^2)^(1/2)))*(1-h^2*x^2/g^2)^(1/2)/h/( -h*x+g)^(1/2)/(h*x+g)^(1/2)
Time = 10.98 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.23 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\frac {\arctan \left (\frac {h x}{\sqrt {g-h x} \sqrt {g+h x}}\right ) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b p q \sqrt {g-h x} \left (2 g h (e+f x) \sqrt {\frac {g+h x}{g-h x}} \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right ) \log (e+f x)+(g+h x) \left (e h+f g \cos \left (2 \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )\right )\right ) \csc \left (2 \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )\right ) \left (2 i \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )^2-4 i \arcsin \left (\frac {\sqrt {1+\frac {e h}{f g}}}{\sqrt {2}}\right ) \arctan \left (\frac {-f g+e h}{\sqrt {-f^2 g^2+e^2 h^2} \sqrt {\frac {g+h x}{g-h x}}}\right )-2 \left (\arcsin \left (\frac {\sqrt {1+\frac {e h}{f g}}}{\sqrt {2}}\right )+\arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )\right ) \log \left (1+\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )} \left (e h-\sqrt {-f^2 g^2+e^2 h^2}\right )}{f g}\right )+2 \left (\arcsin \left (\frac {\sqrt {1+\frac {e h}{f g}}}{\sqrt {2}}\right )-\arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )\right ) \log \left (1+\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )} \left (e h+\sqrt {-f^2 g^2+e^2 h^2}\right )}{f g}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )} \left (-e h+\sqrt {-f^2 g^2+e^2 h^2}\right )}{f g}\right )+\operatorname {PolyLog}\left (2,-\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )} \left (e h+\sqrt {-f^2 g^2+e^2 h^2}\right )}{f g}\right )\right )\right )\right )}{g h^2 (e+f x) \sqrt {g+h x}} \]
(ArcTan[(h*x)/(Sqrt[g - h*x]*Sqrt[g + h*x])]*(a - b*p*q*Log[e + f*x] + b*L og[c*(d*(e + f*x)^p)^q]))/h - (b*p*q*Sqrt[g - h*x]*(2*g*h*(e + f*x)*Sqrt[( g + h*x)/(g - h*x)]*ArcTan[1/Sqrt[(g + h*x)/(g - h*x)]]*Log[e + f*x] + (g + h*x)*(e*h + f*g*Cos[2*ArcTan[1/Sqrt[(g + h*x)/(g - h*x)]]])*Csc[2*ArcTan [1/Sqrt[(g + h*x)/(g - h*x)]]]*((2*I)*ArcTan[1/Sqrt[(g + h*x)/(g - h*x)]]^ 2 - (4*I)*ArcSin[Sqrt[1 + (e*h)/(f*g)]/Sqrt[2]]*ArcTan[(-(f*g) + e*h)/(Sqr t[-(f^2*g^2) + e^2*h^2]*Sqrt[(g + h*x)/(g - h*x)])] - 2*(ArcSin[Sqrt[1 + ( e*h)/(f*g)]/Sqrt[2]] + ArcTan[1/Sqrt[(g + h*x)/(g - h*x)]])*Log[1 + (E^((2 *I)*ArcTan[1/Sqrt[(g + h*x)/(g - h*x)]])*(e*h - Sqrt[-(f^2*g^2) + e^2*h^2] ))/(f*g)] + 2*(ArcSin[Sqrt[1 + (e*h)/(f*g)]/Sqrt[2]] - ArcTan[1/Sqrt[(g + h*x)/(g - h*x)]])*Log[1 + (E^((2*I)*ArcTan[1/Sqrt[(g + h*x)/(g - h*x)]])*( e*h + Sqrt[-(f^2*g^2) + e^2*h^2]))/(f*g)] + I*(PolyLog[2, (E^((2*I)*ArcTan [1/Sqrt[(g + h*x)/(g - h*x)]])*(-(e*h) + Sqrt[-(f^2*g^2) + e^2*h^2]))/(f*g )] + PolyLog[2, -((E^((2*I)*ArcTan[1/Sqrt[(g + h*x)/(g - h*x)]])*(e*h + Sq rt[-(f^2*g^2) + e^2*h^2]))/(f*g))]))))/(g*h^2*(e + f*x)*Sqrt[g + h*x])
Time = 1.88 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.66, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {2895, 2854, 2851, 27, 5240, 5032, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}}dx\) |
| \(\Big \downarrow \) 2854 |
| \(\displaystyle \frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {1-\frac {h^2 x^2}{g^2}}}dx}{\sqrt {g-h x} \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 2851 |
| \(\displaystyle \frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \left (\frac {g \arcsin \left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-b f p q \int \frac {g \arcsin \left (\frac {h x}{g}\right )}{h (e+f x)}dx\right )}{\sqrt {g-h x} \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \left (\frac {g \arcsin \left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f g p q \int \frac {\arcsin \left (\frac {h x}{g}\right )}{e+f x}dx}{h}\right )}{\sqrt {g-h x} \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 5240 |
| \(\displaystyle \frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \left (\frac {g \arcsin \left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f g p q \int \frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right )}{\frac {f x h}{g}+\frac {e h}{g}}d\arcsin \left (\frac {h x}{g}\right )}{h}\right )}{\sqrt {g-h x} \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 5032 |
| \(\displaystyle \frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \left (\frac {g \arcsin \left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f g p q \left (i \int \frac {e^{i \arcsin \left (\frac {h x}{g}\right )} \arcsin \left (\frac {h x}{g}\right )}{e^{i \arcsin \left (\frac {h x}{g}\right )} f+\frac {i e h-\sqrt {f^2 g^2-e^2 h^2}}{g}}d\arcsin \left (\frac {h x}{g}\right )+i \int \frac {e^{i \arcsin \left (\frac {h x}{g}\right )} \arcsin \left (\frac {h x}{g}\right )}{e^{i \arcsin \left (\frac {h x}{g}\right )} f+\frac {i e h+\sqrt {f^2 g^2-e^2 h^2}}{g}}d\arcsin \left (\frac {h x}{g}\right )-\frac {i \arcsin \left (\frac {h x}{g}\right )^2}{2 f}\right )}{h}\right )}{\sqrt {g-h x} \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \left (\frac {g \arcsin \left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f g p q \left (i \left (\frac {i \int \log \left (\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}+1\right )d\arcsin \left (\frac {h x}{g}\right )}{f}-\frac {i \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \arcsin \left (\frac {h x}{g}\right )}}{-\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{f}\right )+i \left (\frac {i \int \log \left (\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}+1\right )d\arcsin \left (\frac {h x}{g}\right )}{f}-\frac {i \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \arcsin \left (\frac {h x}{g}\right )}}{\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{f}\right )-\frac {i \arcsin \left (\frac {h x}{g}\right )^2}{2 f}\right )}{h}\right )}{\sqrt {g-h x} \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \left (\frac {g \arcsin \left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f g p q \left (i \left (\frac {\int e^{-i \arcsin \left (\frac {h x}{g}\right )} \log \left (\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}+1\right )de^{i \arcsin \left (\frac {h x}{g}\right )}}{f}-\frac {i \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \arcsin \left (\frac {h x}{g}\right )}}{-\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{f}\right )+i \left (\frac {\int e^{-i \arcsin \left (\frac {h x}{g}\right )} \log \left (\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}+1\right )de^{i \arcsin \left (\frac {h x}{g}\right )}}{f}-\frac {i \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \arcsin \left (\frac {h x}{g}\right )}}{\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{f}\right )-\frac {i \arcsin \left (\frac {h x}{g}\right )^2}{2 f}\right )}{h}\right )}{\sqrt {g-h x} \sqrt {g+h x}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \left (\frac {g \arcsin \left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f g p q \left (i \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{f}-\frac {i \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \arcsin \left (\frac {h x}{g}\right )}}{-\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{f}\right )+i \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{f}-\frac {i \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \arcsin \left (\frac {h x}{g}\right )}}{\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{f}\right )-\frac {i \arcsin \left (\frac {h x}{g}\right )^2}{2 f}\right )}{h}\right )}{\sqrt {g-h x} \sqrt {g+h x}}\) |
(Sqrt[1 - (h^2*x^2)/g^2]*((g*ArcSin[(h*x)/g]*(a + b*Log[c*(d*(e + f*x)^p)^ q]))/h - (b*f*g*p*q*(((-1/2*I)*ArcSin[(h*x)/g]^2)/f + I*(((-I)*ArcSin[(h*x )/g]*Log[1 + (E^(I*ArcSin[(h*x)/g])*f*g)/(I*e*h - Sqrt[f^2*g^2 - e^2*h^2]) ])/f - PolyLog[2, -((E^(I*ArcSin[(h*x)/g])*f*g)/(I*e*h - Sqrt[f^2*g^2 - e^ 2*h^2]))]/f) + I*(((-I)*ArcSin[(h*x)/g]*Log[1 + (E^(I*ArcSin[(h*x)/g])*f*g )/(I*e*h + Sqrt[f^2*g^2 - e^2*h^2])])/f - PolyLog[2, -((E^(I*ArcSin[(h*x)/ g])*f*g)/(I*e*h + Sqrt[f^2*g^2 - e^2*h^2]))]/f)))/h))/(Sqrt[g - h*x]*Sqrt[ g + h*x])
3.6.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)* (x_)^2], x_Symbol] :> With[{u = IntHide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Simp[b*e*n Int[SimplifyIntegrand[u/(d + e*x) , x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/(Sqrt[(f1_) + (g1_ .)*(x_)]*Sqrt[(f2_) + (g2_.)*(x_)]), x_Symbol] :> Simp[Sqrt[1 + g1*(g2/(f1* f2))*x^2]/(Sqrt[f1 + g1*x]*Sqrt[f2 + g2*x]) Int[(a + b*Log[c*(d + e*x)^n] )/Sqrt[1 + g1*(g2/(f1*f2))*x^2], x], x] /; FreeQ[{a, b, c, d, e, f1, g1, f2 , g2, n}, x] && EqQ[f2*g1 + f1*g2, 0]
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[ (c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1 ))), x] + (Simp[I Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x)))), x], x] + Simp[I Int[(e + f*x)^m*(E^(I*(c + d*x) )/(I*a + Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x)))), x], x]) /; FreeQ[{a, b, c , d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Subst[Int[(a + b*x)^n*(Cos[x]/(c*d + e*Sin[x])), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\sqrt {-h x +g}\, \sqrt {h x +g}}d x\]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + g} \sqrt {-h x + g}} \,d x } \]
integral(-(sqrt(h*x + g)*sqrt(-h*x + g)*b*log(((f*x + e)^p*d)^q*c) + sqrt( h*x + g)*sqrt(-h*x + g)*a)/(h^2*x^2 - g^2), x)
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {g - h x} \sqrt {g + h x}}\, dx \]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + g} \sqrt {-h x + g}} \,d x } \]
b*integrate((q*log(d) + log(((f*x + e)^p)^q) + log(c))/(sqrt(h*x + g)*sqrt (-h*x + g)), x) + a*arcsin(h*x/g)/h
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + g} \sqrt {-h x + g}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {g+h\,x}\,\sqrt {g-h\,x}} \,d x \]