\(\int \frac {a+b \log (c (d (e+f x)^p)^q)}{\sqrt {g-h x} \sqrt {g+h x}} \, dx\) [522]

Optimal result

Integrand size = 38, antiderivative size = 519 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \operatorname {PolyLog}\left (2,-\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \operatorname {PolyLog}\left (2,-\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2454, 222, 2451, 12, 4825, 4617, 2221, 2317, 2438, 2495} \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \operatorname {PolyLog}\left (2,-\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h-\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \operatorname {PolyLog}\left (2,-\frac {e^{i \arcsin \left (\frac {h x}{g}\right )} f g}{i e h+\sqrt {f^2 g^2-e^2 h^2}}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \arcsin \left (\frac {h x}{g}\right )}}{-\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}-\frac {b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right ) \log \left (1+\frac {f g e^{i \arcsin \left (\frac {h x}{g}\right )}}{\sqrt {f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt {g-h x} \sqrt {g+h x}}+\frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \arcsin \left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}} \]

[In]

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])/(Sqrt[g - h*x]*Sqrt[g + h*x]),x]

[Out]

((I/2)*b*g*p*q*Sqrt[1 - (h^2*x^2)/g^2]*ArcSin[(h*x)/g]^2)/(h*Sqrt[g - h*x]*Sqrt[g + h*x]) - (b*g*p*q*Sqrt[1 -
(h^2*x^2)/g^2]*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])])/(h*Sqrt
[g - h*x]*Sqrt[g + h*x]) - (b*g*p*q*Sqrt[1 - (h^2*x^2)/g^2]*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])])/(h*Sqrt[g - h*x]*Sqrt[g + h*x]) + (g*Sqrt[1 - (h^2*x^2)/g^2]*ArcSin[(h*x
)/g]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(h*Sqrt[g - h*x]*Sqrt[g + h*x]) + (I*b*g*p*q*Sqrt[1 - (h^2*x^2)/g^2]*Po
lyLog[2, -((E^(I*ArcSin[(h*x)/g])*f*g)/(I*e*h - Sqrt[f^2*g^2 - e^2*h^2]))])/(h*Sqrt[g - h*x]*Sqrt[g + h*x]) +
(I*b*g*p*q*Sqrt[1 - (h^2*x^2)/g^2]*PolyLog[2, -((E^(I*ArcSin[(h*x)/g])*f*g)/(I*e*h + Sqrt[f^2*g^2 - e^2*h^2]))
])/(h*Sqrt[g - h*x]*Sqrt[g + h*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 2221

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]
- Dist[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]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2438

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]

Rule 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)*(x_)^2], x_Symbol] :> With[{u = Int
Hide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[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]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/(Sqrt[(f1_) + (g1_.)*(x_)]*Sqrt[(f2_) + (g2_.)*(x_)])
, x_Symbol] :> Dist[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]

Rule 2495

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]]

Rule 4617

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] + (Dist[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] + Dist[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]

Rule 4825

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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {\sqrt {1-\frac {h^2 x^2}{g^2}} \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {1-\frac {h^2 x^2}{g^2}}} \, dx}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\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}}-\text {Subst}\left (\frac {\left (b f p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \int \frac {g \sin ^{-1}\left (\frac {h x}{g}\right )}{e h+f h x} \, dx}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\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}}-\text {Subst}\left (\frac {\left (b f g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \int \frac {\sin ^{-1}\left (\frac {h x}{g}\right )}{e h+f h x} \, dx}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\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}}-\text {Subst}\left (\frac {\left (b f g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \frac {x \cos (x)}{\frac {e h^2}{g}+f h \sin (x)} \, dx,x,\sin ^{-1}\left (\frac {h x}{g}\right )\right )}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\left (\frac {h x}{g}\right )^2}{2 h \sqrt {g-h x} \sqrt {g+h x}}+\frac {g \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\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}}-\text {Subst}\left (\frac {\left (i b f g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{e^{i x} f h+\frac {i e h^2}{g}-\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}} \, dx,x,\sin ^{-1}\left (\frac {h x}{g}\right )\right )}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (i b f g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{e^{i x} f h+\frac {i e h^2}{g}+\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}} \, dx,x,\sin ^{-1}\left (\frac {h x}{g}\right )\right )}{\sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\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}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\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}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\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}} \sin ^{-1}\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}}+\text {Subst}\left (\frac {\left (b g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \log \left (1+\frac {e^{i x} f h}{\frac {i e h^2}{g}-\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}}\right ) \, dx,x,\sin ^{-1}\left (\frac {h x}{g}\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (b g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \log \left (1+\frac {e^{i x} f h}{\frac {i e h^2}{g}+\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}}\right ) \, dx,x,\sin ^{-1}\left (\frac {h x}{g}\right )\right )}{h \sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\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}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\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}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\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}} \sin ^{-1}\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}}-\text {Subst}\left (\frac {\left (i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {f h x}{\frac {i e h^2}{g}-\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac {h x}{g}\right )}\right )}{h \sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {f h x}{\frac {i e h^2}{g}+\frac {h \sqrt {f^2 g^2-e^2 h^2}}{g}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac {h x}{g}\right )}\right )}{h \sqrt {g-h x} \sqrt {g+h x}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {i b g p q \sqrt {1-\frac {h^2 x^2}{g^2}} \sin ^{-1}\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}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\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}} \sin ^{-1}\left (\frac {h x}{g}\right ) \log \left (1+\frac {e^{i \sin ^{-1}\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}} \sin ^{-1}\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}} \text {Li}_2\left (-\frac {e^{i \sin ^{-1}\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}} \text {Li}_2\left (-\frac {e^{i \sin ^{-1}\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}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 10.98 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.23 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\frac {\arctan \left (\frac {h x}{\sqrt {g-h x} \sqrt {g+h x}}\right ) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b p q \sqrt {g-h x} \left (2 g h (e+f x) \sqrt {\frac {g+h x}{g-h x}} \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right ) \log (e+f x)+(g+h x) \left (e h+f g \cos \left (2 \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )\right )\right ) \csc \left (2 \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )\right ) \left (2 i \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )^2-4 i \arcsin \left (\frac {\sqrt {1+\frac {e h}{f g}}}{\sqrt {2}}\right ) \arctan \left (\frac {-f g+e h}{\sqrt {-f^2 g^2+e^2 h^2} \sqrt {\frac {g+h x}{g-h x}}}\right )-2 \left (\arcsin \left (\frac {\sqrt {1+\frac {e h}{f g}}}{\sqrt {2}}\right )+\arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )\right ) \log \left (1+\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )} \left (e h-\sqrt {-f^2 g^2+e^2 h^2}\right )}{f g}\right )+2 \left (\arcsin \left (\frac {\sqrt {1+\frac {e h}{f g}}}{\sqrt {2}}\right )-\arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )\right ) \log \left (1+\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )} \left (e h+\sqrt {-f^2 g^2+e^2 h^2}\right )}{f g}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )} \left (-e h+\sqrt {-f^2 g^2+e^2 h^2}\right )}{f g}\right )+\operatorname {PolyLog}\left (2,-\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {g+h x}{g-h x}}}\right )} \left (e h+\sqrt {-f^2 g^2+e^2 h^2}\right )}{f g}\right )\right )\right )\right )}{g h^2 (e+f x) \sqrt {g+h x}} \]

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])/(Sqrt[g - h*x]*Sqrt[g + h*x]),x]

[Out]

(ArcTan[(h*x)/(Sqrt[g - h*x]*Sqrt[g + h*x])]*(a - b*p*q*Log[e + f*x] + b*Log[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)/
(Sqrt[-(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 + Sqrt[-(f^2*g^2) + e^2*h^2]))/(f*g))]))))/(g*h^2*(e + f*x)*Sqrt[g +
h*x])

Maple [F]

\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\sqrt {-h x +g}\, \sqrt {h x +g}}d x\]

[In]

int((a+b*ln(c*(d*(f*x+e)^p)^q))/(-h*x+g)^(1/2)/(h*x+g)^(1/2),x)

[Out]

int((a+b*ln(c*(d*(f*x+e)^p)^q))/(-h*x+g)^(1/2)/(h*x+g)^(1/2),x)

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + g} \sqrt {-h x + g}} \,d x } \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(-h*x+g)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {g - h x} \sqrt {g + h x}}\, dx \]

[In]

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))/(-h*x+g)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Integral((a + b*log(c*(d*(e + f*x)**p)**q))/(sqrt(g - h*x)*sqrt(g + h*x)), x)

Maxima [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + g} \sqrt {-h x + g}} \,d x } \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(-h*x+g)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g-h x} \sqrt {g+h x}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + g} \sqrt {-h x + g}} \,d x } \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(-h*x+g)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

integrate((b*log(((f*x + e)^p*d)^q*c) + a)/(sqrt(h*x + g)*sqrt(-h*x + g)), x)

Mupad [F(-1)]

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

[In]

int((a + b*log(c*(d*(e + f*x)^p)^q))/((g + h*x)^(1/2)*(g - h*x)^(1/2)),x)

[Out]

int((a + b*log(c*(d*(e + f*x)^p)^q))/((g + h*x)^(1/2)*(g - h*x)^(1/2)), x)