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

Optimal. Leaf size=519 \[ \frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \text{PolyLog}\left (2,-\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{-\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \text{PolyLog}\left (2,-\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{-\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right )^2}{2 h \sqrt{g-h x} \sqrt{g+h x}} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 1.4112, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2407, 216, 2404, 12, 4741, 4521, 2190, 2279, 2391, 2445} \[ \frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \text{PolyLog}\left (2,-\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{-\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \text{PolyLog}\left (2,-\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{g \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{-\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}-\frac{b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right ) \log \left (1+\frac{f g e^{i \sin ^{-1}\left (\frac{h x}{g}\right )}}{\sqrt{f^2 g^2-e^2 h^2}+i e h}\right )}{h \sqrt{g-h x} \sqrt{g+h x}}+\frac{i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}} \sin ^{-1}\left (\frac{h x}{g}\right )^2}{2 h \sqrt{g-h x} \sqrt{g+h x}} \]

Antiderivative was successfully verified.

[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 2407

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/(Sqrt[(f1_) + (g1_.)*(x_)]*Sqrt[(f2_) + (g2_.)*(x_)])
, x_Symbol] :> Dist[Sqrt[1 + (g1*g2*x^2)/(f1*f2)]/(Sqrt[f1 + g1*x]*Sqrt[f2 + g2*x]), Int[(a + b*Log[c*(d + e*x
)^n])/Sqrt[1 + (g1*g2*x^2)/(f1*f2)], x], x] /; FreeQ[{a, b, c, d, e, f1, g1, f2, g2, n}, x] && EqQ[f2*g1 + f1*
g2, 0]

Rule 216

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 2404

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 12

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

Rule 4741

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]

Rule 4521

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 2445

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]]

Rubi steps

\begin{align*} \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 &=\operatorname{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 )\\ &=\operatorname{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}}-\operatorname{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}}-\operatorname{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}}-\operatorname{Subst}\left (\frac{\left (b f g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{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}}-\operatorname{Subst}\left (\frac{\left (i b f g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{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 )-\operatorname{Subst}\left (\frac{\left (i b f g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{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}}+\operatorname{Subst}\left (\frac{\left (b g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{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 )+\operatorname{Subst}\left (\frac{\left (b g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{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}}-\operatorname{Subst}\left (\frac{\left (i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{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 )-\operatorname{Subst}\left (\frac{\left (i b g p q \sqrt{1-\frac{h^2 x^2}{g^2}}\right ) \operatorname{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 [B]  time = 4.46077, size = 1083, normalized size = 2.09 \[ \frac{\tan ^{-1}\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{i b p q \sqrt{g-h x} \sqrt{\frac{g+h x}{g-h x}} \left (\log ^2\left (i-\sqrt{\frac{g+h x}{g-h x}}\right )+2 \log (e+f x) \log \left (i-\sqrt{\frac{g+h x}{g-h x}}\right )+2 \log \left (\frac{1}{2} \left (1-i \sqrt{\frac{g+h x}{g-h x}}\right )\right ) \log \left (i-\sqrt{\frac{g+h x}{g-h x}}\right )-2 \log \left (\frac{\sqrt{f g-e h}-\sqrt{f g+e h} \sqrt{\frac{g+h x}{g-h x}}}{\sqrt{f g-e h}-i \sqrt{f g+e h}}\right ) \log \left (i-\sqrt{\frac{g+h x}{g-h x}}\right )-2 \log \left (\frac{\sqrt{f g-e h}+\sqrt{f g+e h} \sqrt{\frac{g+h x}{g-h x}}}{\sqrt{f g-e h}+i \sqrt{f g+e h}}\right ) \log \left (i-\sqrt{\frac{g+h x}{g-h x}}\right )-\log ^2\left (\sqrt{\frac{g+h x}{g-h x}}+i\right )-2 \log (e+f x) \log \left (\sqrt{\frac{g+h x}{g-h x}}+i\right )-2 \log \left (\frac{1}{2} \left (i \sqrt{\frac{g+h x}{g-h x}}+1\right )\right ) \log \left (\sqrt{\frac{g+h x}{g-h x}}+i\right )+2 \log \left (\sqrt{\frac{g+h x}{g-h x}}+i\right ) \log \left (\frac{\sqrt{f g-e h}-\sqrt{f g+e h} \sqrt{\frac{g+h x}{g-h x}}}{\sqrt{f g-e h}+i \sqrt{f g+e h}}\right )+2 \log \left (\sqrt{\frac{g+h x}{g-h x}}+i\right ) \log \left (\frac{\sqrt{f g-e h}+\sqrt{f g+e h} \sqrt{\frac{g+h x}{g-h x}}}{\sqrt{f g-e h}-i \sqrt{f g+e h}}\right )-2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} i \sqrt{\frac{g+h x}{g-h x}}\right )+2 \text{PolyLog}\left (2,\frac{1}{2} i \sqrt{\frac{g+h x}{g-h x}}+\frac{1}{2}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{f g+e h} \left (1-i \sqrt{\frac{g+h x}{g-h x}}\right )}{i \sqrt{f g-e h}+\sqrt{f g+e h}}\right )-2 \text{PolyLog}\left (2,\frac{\sqrt{f g+e h} \left (i \sqrt{\frac{g+h x}{g-h x}}+1\right )}{\sqrt{f g+e h}-i \sqrt{f g-e h}}\right )-2 \text{PolyLog}\left (2,\frac{\sqrt{f g+e h} \left (i \sqrt{\frac{g+h x}{g-h x}}+1\right )}{i \sqrt{f g-e h}+\sqrt{f g+e h}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{f g+e h} \left (\sqrt{\frac{g+h x}{g-h x}}+i\right )}{\sqrt{f g-e h}+i \sqrt{f g+e h}}\right )\right )}{2 h \sqrt{g+h x}} \]

Antiderivative was successfully verified.

[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 - ((I/2)
*b*p*q*Sqrt[g - h*x]*Sqrt[(g + h*x)/(g - h*x)]*(2*Log[e + f*x]*Log[I - Sqrt[(g + h*x)/(g - h*x)]] + Log[I - Sq
rt[(g + h*x)/(g - h*x)]]^2 + 2*Log[I - Sqrt[(g + h*x)/(g - h*x)]]*Log[(1 - I*Sqrt[(g + h*x)/(g - h*x)])/2] - 2
*Log[e + f*x]*Log[I + Sqrt[(g + h*x)/(g - h*x)]] - 2*Log[(1 + I*Sqrt[(g + h*x)/(g - h*x)])/2]*Log[I + Sqrt[(g
+ h*x)/(g - h*x)]] - Log[I + Sqrt[(g + h*x)/(g - h*x)]]^2 - 2*Log[I - Sqrt[(g + h*x)/(g - h*x)]]*Log[(Sqrt[f*g
- e*h] - Sqrt[f*g + e*h]*Sqrt[(g + h*x)/(g - h*x)])/(Sqrt[f*g - e*h] - I*Sqrt[f*g + e*h])] + 2*Log[I + Sqrt[(
g + h*x)/(g - h*x)]]*Log[(Sqrt[f*g - e*h] - Sqrt[f*g + e*h]*Sqrt[(g + h*x)/(g - h*x)])/(Sqrt[f*g - e*h] + I*Sq
rt[f*g + e*h])] + 2*Log[I + Sqrt[(g + h*x)/(g - h*x)]]*Log[(Sqrt[f*g - e*h] + Sqrt[f*g + e*h]*Sqrt[(g + h*x)/(
g - h*x)])/(Sqrt[f*g - e*h] - I*Sqrt[f*g + e*h])] - 2*Log[I - Sqrt[(g + h*x)/(g - h*x)]]*Log[(Sqrt[f*g - e*h]
+ Sqrt[f*g + e*h]*Sqrt[(g + h*x)/(g - h*x)])/(Sqrt[f*g - e*h] + I*Sqrt[f*g + e*h])] - 2*PolyLog[2, 1/2 - (I/2)
*Sqrt[(g + h*x)/(g - h*x)]] + 2*PolyLog[2, 1/2 + (I/2)*Sqrt[(g + h*x)/(g - h*x)]] + 2*PolyLog[2, (Sqrt[f*g + e
*h]*(1 - I*Sqrt[(g + h*x)/(g - h*x)]))/(I*Sqrt[f*g - e*h] + Sqrt[f*g + e*h])] - 2*PolyLog[2, (Sqrt[f*g + e*h]*
(1 + I*Sqrt[(g + h*x)/(g - h*x)]))/((-I)*Sqrt[f*g - e*h] + Sqrt[f*g + e*h])] - 2*PolyLog[2, (Sqrt[f*g + e*h]*(
1 + I*Sqrt[(g + h*x)/(g - h*x)]))/(I*Sqrt[f*g - e*h] + Sqrt[f*g + e*h])] + 2*PolyLog[2, (Sqrt[f*g + e*h]*(I +
Sqrt[(g + h*x)/(g - h*x)]))/(Sqrt[f*g - e*h] + I*Sqrt[f*g + e*h])]))/(h*Sqrt[g + h*x])

________________________________________________________________________________________

Maple [F]  time = 1.06, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) ){\frac{1}{\sqrt{-hx+g}}}{\frac{1}{\sqrt{hx+g}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{h x + g} \sqrt{-h x + g} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt{h x + g} \sqrt{-h x + g} a}{h^{2} x^{2} - g^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)