3.520 \(\int \frac{a+b \log (c (d (e+f x)^p)^q)}{\sqrt{g+h x^2}} \, dx\)

Optimal. Leaf size=515 \[ -\frac{b \sqrt{g} p q \sqrt{\frac{h x^2}{g}+1} \text{PolyLog}\left (2,-\frac{f \sqrt{g} e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}}{e \sqrt{h}-\sqrt{e^2 h+f^2 g}}\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{\frac{h x^2}{g}+1} \text{PolyLog}\left (2,-\frac{f \sqrt{g} e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}}{\sqrt{e^2 h+f^2 g}+e \sqrt{h}}\right )}{\sqrt{h} \sqrt{g+h x^2}}+\frac{\sqrt{g} \sqrt{\frac{h x^2}{g}+1} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{\frac{h x^2}{g}+1} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \log \left (\frac{f \sqrt{g} e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}}{e \sqrt{h}-\sqrt{e^2 h+f^2 g}}+1\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{\frac{h x^2}{g}+1} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \log \left (\frac{f \sqrt{g} e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}}{\sqrt{e^2 h+f^2 g}+e \sqrt{h}}+1\right )}{\sqrt{h} \sqrt{g+h x^2}}+\frac{b \sqrt{g} p q \sqrt{\frac{h x^2}{g}+1} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )^2}{2 \sqrt{h} \sqrt{g+h x^2}} \]

[Out]

(b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]^2)/(2*Sqrt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*
q*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*Log[1 + (E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[
h] - Sqrt[f^2*g + e^2*h])])/(Sqrt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)
/Sqrt[g]]*Log[1 + (E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[h] + Sqrt[f^2*g + e^2*h])])/(Sqrt[h]*Sqrt
[g + h*x^2]) + (Sqrt[g]*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(Sq
rt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*PolyLog[2, -((E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqr
t[g])/(e*Sqrt[h] - Sqrt[f^2*g + e^2*h]))])/(Sqrt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*Poly
Log[2, -((E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[h] + Sqrt[f^2*g + e^2*h]))])/(Sqrt[h]*Sqrt[g + h*x
^2])

________________________________________________________________________________________

Rubi [A]  time = 1.2005, antiderivative size = 515, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2406, 215, 2404, 12, 5799, 5561, 2190, 2279, 2391, 2445} \[ -\frac{b \sqrt{g} p q \sqrt{\frac{h x^2}{g}+1} \text{PolyLog}\left (2,-\frac{f \sqrt{g} e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}}{e \sqrt{h}-\sqrt{e^2 h+f^2 g}}\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{\frac{h x^2}{g}+1} \text{PolyLog}\left (2,-\frac{f \sqrt{g} e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}}{\sqrt{e^2 h+f^2 g}+e \sqrt{h}}\right )}{\sqrt{h} \sqrt{g+h x^2}}+\frac{\sqrt{g} \sqrt{\frac{h x^2}{g}+1} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{\frac{h x^2}{g}+1} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \log \left (\frac{f \sqrt{g} e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}}{e \sqrt{h}-\sqrt{e^2 h+f^2 g}}+1\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{\frac{h x^2}{g}+1} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \log \left (\frac{f \sqrt{g} e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}}{\sqrt{e^2 h+f^2 g}+e \sqrt{h}}+1\right )}{\sqrt{h} \sqrt{g+h x^2}}+\frac{b \sqrt{g} p q \sqrt{\frac{h x^2}{g}+1} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )^2}{2 \sqrt{h} \sqrt{g+h x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])/Sqrt[g + h*x^2],x]

[Out]

(b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]^2)/(2*Sqrt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*
q*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*Log[1 + (E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[
h] - Sqrt[f^2*g + e^2*h])])/(Sqrt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)
/Sqrt[g]]*Log[1 + (E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[h] + Sqrt[f^2*g + e^2*h])])/(Sqrt[h]*Sqrt
[g + h*x^2]) + (Sqrt[g]*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(Sq
rt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*PolyLog[2, -((E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqr
t[g])/(e*Sqrt[h] - Sqrt[f^2*g + e^2*h]))])/(Sqrt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*Poly
Log[2, -((E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[h] + Sqrt[f^2*g + e^2*h]))])/(Sqrt[h]*Sqrt[g + h*x
^2])

Rule 2406

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 +
(g*x^2)/f]/Sqrt[f + g*x^2], Int[(a + b*Log[c*(d + e*x)^n])/Sqrt[1 + (g*x^2)/f], x], x] /; FreeQ[{a, b, c, d,
e, f, g, n}, x] &&  !GtQ[f, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[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 5799

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Cosh[x
])/(c*d + e*Sinh[x]), x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

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^2}} \, dx &=\operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt{g+h x^2}} \, 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 x^2}{g}} \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt{1+\frac{h x^2}{g}}} \, dx}{\sqrt{g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\sqrt{g} \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{h} \sqrt{g+h x^2}}-\operatorname{Subst}\left (\frac{\left (b f p q \sqrt{1+\frac{h x^2}{g}}\right ) \int \frac{\sqrt{g} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}{\sqrt{h} (e+f x)} \, dx}{\sqrt{g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\sqrt{g} \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{h} \sqrt{g+h x^2}}-\operatorname{Subst}\left (\frac{\left (b f \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}{e+f x} \, dx}{\sqrt{h} \sqrt{g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\sqrt{g} \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{h} \sqrt{g+h x^2}}-\operatorname{Subst}\left (\frac{\left (b f \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}}\right ) \operatorname{Subst}\left (\int \frac{x \cosh (x)}{\frac{e \sqrt{h}}{\sqrt{g}}+f \sinh (x)} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )\right )}{\sqrt{h} \sqrt{g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )^2}{2 \sqrt{h} \sqrt{g+h x^2}}+\frac{\sqrt{g} \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{h} \sqrt{g+h x^2}}-\operatorname{Subst}\left (\frac{\left (b f \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{e^x f+\frac{e \sqrt{h}}{\sqrt{g}}-\frac{\sqrt{f^2 g+e^2 h}}{\sqrt{g}}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )\right )}{\sqrt{h} \sqrt{g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (b f \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{e^x f+\frac{e \sqrt{h}}{\sqrt{g}}+\frac{\sqrt{f^2 g+e^2 h}}{\sqrt{g}}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )\right )}{\sqrt{h} \sqrt{g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )^2}{2 \sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \log \left (1+\frac{e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )} f \sqrt{g}}{e \sqrt{h}-\sqrt{f^2 g+e^2 h}}\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \log \left (1+\frac{e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )} f \sqrt{g}}{e \sqrt{h}+\sqrt{f^2 g+e^2 h}}\right )}{\sqrt{h} \sqrt{g+h x^2}}+\frac{\sqrt{g} \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{h} \sqrt{g+h x^2}}+\operatorname{Subst}\left (\frac{\left (b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{e^x f}{\frac{e \sqrt{h}}{\sqrt{g}}-\frac{\sqrt{f^2 g+e^2 h}}{\sqrt{g}}}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )\right )}{\sqrt{h} \sqrt{g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{e^x f}{\frac{e \sqrt{h}}{\sqrt{g}}+\frac{\sqrt{f^2 g+e^2 h}}{\sqrt{g}}}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )\right )}{\sqrt{h} \sqrt{g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )^2}{2 \sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \log \left (1+\frac{e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )} f \sqrt{g}}{e \sqrt{h}-\sqrt{f^2 g+e^2 h}}\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \log \left (1+\frac{e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )} f \sqrt{g}}{e \sqrt{h}+\sqrt{f^2 g+e^2 h}}\right )}{\sqrt{h} \sqrt{g+h x^2}}+\frac{\sqrt{g} \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{h} \sqrt{g+h x^2}}+\operatorname{Subst}\left (\frac{\left (b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{f x}{\frac{e \sqrt{h}}{\sqrt{g}}-\frac{\sqrt{f^2 g+e^2 h}}{\sqrt{g}}}\right )}{x} \, dx,x,e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}\right )}{\sqrt{h} \sqrt{g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{f x}{\frac{e \sqrt{h}}{\sqrt{g}}+\frac{\sqrt{f^2 g+e^2 h}}{\sqrt{g}}}\right )}{x} \, dx,x,e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )}\right )}{\sqrt{h} \sqrt{g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )^2}{2 \sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \log \left (1+\frac{e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )} f \sqrt{g}}{e \sqrt{h}-\sqrt{f^2 g+e^2 h}}\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \log \left (1+\frac{e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )} f \sqrt{g}}{e \sqrt{h}+\sqrt{f^2 g+e^2 h}}\right )}{\sqrt{h} \sqrt{g+h x^2}}+\frac{\sqrt{g} \sqrt{1+\frac{h x^2}{g}} \sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \text{Li}_2\left (-\frac{e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )} f \sqrt{g}}{e \sqrt{h}-\sqrt{f^2 g+e^2 h}}\right )}{\sqrt{h} \sqrt{g+h x^2}}-\frac{b \sqrt{g} p q \sqrt{1+\frac{h x^2}{g}} \text{Li}_2\left (-\frac{e^{\sinh ^{-1}\left (\frac{\sqrt{h} x}{\sqrt{g}}\right )} f \sqrt{g}}{e \sqrt{h}+\sqrt{f^2 g+e^2 h}}\right )}{\sqrt{h} \sqrt{g+h x^2}}\\ \end{align*}

Mathematica [F]  time = 3.59329, size = 0, normalized size = 0. \[ \int \frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt{g+h x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])/Sqrt[g + h*x^2],x]

[Out]

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])/Sqrt[g + h*x^2], x]

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Maple [F]  time = 0.664, 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{h{x}^{2}+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^2+g)^(1/2),x)

[Out]

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

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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^2+g)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{h x^{2} + g} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt{h x^{2} + g} a}{h x^{2} + g}, 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^2+g)^(1/2),x, algorithm="fricas")

[Out]

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

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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^{2}}}\, 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**2+g)**(1/2),x)

[Out]

Integral((a + b*log(c*(d*(e + f*x)**p)**q))/sqrt(g + h*x**2), x)

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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^{2} + 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^2+g)^(1/2),x, algorithm="giac")

[Out]

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