3.276 \(\int \frac{a+b \log (c (d+e x)^n)}{\sqrt{f+g x^2}} \, dx\)

Optimal. Leaf size=506 \[ -\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \text{PolyLog}\left (2,-\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+e^2 f}}\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \text{PolyLog}\left (2,-\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{\sqrt{d^2 g+e^2 f}+d \sqrt{g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\sqrt{f} \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+e^2 f}}+1\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{\sqrt{d^2 g+e^2 f}+d \sqrt{g}}+1\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )^2}{2 \sqrt{g} \sqrt{f+g x^2}} \]

[Out]

(b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[f]]^2)/(2*Sqrt[g]*Sqrt[f + g*x^2]) - (b*Sqrt[f]*n*Sq
rt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Log[1 + (e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[g] -
Sqrt[e^2*f + d^2*g])])/(Sqrt[g]*Sqrt[f + g*x^2]) - (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[
f]]*Log[1 + (e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[g] + Sqrt[e^2*f + d^2*g])])/(Sqrt[g]*Sqrt[f + g
*x^2]) + (Sqrt[f]*Sqrt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*(a + b*Log[c*(d + e*x)^n]))/(Sqrt[g]*Sqrt[f
+ g*x^2]) - (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*PolyLog[2, -((e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[
g] - Sqrt[e^2*f + d^2*g]))])/(Sqrt[g]*Sqrt[f + g*x^2]) - (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*PolyLog[2, -((e*E^Ar
cSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[g] + Sqrt[e^2*f + d^2*g]))])/(Sqrt[g]*Sqrt[f + g*x^2])

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Rubi [A]  time = 0.556819, antiderivative size = 506, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {2406, 215, 2404, 12, 5799, 5561, 2190, 2279, 2391} \[ -\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \text{PolyLog}\left (2,-\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+e^2 f}}\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \text{PolyLog}\left (2,-\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{\sqrt{d^2 g+e^2 f}+d \sqrt{g}}\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{\sqrt{f} \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+e^2 f}}+1\right )}{\sqrt{g} \sqrt{f+g x^2}}-\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{e \sqrt{f} e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}}{\sqrt{d^2 g+e^2 f}+d \sqrt{g}}+1\right )}{\sqrt{g} \sqrt{f+g x^2}}+\frac{b \sqrt{f} n \sqrt{\frac{g x^2}{f}+1} \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )^2}{2 \sqrt{g} \sqrt{f+g x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[f]]^2)/(2*Sqrt[g]*Sqrt[f + g*x^2]) - (b*Sqrt[f]*n*Sq
rt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Log[1 + (e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[g] -
Sqrt[e^2*f + d^2*g])])/(Sqrt[g]*Sqrt[f + g*x^2]) - (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[
f]]*Log[1 + (e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[g] + Sqrt[e^2*f + d^2*g])])/(Sqrt[g]*Sqrt[f + g
*x^2]) + (Sqrt[f]*Sqrt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*(a + b*Log[c*(d + e*x)^n]))/(Sqrt[g]*Sqrt[f
+ g*x^2]) - (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*PolyLog[2, -((e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[
g] - Sqrt[e^2*f + d^2*g]))])/(Sqrt[g]*Sqrt[f + g*x^2]) - (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*PolyLog[2, -((e*E^Ar
cSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[g] + Sqrt[e^2*f + d^2*g]))])/(Sqrt[g]*Sqrt[f + g*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]

Rubi steps

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

Mathematica [F]  time = 3.60508, size = 0, normalized size = 0. \[ \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{f+g x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.822, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) ){\frac{1}{\sqrt{g{x}^{2}+f}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*ln(c*(e*x+d)^n))/(g*x^2+f)^(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*(e*x+d)^n))/(g*x^2+f)^(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{g x^{2} + f} b \log \left ({\left (e x + d\right )}^{n} c\right ) + \sqrt{g x^{2} + f} a}{g x^{2} + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((sqrt(g*x^2 + f)*b*log((e*x + d)^n*c) + sqrt(g*x^2 + f)*a)/(g*x^2 + f), 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 + e x\right )^{n} \right )}}{\sqrt{f + g x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))/(g*x**2+f)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{\sqrt{g x^{2} + f}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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