3.3.76 \(\int \frac {a+b \log (c (d+e x)^n)}{\sqrt {f+g x^2}} \, dx\) [276]

Optimal. Leaf size=506 \[ \frac {b \sqrt {f} n \sqrt {1+\frac {g x^2}{f}} \sinh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )^2}{2 \sqrt {g} \sqrt {f+g x^2}}-\frac {b \sqrt {f} n \sqrt {1+\frac {g x^2}{f}} \sinh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {e e^{\sinh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )} \sqrt {f}}{d \sqrt {g}-\sqrt {e^2 f+d^2 g}}\right )}{\sqrt {g} \sqrt {f+g x^2}}-\frac {b \sqrt {f} n \sqrt {1+\frac {g x^2}{f}} \sinh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {e e^{\sinh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )} \sqrt {f}}{d \sqrt {g}+\sqrt {e^2 f+d^2 g}}\right )}{\sqrt {g} \sqrt {f+g x^2}}+\frac {\sqrt {f} \sqrt {1+\frac {g x^2}{f}} \sinh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {g} \sqrt {f+g x^2}}-\frac {b \sqrt {f} n \sqrt {1+\frac {g x^2}{f}} \text {Li}_2\left (-\frac {e e^{\sinh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )} \sqrt {f}}{d \sqrt {g}-\sqrt {e^2 f+d^2 g}}\right )}{\sqrt {g} \sqrt {f+g x^2}}-\frac {b \sqrt {f} n \sqrt {1+\frac {g x^2}{f}} \text {Li}_2\left (-\frac {e e^{\sinh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )} \sqrt {f}}{d \sqrt {g}+\sqrt {e^2 f+d^2 g}}\right )}{\sqrt {g} \sqrt {f+g x^2}} \]

[Out]

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

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Rubi [A]
time = 0.39, antiderivative size = 506, normalized size of antiderivative = 1.00, 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 = {2453, 221, 2451, 12, 5827, 5680, 2221, 2317, 2438} \begin {gather*} -\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}} \end {gather*}

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 12

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

Rule 221

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 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 2453

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

Rule 5680

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 5827

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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [F]
time = 2.82, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x^2}} \, dx \end {gather*}

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 = 0.14, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{\sqrt {g \,x^{2}+f}}\, dx\]

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{\sqrt {f + g x^{2}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\sqrt {g\,x^2+f}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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