3.6.20 \(\int \frac {a+b \log (c (d (e+f x)^p)^q)}{\sqrt {g+h x^2}} \, dx\) [520]

Optimal. Leaf size=515 \[ \frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 \sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (1+\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}-\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (1+\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}+\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}+\frac {\sqrt {g} \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}-\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}+\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}} \]

[Out]

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

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Rubi [A]
time = 0.89, antiderivative size = 515, normalized size of antiderivative = 1.00, 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 = {2453, 221, 2451, 12, 5827, 5680, 2221, 2317, 2438, 2495} \begin {gather*} -\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}} \end {gather*}

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 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 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 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 \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx &=\text {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 )\\ &=\text {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}}-\text {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}}-\text {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}}-\text {Subst}\left (\frac {\left (b f \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \text {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}}-\text {Subst}\left (\frac {\left (b f \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \text {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 )-\text {Subst}\left (\frac {\left (b f \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \text {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}}+\text {Subst}\left (\frac {\left (b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \text {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 )+\text {Subst}\left (\frac {\left (b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \text {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}}+\text {Subst}\left (\frac {\left (b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \text {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 )+\text {Subst}\left (\frac {\left (b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \text {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*}

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Mathematica [F]
time = 2.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx \end {gather*}

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 = 0.21, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\sqrt {h \,x^{2}+g}}\, dx\]

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]
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*(d*(f*x+e)^p)^q))/(h*x^2+g)^(1/2),x, algorithm="maxima")

[Out]

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

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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*(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, 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*(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)

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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\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {h\,x^2+g}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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