Integrand size = 30, antiderivative size = 515 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx=\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 \sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (1+\frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}-\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (1+\frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}+\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}+\frac {\sqrt {g} \sqrt {1+\frac {h x^2}{g}} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}-\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}+\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}} \]
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*a rcsinh(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)
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx=\int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx \]
Time = 1.54 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.69, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2895, 2853, 2851, 27, 6242, 6095, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}}dx\) |
| \(\Big \downarrow \) 2853 |
| \(\displaystyle \frac {\sqrt {\frac {h x^2}{g}+1} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {\frac {h x^2}{g}+1}}dx}{\sqrt {g+h x^2}}\) |
| \(\Big \downarrow \) 2851 |
| \(\displaystyle \frac {\sqrt {\frac {h x^2}{g}+1} \left (\frac {\sqrt {g} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-b f p q \int \frac {\sqrt {g} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}{\sqrt {h} (e+f x)}dx\right )}{\sqrt {g+h x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\frac {h x^2}{g}+1} \left (\frac {\sqrt {g} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b f \sqrt {g} p q \int \frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}{e+f x}dx}{\sqrt {h}}\right )}{\sqrt {g+h x^2}}\) |
| \(\Big \downarrow \) 6242 |
| \(\displaystyle \frac {\sqrt {\frac {h x^2}{g}+1} \left (\frac {\sqrt {g} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b f \sqrt {g} p q \int \frac {\sqrt {\frac {h x^2}{g}+1} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}{\frac {\sqrt {h} e}{\sqrt {g}}+\frac {f \sqrt {h} x}{\sqrt {g}}}d\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}{\sqrt {h}}\right )}{\sqrt {g+h x^2}}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {\sqrt {\frac {h x^2}{g}+1} \left (\frac {\sqrt {g} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b f \sqrt {g} p q \left (\int \frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}{\frac {\sqrt {h} e}{\sqrt {g}}+e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f-\frac {\sqrt {h e^2+f^2 g}}{\sqrt {g}}}d\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )+\int \frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}{\frac {\sqrt {h} e}{\sqrt {g}}+e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f+\frac {\sqrt {h e^2+f^2 g}}{\sqrt {g}}}d\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )-\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 f}\right )}{\sqrt {h}}\right )}{\sqrt {g+h x^2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\sqrt {\frac {h x^2}{g}+1} \left (\frac {\sqrt {g} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b f \sqrt {g} p q \left (-\frac {\int \log \left (\frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} \sqrt {g} f}{e \sqrt {h}-\sqrt {h e^2+f^2 g}}+1\right )d\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}{f}-\frac {\int \log \left (\frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} \sqrt {g} f}{\sqrt {h} e+\sqrt {h e^2+f^2 g}}+1\right )d\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}{f}+\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (\frac {f \sqrt {g} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+f^2 g}}+1\right )}{f}+\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (\frac {f \sqrt {g} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{\sqrt {e^2 h+f^2 g}+e \sqrt {h}}+1\right )}{f}-\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 f}\right )}{\sqrt {h}}\right )}{\sqrt {g+h x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\sqrt {\frac {h x^2}{g}+1} \left (\frac {\sqrt {g} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b f \sqrt {g} p q \left (-\frac {\int e^{-\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} \log \left (\frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} \sqrt {g} f}{e \sqrt {h}-\sqrt {h e^2+f^2 g}}+1\right )de^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{f}-\frac {\int e^{-\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} \log \left (\frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} \sqrt {g} f}{\sqrt {h} e+\sqrt {h e^2+f^2 g}}+1\right )de^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{f}+\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (\frac {f \sqrt {g} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+f^2 g}}+1\right )}{f}+\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (\frac {f \sqrt {g} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{\sqrt {e^2 h+f^2 g}+e \sqrt {h}}+1\right )}{f}-\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 f}\right )}{\sqrt {h}}\right )}{\sqrt {g+h x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\sqrt {\frac {h x^2}{g}+1} \left (\frac {\sqrt {g} \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b f \sqrt {g} p q \left (\frac {\operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}-\sqrt {h e^2+f^2 g}}\right )}{f}+\frac {\operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{\sqrt {h} e+\sqrt {h e^2+f^2 g}}\right )}{f}+\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (\frac {f \sqrt {g} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+f^2 g}}+1\right )}{f}+\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (\frac {f \sqrt {g} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{\sqrt {e^2 h+f^2 g}+e \sqrt {h}}+1\right )}{f}-\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 f}\right )}{\sqrt {h}}\right )}{\sqrt {g+h x^2}}\) |
(Sqrt[1 + (h*x^2)/g]*((Sqrt[g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*(a + b*Log[c*( d*(e + f*x)^p)^q]))/Sqrt[h] - (b*f*Sqrt[g]*p*q*(-1/2*ArcSinh[(Sqrt[h]*x)/S qrt[g]]^2/f + (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])])/f + (ArcSinh[(Sqr t[h]*x)/Sqrt[g]]*Log[1 + (E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqr t[h] + Sqrt[f^2*g + e^2*h])])/f + PolyLog[2, -((E^ArcSinh[(Sqrt[h]*x)/Sqrt [g]]*f*Sqrt[g])/(e*Sqrt[h] - Sqrt[f^2*g + e^2*h]))]/f + PolyLog[2, -((E^Ar cSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[h] + Sqrt[f^2*g + e^2*h]))]/ f))/Sqrt[h]))/Sqrt[g + h*x^2]
3.6.20.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Si mp[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]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)* (x_)^2], x_Symbol] :> With[{u = IntHide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Simp[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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)* (x_)^2], x_Symbol] :> Simp[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]
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]]
Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin h[(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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbo l] :> 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]
\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\sqrt {h \,x^{2}+g}}d x\]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x^{2} + g}} \,d x } \]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {g + h x^{2}}}\, dx \]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x^{2} + g}} \,d x } \]
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)
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x^{2} + g}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {h\,x^2+g}} \,d x \]