Integrand size = 22, antiderivative size = 330 \[ \int \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} b n x \sqrt {d+e x^2}+\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 \sqrt {e} \sqrt {d+e x^2}}-\frac {b d n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {e}}-\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{3/2} \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 \sqrt {e} \sqrt {d+e x^2}} \]
-1/4*b*d*n*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(1/2)-1/4*b*n*x*(e*x^2+d)^ (1/2)+1/2*x*(a+b*ln(c*x^n))*(e*x^2+d)^(1/2)+1/4*b*d^(3/2)*n*arcsinh(x*e^(1 /2)/d^(1/2))^2*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)-1/2*b*d^(3/2)*n*a rcsinh(x*e^(1/2)/d^(1/2))*ln(1-(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*(1 +e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)+1/2*d^(3/2)*arcsinh(x*e^(1/2)/d^(1 /2))*(a+b*ln(c*x^n))*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)-1/4*b*d^(3/ 2)*n*polylog(2,(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*(1+e*x^2/d)^(1/2)/ e^(1/2)/(e*x^2+d)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.23 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.72 \[ \int \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {-2 b \sqrt {e} n x \sqrt {d+e x^2} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {e x^2}{d}\right )+b \sqrt {d} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (-1+2 \log (x))+\sqrt {1+\frac {e x^2}{d}} \left (\sqrt {e} (2 a-b n) x \sqrt {d+e x^2}+2 d (a-b n \log (x)) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )+2 b \log \left (c x^n\right ) \left (\sqrt {e} x \sqrt {d+e x^2}+d \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )\right )\right )}{4 \sqrt {e} \sqrt {1+\frac {e x^2}{d}}} \]
(-2*b*Sqrt[e]*n*x*Sqrt[d + e*x^2]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, -((e*x^2)/d)] + b*Sqrt[d]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqr t[d]]*(-1 + 2*Log[x]) + Sqrt[1 + (e*x^2)/d]*(Sqrt[e]*(2*a - b*n)*x*Sqrt[d + e*x^2] + 2*d*(a - b*n*Log[x])*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]] + 2*b*L og[c*x^n]*(Sqrt[e]*x*Sqrt[d + e*x^2] + d*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2] ])))/(4*Sqrt[e]*Sqrt[1 + (e*x^2)/d])
Result contains complex when optimal does not.
Time = 0.83 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.75, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {2758, 211, 224, 219, 2764, 2762, 6190, 3042, 26, 4199, 25, 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.
| \(\displaystyle \int \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2758 |
| \(\displaystyle \frac {1}{2} d \int \frac {a+b \log \left (c x^n\right )}{\sqrt {e x^2+d}}dx-\frac {1}{2} b n \int \sqrt {e x^2+d}dx+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {1}{2} d \int \frac {a+b \log \left (c x^n\right )}{\sqrt {e x^2+d}}dx-\frac {1}{2} b n \left (\frac {1}{2} d \int \frac {1}{\sqrt {e x^2+d}}dx+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{2} d \int \frac {a+b \log \left (c x^n\right )}{\sqrt {e x^2+d}}dx-\frac {1}{2} b n \left (\frac {1}{2} d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} d \int \frac {a+b \log \left (c x^n\right )}{\sqrt {e x^2+d}}dx+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
| \(\Big \downarrow \) 2764 |
| \(\displaystyle \frac {d \sqrt {\frac {e x^2}{d}+1} \int \frac {a+b \log \left (c x^n\right )}{\sqrt {\frac {e x^2}{d}+1}}dx}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
| \(\Big \downarrow \) 2762 |
| \(\displaystyle \frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b \sqrt {d} n \int \frac {\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}dx}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle \frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b \sqrt {d} n \int \frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} x}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b \sqrt {d} n \int -i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \tan \left (i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \int \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \tan \left (i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (2 i \int -\frac {e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \int \frac {e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )de^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\) |
-1/2*(b*n*((x*Sqrt[d + e*x^2])/2 + (d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] )/(2*Sqrt[e]))) + (x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/2 + (d*Sqrt[1 + ( e*x^2)/d]*((Sqrt[d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/Sqrt[ e] + (I*b*Sqrt[d]*n*((-1/2*I)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2 - (2*I)*(-1/2 *(ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])] ) - PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])]/4)))/Sqrt[e]))/(2*Sqrt[ d + e*x^2])
3.3.58.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Sy mbol] :> Simp[x*(d + e*x^2)^q*((a + b*Log[c*x^n])/(2*q + 1)), x] + (-Simp[b *(n/(2*q + 1)) Int[(d + e*x^2)^q, x], x] + Simp[2*d*(q/(2*q + 1)) Int[( d + e*x^2)^(q - 1)*(a + b*Log[c*x^n]), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] && GtQ[q, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symb ol] :> Simp[ArcSinh[Rt[e, 2]*(x/Sqrt[d])]*((a + b*Log[c*x^n])/Rt[e, 2]), x] - Simp[b*(n/Rt[e, 2]) Int[ArcSinh[Rt[e, 2]*(x/Sqrt[d])]/x, x], x] /; Fre eQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && PosQ[e]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symb ol] :> Simp[Sqrt[1 + (e/d)*x^2]/Sqrt[d + e*x^2] Int[(a + b*Log[c*x^n])/Sq rt[1 + (e/d)*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && !GtQ[d, 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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
\[\int \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e \,x^{2}+d}d x\]
\[ \int \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]
\[ \int \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
Exception generated. \[ \int \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]
Timed out. \[ \int \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]