Integrand size = 20, antiderivative size = 98 \[ \int \frac {\sqrt {d x}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {3 e^{-\frac {3 a}{2 b n}} (d x)^{3/2} \left (c x^n\right )^{\left .-\frac {3}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {(d x)^{3/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
3/2*(d*x)^(3/2)*Ei(3/2*(a+b*ln(c*x^n))/b/n)/b^2/d/exp(3/2*a/b/n)/n^2/((c*x ^n)^(3/2/n))-(d*x)^(3/2)/b/d/n/(a+b*ln(c*x^n))
Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {d x}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {x \sqrt {d x} \left (3 e^{-\frac {3 a}{2 b n}} \left (c x^n\right )^{\left .-\frac {3}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )-\frac {2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2} \]
(x*Sqrt[d*x]*((3*ExpIntegralEi[(3*(a + b*Log[c*x^n]))/(2*b*n)])/(E^((3*a)/ (2*b*n))*(c*x^n)^(3/(2*n))) - (2*b*n)/(a + b*Log[c*x^n])))/(2*b^2*n^2)
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2743, 2747, 2609}
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.
| \(\displaystyle \int \frac {\sqrt {d x}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle \frac {3 \int \frac {\sqrt {d x}}{a+b \log \left (c x^n\right )}dx}{2 b n}-\frac {(d x)^{3/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}\) |
| \(\Big \downarrow \) 2747 |
| \(\displaystyle \frac {3 (d x)^{3/2} \left (c x^n\right )^{\left .-\frac {3}{2}\right /n} \int \frac {\left (c x^n\right )^{\left .\frac {3}{2}\right /n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 b d n^2}-\frac {(d x)^{3/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle \frac {3 (d x)^{3/2} e^{-\frac {3 a}{2 b n}} \left (c x^n\right )^{\left .-\frac {3}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {(d x)^{3/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}\) |
(3*(d*x)^(3/2)*ExpIntegralEi[(3*(a + b*Log[c*x^n]))/(2*b*n)])/(2*b^2*d*E^( (3*a)/(2*b*n))*n^2*(c*x^n)^(3/(2*n))) - (d*x)^(3/2)/(b*d*n*(a + b*Log[c*x^ n]))
3.2.9.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^(p + 1)/(b*d*n*(p + 1))), x] - Simp[(m + 1)/(b*n*(p + 1)) Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n)) Subst[Int[E^(((m + 1)/n )*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.89 (sec) , antiderivative size = 430, normalized size of antiderivative = 4.39
| method | result | size |
| risch | \(-\frac {2 x^{2} d}{b n \sqrt {d x}\, \left (2 a +2 b \ln \left (c \right )+2 b \ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}\right )}-\frac {3 \,{\mathrm e}^{\frac {3 i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}} \operatorname {Ei}_{1}\left (-\frac {3 \ln \left (d x \right )}{2}+\frac {3 i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}\right )}{2 d \,b^{2} n^{2}}\) | \(430\) |
-2/b/n*x^2/(d*x)^(1/2)/(2*a+2*b*ln(c)+2*b*ln(exp(n*ln(x)))-I*b*Pi*csgn(I*c )*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))+I*b*Pi*csgn(I*c)*csgn(I*c*ex p(n*ln(x)))^2+I*b*Pi*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2-I*b*Pi* csgn(I*c*exp(n*ln(x)))^3)*d-3/2/d/b^2/n^2*exp(3/4*I*(b*Pi*csgn(I*c)*csgn(I *exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))-b*Pi*csgn(I*c)*csgn(I*c*exp(n*ln(x)) )^2-b*Pi*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2+b*Pi*csgn(I*c*exp(n *ln(x)))^3+2*I*b*n*(ln(x)-ln(d*x))+2*I*b*ln(c)+2*I*b*(ln(exp(n*ln(x)))-n*l n(x))+2*I*a)/b/n)*Ei(1,-3/2*ln(d*x)+3/4*I*(b*Pi*csgn(I*c)*csgn(I*exp(n*ln( x)))*csgn(I*c*exp(n*ln(x)))-b*Pi*csgn(I*c)*csgn(I*c*exp(n*ln(x)))^2-b*Pi*c sgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2+b*Pi*csgn(I*c*exp(n*ln(x)))^3 +2*I*b*n*(ln(x)-ln(d*x))+2*I*b*ln(c)+2*I*b*(ln(exp(n*ln(x)))-n*ln(x))+2*I* a)/b/n)
\[ \int \frac {\sqrt {d x}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {\sqrt {d x}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {\sqrt {d x}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {\sqrt {d x}}{\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {d x}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {\sqrt {d x}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
4*b*sqrt(d)*n*integrate(1/3*sqrt(x)/(b^3*log(c)^3 + b^3*log(x^n)^3 + 3*a*b ^2*log(c)^2 + 3*a^2*b*log(c) + a^3 + 3*(b^3*log(c) + a*b^2)*log(x^n)^2 + 3 *(b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*log(x^n)), x) + 2/3*sqrt(d)*x^(3/ 2)/(b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + a^2 + 2*(b^2*log(c) + a *b)*log(x^n))
\[ \int \frac {\sqrt {d x}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {\sqrt {d x}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d x}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {\sqrt {d\,x}}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]