Integrand size = 20, antiderivative size = 98 \[ \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {3 e^{\frac {3 a}{2 b n}} \left (c x^n\right )^{\left .\frac {3}{2}\right /n} \operatorname {ExpIntegralEi}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2 (d x)^{3/2}}-\frac {1}{b d n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \] Output:
-3/2*exp(3/2*a/b/n)*(c*x^n)^(3/2/n)*Ei(1/2*(-3*a-3*b*ln(c*x^n))/b/n)/b^2/d /n^2/(d*x)^(3/2)-1/b/d/n/(d*x)^(3/2)/(a+b*ln(c*x^n))
Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {x \left (2 b n+3 e^{\frac {3 a}{2 b n}} \left (c x^n\right )^{\left .\frac {3}{2}\right /n} \operatorname {ExpIntegralEi}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{2 b^2 n^2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )} \] Input:
Integrate[1/((d*x)^(5/2)*(a + b*Log[c*x^n])^2),x]
Output:
-1/2*(x*(2*b*n + 3*E^((3*a)/(2*b*n))*(c*x^n)^(3/(2*n))*ExpIntegralEi[(-3*( a + b*Log[c*x^n]))/(2*b*n)]*(a + b*Log[c*x^n])))/(b^2*n^2*(d*x)^(5/2)*(a + b*Log[c*x^n]))
Time = 0.33 (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.
\(\displaystyle \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx\) |
\(\Big \downarrow \) 2743 |
\(\displaystyle -\frac {3 \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}dx}{2 b n}-\frac {1}{b d n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle -\frac {3 \left (c x^n\right )^{\left .\frac {3}{2}\right /n} \int \frac {\left (c x^n\right )^{\left .-\frac {3}{2}\right /n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 b d n^2 (d x)^{3/2}}-\frac {1}{b d n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle -\frac {3 e^{\frac {3 a}{2 b n}} \left (c x^n\right )^{\left .\frac {3}{2}\right /n} \operatorname {ExpIntegralEi}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2 (d x)^{3/2}}-\frac {1}{b d n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}\) |
Input:
Int[1/((d*x)^(5/2)*(a + b*Log[c*x^n])^2),x]
Output:
(-3*E^((3*a)/(2*b*n))*(c*x^n)^(3/(2*n))*ExpIntegralEi[(-3*(a + b*Log[c*x^n ]))/(2*b*n)])/(2*b^2*d*n^2*(d*x)^(3/2)) - 1/(b*d*n*(d*x)^(3/2)*(a + b*Log[ c*x^n]))
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 = 2.80 (sec) , antiderivative size = 432, normalized size of antiderivative = 4.41
method | result | size |
risch | \(-\frac {2}{b n x \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 \pi b \,\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 \pi b \,\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 ) \operatorname {csgn}\left (i c \right )-i \pi b \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+i \pi b \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i c \right )\right ) d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {3 i \left (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 ) \operatorname {csgn}\left (i c \right )-b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i c \right )-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 \left (\ln \left (x \right )-\ln \left (d x \right )\right ) b n +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 n b}} \operatorname {expIntegral}_{1}\left (\frac {3 \ln \left (d x \right )}{2}-\frac {3 i \left (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 ) \operatorname {csgn}\left (i c \right )-b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i c \right )-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 \left (\ln \left (x \right )-\ln \left (d x \right )\right ) b n +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 n b}\right )}{2 d \,b^{2} n^{2}}\) | \(432\) |
Input:
int(1/(d*x)^(5/2)/(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)
Output:
-2/b/n/x/(d*x)^(1/2)/(2*a+2*b*ln(c)+2*b*ln(exp(n*ln(x)))+I*Pi*b*csgn(I*exp (n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2-I*Pi*b*csgn(I*exp(n*ln(x)))*csgn(I*c*e xp(n*ln(x)))*csgn(I*c)-I*Pi*b*csgn(I*c*exp(n*ln(x)))^3+I*Pi*b*csgn(I*c*exp (n*ln(x)))^2*csgn(I*c))/d^2+3/2/d/b^2/n^2*exp(-3/4*I*(b*Pi*csgn(I*exp(n*ln (x)))*csgn(I*c*exp(n*ln(x)))*csgn(I*c)-b*Pi*csgn(I*c*exp(n*ln(x)))^2*csgn( I*c)-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*(ln(x)-ln(d*x))*b*n+2*I*b*ln(c)+2*I*b*(ln(exp(n*ln(x)))-n* ln(x))+2*I*a)/n/b)*Ei(1,3/2*ln(d*x)-3/4*I*(b*Pi*csgn(I*exp(n*ln(x)))*csgn( I*c*exp(n*ln(x)))*csgn(I*c)-b*Pi*csgn(I*c*exp(n*ln(x)))^2*csgn(I*c)-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*(ln(x)-ln(d*x))*b*n+2*I*b*ln(c)+2*I*b*(ln(exp(n*ln(x)))-n*ln(x))+2*I* a)/n/b)
\[ \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {5}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(1/(d*x)^(5/2)/(a+b*log(c*x^n))^2,x, algorithm="fricas")
Output:
integral(sqrt(d*x)/(b^2*d^3*x^3*log(c*x^n)^2 + 2*a*b*d^3*x^3*log(c*x^n) + a^2*d^3*x^3), x)
\[ \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\left (d x\right )^{\frac {5}{2}} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \] Input:
integrate(1/(d*x)**(5/2)/(a+b*ln(c*x**n))**2,x)
Output:
Integral(1/((d*x)**(5/2)*(a + b*log(c*x**n))**2), x)
\[ \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {5}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(1/(d*x)^(5/2)/(a+b*log(c*x^n))^2,x, algorithm="maxima")
Output:
-4*b*n*integrate(1/3/((b^3*d^(5/2)*log(c)^3 + b^3*d^(5/2)*log(x^n)^3 + 3*a *b^2*d^(5/2)*log(c)^2 + 3*a^2*b*d^(5/2)*log(c) + a^3*d^(5/2) + 3*(b^3*d^(5 /2)*log(c) + a*b^2*d^(5/2))*log(x^n)^2 + 3*(b^3*d^(5/2)*log(c)^2 + 2*a*b^2 *d^(5/2)*log(c) + a^2*b*d^(5/2))*log(x^n))*x^(5/2)), x) - 2/3/((b^2*d^(5/2 )*log(c)^2 + b^2*d^(5/2)*log(x^n)^2 + 2*a*b*d^(5/2)*log(c) + a^2*d^(5/2) + 2*(b^2*d^(5/2)*log(c) + a*b*d^(5/2))*log(x^n))*x^(3/2))
\[ \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {5}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(1/(d*x)^(5/2)/(a+b*log(c*x^n))^2,x, algorithm="giac")
Output:
integrate(1/((d*x)^(5/2)*(b*log(c*x^n) + a)^2), x)
Timed out. \[ \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{{\left (d\,x\right )}^{5/2}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \] Input:
int(1/((d*x)^(5/2)*(a + b*log(c*x^n))^2),x)
Output:
int(1/((d*x)^(5/2)*(a + b*log(c*x^n))^2), x)
\[ \int \frac {1}{(d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {\sqrt {d}\, \left (\int \frac {1}{\sqrt {x}\, \mathrm {log}\left (x^{n} c \right )^{2} b^{2} x^{2}+2 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) a b \,x^{2}+\sqrt {x}\, a^{2} x^{2}}d x \right )}{d^{3}} \] Input:
int(1/(d*x)^(5/2)/(a+b*log(c*x^n))^2,x)
Output:
(sqrt(d)*int(1/(sqrt(x)*log(x**n*c)**2*b**2*x**2 + 2*sqrt(x)*log(x**n*c)*a *b*x**2 + sqrt(x)*a**2*x**2),x))/d**3