Integrand size = 22, antiderivative size = 147 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {a}{b m n}} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{b^{3/2} f m^{3/2} n^{3/2}}-\frac {2 (e+f x)}{b f m n \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}} \]
2*(f*x+e)*erfi((a+b*ln(c*(d*(f*x+e)^m)^n))^(1/2)/b^(1/2)/m^(1/2)/n^(1/2))* Pi^(1/2)/b^(3/2)/exp(a/b/m/n)/f/m^(3/2)/n^(3/2)/((c*(d*(f*x+e)^m)^n)^(1/m/ n))-2*(f*x+e)/b/f/m/n/(a+b*ln(c*(d*(f*x+e)^m)^n))^(1/2)
Time = 0.17 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=-\frac {2 e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \left (e^{\frac {a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{\frac {1}{m n}}-\Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right ) \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}}\right )}{b f m n \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}} \]
(-2*(e + f*x)*(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n)) - Gamma[1/2, -((a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n))]*Sqrt[-((a + b*Log[c*(d*(e + f *x)^m)^n])/(b*m*n))]))/(b*E^(a/(b*m*n))*f*m*n*(c*(d*(e + f*x)^m)^n)^(1/(m* n))*Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]])
Time = 0.65 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2895, 2836, 2734, 2737, 2611, 2633}
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.
\(\displaystyle \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 2836 |
\(\displaystyle \frac {\int \frac {1}{\left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}}d(e+f x)}{f}\) |
\(\Big \downarrow \) 2734 |
\(\displaystyle \frac {\frac {2 \int \frac {1}{\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}d(e+f x)}{b m n}-\frac {2 (e+f x)}{b m n \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}}{f}\) |
\(\Big \downarrow \) 2737 |
\(\displaystyle \frac {\frac {2 (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \int \frac {\left (c d^n (e+f x)^{m n}\right )^{\frac {1}{m n}}}{\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}d\log \left (c d^n (e+f x)^{m n}\right )}{b m^2 n^2}-\frac {2 (e+f x)}{b m n \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}}{f}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {\frac {4 (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \int \exp \left (\frac {a+b \log \left (c d^n (e+f x)^{m n}\right )}{b m n}-\frac {a}{b m n}\right )d\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}{b^2 m^2 n^2}-\frac {2 (e+f x)}{b m n \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}}{f}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {\frac {2 \sqrt {\pi } (e+f x) e^{-\frac {a}{b m n}} \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{b^{3/2} m^{3/2} n^{3/2}}-\frac {2 (e+f x)}{b m n \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}}{f}\) |
((2*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*d^n*(e + f*x)^(m*n)]]/(Sqrt[b ]*Sqrt[m]*Sqrt[n])])/(b^(3/2)*E^(a/(b*m*n))*m^(3/2)*n^(3/2)*(c*d^n*(e + f* x)^(m*n))^(1/(m*n))) - (2*(e + f*x))/(b*m*n*Sqrt[a + b*Log[c*d^n*(e + f*x) ^(m*n)]]))/f
3.5.15.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b *Log[c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1)) Int[(a + b *Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] && Int egerQ[2*p]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x/(n*(c*x ^n)^(1/n)) Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ [{a, b, c, n, p}, x]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : > Simp[1/e Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ a, b, c, d, e, n, p}, x]
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]]
\[\int \frac {1}{{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )\right )}^{\frac {3}{2}}}d x\]
Exception generated. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\right )}^{3/2}} \,d x \]