Integrand size = 18, antiderivative size = 156 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\frac {4 e^{-\frac {a}{b n}} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e n^{5/2}}-\frac {2 (d+e x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}-\frac {4 (d+e x)}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
-2/3*(e*x+d)/b/e/n/(a+b*ln(c*(e*x+d)^n))^(3/2)+4/3*(e*x+d)*erfi((a+b*ln(c* (e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*Pi^(1/2)/b^(5/2)/e/exp(a/b/n)/n^(5/2)/( (c*(e*x+d)^n)^(1/n))-4/3*(e*x+d)/b^2/e/n^2/(a+b*ln(c*(e*x+d)^n))^(1/2)
Time = 0.12 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=-\frac {2 e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (2 b n \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{3/2}+e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (2 a+b n+2 b \log \left (c (d+e x)^n\right )\right )\right )}{3 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]
(-2*(d + e*x)*(2*b*n*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^(3/2) + E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1 )*(2*a + b*n + 2*b*Log[c*(d + e*x)^n])))/(3*b^2*e*E^(a/(b*n))*n^2*(c*(d + e*x)^n)^n^(-1)*(a + b*Log[c*(d + e*x)^n])^(3/2))
Time = 0.42 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2836, 2734, 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.
\(\displaystyle \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 2836 |
\(\displaystyle \frac {\int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}d(d+e x)}{e}\) |
\(\Big \downarrow \) 2734 |
\(\displaystyle \frac {\frac {2 \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}d(d+e x)}{3 b n}-\frac {2 (d+e x)}{3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}}{e}\) |
\(\Big \downarrow \) 2734 |
\(\displaystyle \frac {\frac {2 \left (\frac {2 \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}d(d+e x)}{b n}-\frac {2 (d+e x)}{b n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b n}-\frac {2 (d+e x)}{3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}}{e}\) |
\(\Big \downarrow \) 2737 |
\(\displaystyle \frac {\frac {2 \left (\frac {2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \int \frac {\left (c (d+e x)^n\right )^{\frac {1}{n}}}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}d\log \left (c (d+e x)^n\right )}{b n^2}-\frac {2 (d+e x)}{b n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b n}-\frac {2 (d+e x)}{3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}}{e}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {\frac {2 \left (\frac {4 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \int e^{\frac {a+b \log \left (c (d+e x)^n\right )}{b n}-\frac {a}{b n}}d\sqrt {a+b \log \left (c (d+e x)^n\right )}}{b^2 n^2}-\frac {2 (d+e x)}{b n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b n}-\frac {2 (d+e x)}{3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}}{e}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {\frac {2 \left (\frac {2 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} n^{3/2}}-\frac {2 (d+e x)}{b n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b n}-\frac {2 (d+e x)}{3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}}{e}\) |
((-2*(d + e*x))/(3*b*n*(a + b*Log[c*(d + e*x)^n])^(3/2)) + (2*((2*Sqrt[Pi] *(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(b^(3/2 )*E^(a/(b*n))*n^(3/2)*(c*(d + e*x)^n)^n^(-1)) - (2*(d + e*x))/(b*n*Sqrt[a + b*Log[c*(d + e*x)^n]])))/(3*b*n))/e
3.1.29.3.1 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]
\[\int \frac {1}{{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2}} \,d x \]