Integrand size = 18, antiderivative size = 179 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=-\frac {15 b^{5/2} e^{-\frac {a}{b n}} n^{5/2} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{8 e}+\frac {15 b^2 n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e}-\frac {5 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e} \]
-5/2*b*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^(3/2)/e+(e*x+d)*(a+b*ln(c*(e*x+d)^n ))^(5/2)/e-15/8*b^(5/2)*n^(5/2)*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(1/2)/b ^(1/2)/n^(1/2))*Pi^(1/2)/e/exp(a/b/n)/((c*(e*x+d)^n)^(1/n))+15/4*b^2*n^2*( e*x+d)*(a+b*ln(c*(e*x+d)^n))^(1/2)/e
Time = 0.12 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.85 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\frac {(d+e x) \left (8 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-5 b n \left (3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \sqrt {\pi } \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (2 a-3 b n+2 b \log \left (c (d+e x)^n\right )\right )\right )\right )}{8 e} \]
((d + e*x)*(8*(a + b*Log[c*(d + e*x)^n])^(5/2) - 5*b*n*((3*b^(3/2)*n^(3/2) *Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b *n))*(c*(d + e*x)^n)^n^(-1)) + 2*Sqrt[a + b*Log[c*(d + e*x)^n]]*(2*a - 3*b *n + 2*b*Log[c*(d + e*x)^n]))))/(8*e)
Time = 0.45 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {2836, 2733, 2733, 2733, 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.
\(\displaystyle \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 2836 |
\(\displaystyle \frac {\int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}d(d+e x)}{e}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-\frac {5}{2} b n \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}d(d+e x)}{e}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-\frac {5}{2} b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}-\frac {3}{2} b n \int \sqrt {a+b \log \left (c (d+e x)^n\right )}d(d+e x)\right )}{e}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-\frac {5}{2} b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}-\frac {3}{2} b n \left ((d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}-\frac {1}{2} b n \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}d(d+e x)\right )\right )}{e}\) |
\(\Big \downarrow \) 2737 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-\frac {5}{2} b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}-\frac {3}{2} b n \left ((d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}-\frac {1}{2} b (d+e x) \left (c (d+e x)^n\right )^{-1/n} \int \frac {\left (c (d+e x)^n\right )^{\frac {1}{n}}}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}d\log \left (c (d+e x)^n\right )\right )\right )}{e}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-\frac {5}{2} b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}-\frac {3}{2} b n \left ((d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}-(d+e x) \left (c (d+e x)^n\right )^{-1/n} \int e^{\frac {a+b \log \left (c (d+e x)^n\right )}{b n}-\frac {a}{b n}}d\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )\right )}{e}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-\frac {5}{2} b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}-\frac {3}{2} b n \left ((d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sqrt {n} e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )\right )}{e}\) |
((d + e*x)*(a + b*Log[c*(d + e*x)^n])^(5/2) - (5*b*n*((d + e*x)*(a + b*Log [c*(d + e*x)^n])^(3/2) - (3*b*n*(-1/2*(Sqrt[b]*Sqrt[n]*Sqrt[Pi]*(d + e*x)* Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) + (d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]))/2))/2)/e
3.2.19.3.1 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, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[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]
\[\int {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {5}{2}}d x\]
Exception generated. \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {5}{2}}\, dx \]
\[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
\[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2} \,d x \]