Integrand size = 22, antiderivative size = 176 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2} \, dx=\frac {3 b^{3/2} e^{-\frac {a}{b m n}} m^{3/2} n^{3/2} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{4 f}-\frac {3 b m n (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{2 f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}{f} \]
(f*x+e)*(a+b*ln(c*(d*(f*x+e)^m)^n))^(3/2)/f+3/4*b^(3/2)*m^(3/2)*n^(3/2)*(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)/exp(a/b/m/n)/f/((c*(d*(f*x+e)^m)^n)^(1/m/n))-3/2*b*m*n*(f*x+e)*(a+b*l n(c*(d*(f*x+e)^m)^n))^(1/2)/f
Time = 0.05 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.91 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2} \, dx=\frac {(e+f x) \left (3 b^{3/2} e^{-\frac {a}{b m n}} m^{3/2} n^{3/2} \sqrt {\pi } \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \left (2 a-3 b m n+2 b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )\right )}{4 f} \]
((e + f*x)*((3*b^(3/2)*m^(3/2)*n^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c*(d*( e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(E^(a/(b*m*n))*(c*(d*(e + f*x) ^m)^n)^(1/(m*n))) + 2*Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]*(2*a - 3*b*m*n + 2*b*Log[c*(d*(e + f*x)^m)^n])))/(4*f)
Time = 0.70 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2895, 2836, 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.
\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}dx\) |
\(\Big \downarrow \) 2836 |
\(\displaystyle \frac {\int \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}d(e+f x)}{f}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}-\frac {3}{2} b m n \int \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}d(e+f x)}{f}\) |
\(\Big \downarrow \) 2733 |
\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}-\frac {3}{2} b m n \left ((e+f x) \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}-\frac {1}{2} b m n \int \frac {1}{\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}d(e+f x)\right )}{f}\) |
\(\Big \downarrow \) 2737 |
\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}-\frac {3}{2} b m n \left ((e+f x) \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}-\frac {1}{2} b (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \int \frac {\left (c d^n (e+f x)^{m n}\right )^{\frac {1}{m n}}}{\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}d\log \left (c d^n (e+f x)^{m n}\right )\right )}{f}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}-\frac {3}{2} b m n \left ((e+f x) \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}-(e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \int \exp \left (\frac {a+b \log \left (c d^n (e+f x)^{m n}\right )}{b m n}-\frac {a}{b m n}\right )d\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}\right )}{f}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}-\frac {3}{2} b m n \left ((e+f x) \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sqrt {m} \sqrt {n} (e+f x) e^{-\frac {a}{b m n}} \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )\right )}{f}\) |
((e + f*x)*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^(3/2) - (3*b*m*n*(-1/2*(Sqrt [b]*Sqrt[m]*Sqrt[n]*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*d^n*(e + f*x) ^(m*n)]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(E^(a/(b*m*n))*(c*d^n*(e + f*x)^(m*n) )^(1/(m*n))) + (e + f*x)*Sqrt[a + b*Log[c*d^n*(e + f*x)^(m*n)]]))/2)/f
3.5.12.3.1 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[((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]]
\[\int {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )\right )}^{\frac {3}{2}}d x\]
Exception generated. \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2} \, dx=\int \left (a + b \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2} \, dx=\int { {\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2} \, dx=\int { {\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2} \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\right )}^{3/2} \,d x \]