3.5.11 \(\int (a+b \log (c (d (e+f x)^m)^n))^{5/2} \, dx\) [411]

3.5.11.1 Optimal result

Integrand size = 22, antiderivative size = 219 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2} \, dx=-\frac {15 b^{5/2} e^{-\frac {a}{b m n}} m^{5/2} n^{5/2} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{8 f}+\frac {15 b^2 m^2 n^2 (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{4 f}-\frac {5 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}{2 f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}{f} \]

output

-5/2*b*m*n*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^m)^n))^(3/2)/f+(f*x+e)*(a+b*ln(c*( 
d*(f*x+e)^m)^n))^(5/2)/f-15/8*b^(5/2)*m^(5/2)*n^(5/2)*(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))+15/4*b^2*m^2*n^2*(f*x+e)*(a+b*ln(c*(d*(f*x 
+e)^m)^n))^(1/2)/f
 

3.5.11.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.87 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2} \, dx=\frac {(e+f x) \left (8 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}-5 b m n \left (3 b^{3/2} e^{-\frac {a}{b m n}} m^{3/2} n^{3/2} \sqrt {\pi } \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \left (2 a-3 b m n+2 b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )\right )\right )}{8 f} \]

input

Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2),x]
 

output

((e + f*x)*(8*(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2) - 5*b*m*n*((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]))))/(8*f)
 

3.5.11.3 Rubi [A] (warning: unable to verify)

Time = 0.82 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2895, 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.

\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2} \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}dx\)

\(\Big \downarrow \) 2836

\(\displaystyle \frac {\int \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{5/2}d(e+f x)}{f}\)

\(\Big \downarrow \) 2733

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{5/2}-\frac {5}{2} b m n \int \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}d(e+f x)}{f}\)

\(\Big \downarrow \) 2733

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{5/2}-\frac {5}{2} b m n \left ((e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}-\frac {3}{2} b m n \int \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}d(e+f x)\right )}{f}\)

\(\Big \downarrow \) 2733

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{5/2}-\frac {5}{2} b m n \left ((e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}-\frac {3}{2} b m n \left ((e+f x) \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}-\frac {1}{2} b m n \int \frac {1}{\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}d(e+f x)\right )\right )}{f}\)

\(\Big \downarrow \) 2737

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{5/2}-\frac {5}{2} b m n \left ((e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}-\frac {3}{2} b m n \left ((e+f x) \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}-\frac {1}{2} b (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \int \frac {\left (c d^n (e+f x)^{m n}\right )^{\frac {1}{m n}}}{\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}d\log \left (c d^n (e+f x)^{m n}\right )\right )\right )}{f}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{5/2}-\frac {5}{2} b m n \left ((e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}-\frac {3}{2} b m n \left ((e+f x) \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}-(e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \int \exp \left (\frac {a+b \log \left (c d^n (e+f x)^{m n}\right )}{b m n}-\frac {a}{b m n}\right )d\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}\right )\right )}{f}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{5/2}-\frac {5}{2} b m n \left ((e+f x) \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}-\frac {3}{2} b m n \left ((e+f x) \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sqrt {m} \sqrt {n} (e+f x) e^{-\frac {a}{b m n}} \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )\right )\right )}{f}\)

input

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2),x]
 

output

((e + f*x)*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^(5/2) - (5*b*m*n*((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)]]/(S 
qrt[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))/2)/f
 

3.5.11.3.1 Defintions of rubi rules used

rule 2611

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]
 

rule 2633

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]
 

rule 2733

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]
 

rule 2737

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]
 

rule 2836

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]
 

rule 2895

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]]
 

3.5.11.4 Maple [F]

\[\int {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )\right )}^{\frac {5}{2}}d x\]

input

int((a+b*ln(c*(d*(f*x+e)^m)^n))^(5/2),x)
 

output

int((a+b*ln(c*(d*(f*x+e)^m)^n))^(5/2),x)
 

3.5.11.5 Fricas [F(-2)]

Exception generated. \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^(5/2),x, algorithm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.5.11.6 Sympy [F(-1)]

Timed out. \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2} \, dx=\text {Timed out} \]

input

integrate((a+b*ln(c*(d*(f*x+e)**m)**n))**(5/2),x)
 

output

Timed out
 

3.5.11.7 Maxima [F]

\[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2} \, dx=\int { {\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

input

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^(5/2),x, algorithm="maxima")
 

output

integrate((b*log(((f*x + e)^m*d)^n*c) + a)^(5/2), x)
 

3.5.11.8 Giac [F]

\[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2} \, dx=\int { {\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

input

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^(5/2),x, algorithm="giac")
 

output

integrate((b*log(((f*x + e)^m*d)^n*c) + a)^(5/2), x)
 

3.5.11.9 Mupad [F(-1)]

Timed out. \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2} \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\right )}^{5/2} \,d x \]

input

int((a + b*log(c*(d*(e + f*x)^m)^n))^(5/2),x)
 

output

int((a + b*log(c*(d*(e + f*x)^m)^n))^(5/2), x)