\(\int (a+b \log (c (d (e+f x)^m)^n))^{5/2} \, dx\) [430]

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:

-15/8*b^(5/2)*m^(5/2)*n^(5/2)*Pi^(1/2)*(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))/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-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
 

Mathematica [A] (verified)

Time = 0.35 (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)
 

Rubi [A] (warning: unable to verify)

Time = 1.48 (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
 

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]]
 

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)
 

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)
 

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
 

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)
 

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)
 

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)
 

Reduce [F]

\[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*log(d**n*(e + f*x)**(m*n)*c)** 
2*a*b**2*e + 8*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*log(d**n*(e + f*x) 
**(m*n)*c)**2*a*b**2*f*x + 4*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*log( 
d**n*(e + f*x)**(m*n)*c)**2*b**3*e*m*n + 4*sqrt(log(d**n*(e + f*x)**(m*n)* 
c)*b + a)*log(d**n*(e + f*x)**(m*n)*c)**2*b**3*f*m*n*x + 16*sqrt(log(d**n* 
(e + f*x)**(m*n)*c)*b + a)*log(d**n*(e + f*x)**(m*n)*c)*a**2*b*e + 16*sqrt 
(log(d**n*(e + f*x)**(m*n)*c)*b + a)*log(d**n*(e + f*x)**(m*n)*c)*a**2*b*f 
*x - 12*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*log(d**n*(e + f*x)**(m*n) 
*c)*a*b**2*e*m*n - 12*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*log(d**n*(e 
 + f*x)**(m*n)*c)*a*b**2*f*m*n*x - 10*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b 
+ a)*log(d**n*(e + f*x)**(m*n)*c)*b**3*f*m**2*n**2*x + 8*sqrt(log(d**n*(e 
+ f*x)**(m*n)*c)*b + a)*a**3*e + 8*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a 
)*a**3*f*x - 16*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*a**2*b*e*m*n - 16 
*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*a**2*b*f*m*n*x + 20*sqrt(log(d** 
n*(e + f*x)**(m*n)*c)*b + a)*a*b**2*f*m**2*n**2*x + 30*int((sqrt(log(d**n* 
(e + f*x)**(m*n)*c)*b + a)*log(d**n*(e + f*x)**(m*n)*c)*x)/(2*log(d**n*(e 
+ f*x)**(m*n)*c)*a*b*e + 2*log(d**n*(e + f*x)**(m*n)*c)*a*b*f*x + log(d**n 
*(e + f*x)**(m*n)*c)*b**2*e*m*n + log(d**n*(e + f*x)**(m*n)*c)*b**2*f*m*n* 
x + 2*a**2*e + 2*a**2*f*x + a*b*e*m*n + a*b*f*m*n*x),x)*a*b**4*f**2*m**3*n 
**3 + 15*int((sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*log(d**n*(e + f*...