\(\int \frac {1}{(a+b \log (c (d (e+f x)^m)^n))^{7/2}} \, dx\) [436]

Optimal result

Integrand size = 22, antiderivative size = 237 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\frac {8 e^{-\frac {a}{b m n}} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{15 b^{7/2} f m^{7/2} n^{7/2}}-\frac {2 (e+f x)}{5 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}-\frac {4 (e+f x)}{15 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}-\frac {8 (e+f x)}{15 b^3 f m^3 n^3 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}} \] Output:

8/15*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))/b^(7/2)/exp(a/b/m/n)/f/m^(7/2)/n^(7/2)/((c*(d*(f*x+e)^m)^n)^(1 
/m/n))-2/5*(f*x+e)/b/f/m/n/(a+b*ln(c*(d*(f*x+e)^m)^n))^(5/2)-4/15*(f*x+e)/ 
b^2/f/m^2/n^2/(a+b*ln(c*(d*(f*x+e)^m)^n))^(3/2)-8/15*(f*x+e)/b^3/f/m^3/n^3 
/(a+b*ln(c*(d*(f*x+e)^m)^n))^(1/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=-\frac {2 e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \left (-4 \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right ) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2 \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}}+e^{\frac {a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{\frac {1}{m n}} \left (4 a^2+2 a b m n+3 b^2 m^2 n^2+2 b (4 a+b m n) \log \left (c \left (d (e+f x)^m\right )^n\right )+4 b^2 \log ^2\left (c \left (d (e+f x)^m\right )^n\right )\right )\right )}{15 b^3 f m^3 n^3 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}} \] Input:

Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-7/2),x]
 

Output:

(-2*(e + f*x)*(-4*Gamma[1/2, -((a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n))]* 
(a + b*Log[c*(d*(e + f*x)^m)^n])^2*Sqrt[-((a + b*Log[c*(d*(e + f*x)^m)^n]) 
/(b*m*n))] + E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(4*a^2 + 2*a*b* 
m*n + 3*b^2*m^2*n^2 + 2*b*(4*a + b*m*n)*Log[c*(d*(e + f*x)^m)^n] + 4*b^2*L 
og[c*(d*(e + f*x)^m)^n]^2)))/(15*b^3*E^(a/(b*m*n))*f*m^3*n^3*(c*(d*(e + f* 
x)^m)^n)^(1/(m*n))*(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2))
 

Rubi [A] (warning: unable to verify)

Time = 1.68 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2895, 2836, 2734, 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.

\(\displaystyle \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}}dx\)

\(\Big \downarrow \) 2836

\(\displaystyle \frac {\int \frac {1}{\left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{7/2}}d(e+f x)}{f}\)

\(\Big \downarrow \) 2734

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

\(\Big \downarrow \) 2734

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

\(\Big \downarrow \) 2734

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2611

\(\displaystyle \frac {\frac {2 \left (\frac {2 \left (\frac {4 (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \int \exp \left (\frac {a+b \log \left (c d^n (e+f x)^{m n}\right )}{b m n}-\frac {a}{b m n}\right )d\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}{b^2 m^2 n^2}-\frac {2 (e+f x)}{b m n \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}}\right )}{3 b m n}-\frac {2 (e+f x)}{3 b m n \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}}\right )}{5 b m n}-\frac {2 (e+f x)}{5 b m n \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{5/2}}}{f}\)

\(\Big \downarrow \) 2633

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

Input:

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-7/2),x]
 

Output:

((-2*(e + f*x))/(5*b*m*n*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^(5/2)) + (2*(( 
-2*(e + f*x))/(3*b*m*n*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^(3/2)) + (2*((2* 
Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*d^n*(e + f*x)^(m*n)]]/(Sqrt[b]*Sq 
rt[m]*Sqrt[n])])/(b^(3/2)*E^(a/(b*m*n))*m^(3/2)*n^(3/2)*(c*d^n*(e + f*x)^( 
m*n))^(1/(m*n))) - (2*(e + f*x))/(b*m*n*Sqrt[a + b*Log[c*d^n*(e + f*x)^(m* 
n)]])))/(3*b*m*n)))/(5*b*m*n))/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 2734

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]
 

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]

\[\int \frac {1}{{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )\right )}^{\frac {7}{2}}}d x\]

Input:

int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^(7/2),x)
 

Output:

int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^(7/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^(7/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*ln(c*(d*(f*x+e)**m)**n))**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^(7/2),x, algorithm="maxima")
 

Output:

integrate((b*log(((f*x + e)^m*d)^n*c) + a)^(-7/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^(7/2),x, algorithm="giac")
 

Output:

integrate((b*log(((f*x + e)^m*d)^n*c) + a)^(-7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\right )}^{7/2}} \,d x \] Input:

int(1/(a + b*log(c*(d*(e + f*x)^m)^n))^(7/2),x)
 

Output:

int(1/(a + b*log(c*(d*(e + f*x)^m)^n))^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

int(1/(a+b*log(c*(d*(f*x+e)^m)^n))^(7/2),x)
                                                                                    
                                                                                    
 

Output:

( - 2*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*e + 5*int((sqrt(log(d**n*(e 
 + f*x)**(m*n)*c)*b + a)*x)/(log(d**n*(e + f*x)**(m*n)*c)**4*b**4*e + log( 
d**n*(e + f*x)**(m*n)*c)**4*b**4*f*x + 4*log(d**n*(e + f*x)**(m*n)*c)**3*a 
*b**3*e + 4*log(d**n*(e + f*x)**(m*n)*c)**3*a*b**3*f*x + 6*log(d**n*(e + f 
*x)**(m*n)*c)**2*a**2*b**2*e + 6*log(d**n*(e + f*x)**(m*n)*c)**2*a**2*b**2 
*f*x + 4*log(d**n*(e + f*x)**(m*n)*c)*a**3*b*e + 4*log(d**n*(e + f*x)**(m* 
n)*c)*a**3*b*f*x + a**4*e + a**4*f*x),x)*log(d**n*(e + f*x)**(m*n)*c)**3*b 
**4*f**2*m*n + 15*int((sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*x)/(log(d* 
*n*(e + f*x)**(m*n)*c)**4*b**4*e + log(d**n*(e + f*x)**(m*n)*c)**4*b**4*f* 
x + 4*log(d**n*(e + f*x)**(m*n)*c)**3*a*b**3*e + 4*log(d**n*(e + f*x)**(m* 
n)*c)**3*a*b**3*f*x + 6*log(d**n*(e + f*x)**(m*n)*c)**2*a**2*b**2*e + 6*lo 
g(d**n*(e + f*x)**(m*n)*c)**2*a**2*b**2*f*x + 4*log(d**n*(e + f*x)**(m*n)* 
c)*a**3*b*e + 4*log(d**n*(e + f*x)**(m*n)*c)*a**3*b*f*x + a**4*e + a**4*f* 
x),x)*log(d**n*(e + f*x)**(m*n)*c)**2*a*b**3*f**2*m*n + 15*int((sqrt(log(d 
**n*(e + f*x)**(m*n)*c)*b + a)*x)/(log(d**n*(e + f*x)**(m*n)*c)**4*b**4*e 
+ log(d**n*(e + f*x)**(m*n)*c)**4*b**4*f*x + 4*log(d**n*(e + f*x)**(m*n)*c 
)**3*a*b**3*e + 4*log(d**n*(e + f*x)**(m*n)*c)**3*a*b**3*f*x + 6*log(d**n* 
(e + f*x)**(m*n)*c)**2*a**2*b**2*e + 6*log(d**n*(e + f*x)**(m*n)*c)**2*a** 
2*b**2*f*x + 4*log(d**n*(e + f*x)**(m*n)*c)*a**3*b*e + 4*log(d**n*(e + f*x 
)**(m*n)*c)*a**3*b*f*x + a**4*e + a**4*f*x),x)*log(d**n*(e + f*x)**(m*n...