\(\int \sqrt {a+b \log (c (d (e+f x)^m)^n)} \, dx\) [432]

Optimal result

Integrand size = 22, antiderivative size = 139 \[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=-\frac {\sqrt {b} e^{-\frac {a}{b m n}} \sqrt {m} \sqrt {n} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{2 f}+\frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96 \[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\frac {(e+f x) \left (-\sqrt {b} e^{-\frac {a}{b m n}} \sqrt {m} \sqrt {n} \sqrt {\pi } \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}\right )}{2 f} \] Input:

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

Output:

((e + f*x)*(-((Sqrt[b]*Sqrt[m]*Sqrt[n]*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*f)
 

Rubi [A] (warning: unable to verify)

Time = 1.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2895, 2836, 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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2836

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

Input:

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

Output:

(-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)]] 
)/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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(sqrt(a + b*log(c*(d*(e + f*x)**m)**n)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\frac {2 \sqrt {\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b +a}\, \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b e +2 \sqrt {\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b +a}\, a e +6 \sqrt {\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b +a}\, a f x +6 \left (\int \frac {\sqrt {\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b +a}\, \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) x}{2 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a b e +2 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a b f x +\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b^{2} e m n +\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b^{2} f m n x +2 a^{2} e +2 a^{2} f x +a b e m n +a b f m n x}d x \right ) a \,b^{2} f^{2} m n +3 \left (\int \frac {\sqrt {\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b +a}\, \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) x}{2 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a b e +2 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a b f x +\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b^{2} e m n +\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b^{2} f m n x +2 a^{2} e +2 a^{2} f x +a b e m n +a b f m n x}d x \right ) b^{3} f^{2} m^{2} n^{2}}{3 f \left (b m n +2 a \right )} \] Input:

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

Output:

(2*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*log(d**n*(e + f*x)**(m*n)*c)*b 
*e + 2*sqrt(log(d**n*(e + f*x)**(m*n)*c)*b + a)*a*e + 6*sqrt(log(d**n*(e + 
 f*x)**(m*n)*c)*b + a)*a*f*x + 6*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**2*f**2*m*n + 3*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)*b**3*f**2*m**2*n**2 
)/(3*f*(2*a + b*m*n))