\(\int \sqrt {a+b \log (c (d+e x)^n)} \, dx\) [107]

Optimal result

Integrand size = 18, antiderivative size = 111 \[ \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=-\frac {\sqrt {b} e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e}+\frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e} \] Output:

-1/2*b^(1/2)*n^(1/2)*Pi^(1/2)*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(1/2)/b^( 
1/2)/n^(1/2))/e/exp(a/b/n)/((c*(e*x+d)^n)^(1/n))+(e*x+d)*(a+b*ln(c*(e*x+d) 
^n))^(1/2)/e
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95 \[ \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\frac {(d+e x) \left (-\sqrt {b} e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{2 e} \] Input:

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

Output:

((d + e*x)*(-((Sqrt[b]*Sqrt[n]*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n] 
]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1))) + 2*Sqrt[a + b 
*Log[c*(d + e*x)^n]]))/(2*e)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {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.

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

\(\Big \downarrow \) 2836

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

Input:

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

Output:

(-1/2*(Sqrt[b]*Sqrt[n]*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^ 
n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) + (d + e*x)*S 
qrt[a + b*Log[c*(d + e*x)^n]])/e
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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