\(\int (a+b \log (c \sqrt {d \sqrt {e+f x}}))^p \, dx\) [438]

Optimal result

Integrand size = 24, antiderivative size = 109 \[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=\frac {4^{-p} e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )}{b}\right )^{-p}}{c^4 d^2 f} \] Output:

GAMMA(p+1,(-4*a-4*b*ln(c*(d*(f*x+e)^(1/2))^(1/2)))/b)*(a+b*ln(c*(d*(f*x+e) 
^(1/2))^(1/2)))^p/(4^p)/c^4/d^2/exp(4*a/b)/f/((-(a+b*ln(c*(d*(f*x+e)^(1/2) 
)^(1/2)))/b)^p)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=\frac {2^{-2 p} e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )}{b}\right )^{-p}}{c^4 d^2 f} \] Input:

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

Output:

(Gamma[1 + p, (-4*(a + b*Log[c*Sqrt[d*Sqrt[e + f*x]]]))/b]*(a + b*Log[c*Sq 
rt[d*Sqrt[e + f*x]]])^p)/(2^(2*p)*c^4*d^2*E^((4*a)/b)*f*(-((a + b*Log[c*Sq 
rt[d*Sqrt[e + f*x]]])/b))^p)
 

Rubi [A] (warning: unable to verify)

Time = 0.73 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2895, 2836, 2736, 2612}

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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2836

\(\displaystyle \frac {\int \left (a+b \log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )\right )^pd(e+f x)}{f}\)

\(\Big \downarrow \) 2736

\(\displaystyle \frac {4 \int c^4 d^2 (e+f x) \left (a+b \log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )\right )^pd\log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )}{c^4 d^2 f}\)

\(\Big \downarrow \) 2612

\(\displaystyle \frac {4^{-p} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )\right )}{b}\right )}{c^4 d^2 f}\)

Input:

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

Output:

(Gamma[1 + p, (-4*(a + b*Log[c*Sqrt[d]*(e + f*x)^(1/4)]))/b]*(a + b*Log[c* 
Sqrt[d]*(e + f*x)^(1/4)])^p)/(4^p*c^4*d^2*E^((4*a)/b)*f*(-((a + b*Log[c*Sq 
rt[d]*(e + f*x)^(1/4)])/b))^p)
 

Defintions of rubi rules used

rule 2612

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] 
:> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) 
)^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, 
 ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && 
!IntegerQ[m]
 

rule 2736

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[1/(n*c^(1 
/n))   Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b 
, c, p}, x] && IntegerQ[1/n]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*log(sqrt(sqrt(f*x + e)*d)*c) + a)^p, x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64 \[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=-\frac {4 \, {\left (b \log \left (\sqrt {\sqrt {f x + e} d} c\right ) + a\right )}^{p + 1} e^{\left (-\frac {4 \, a}{b}\right )} E_{-p}\left (-\frac {4 \, {\left (b \log \left (\sqrt {\sqrt {f x + e} d} c\right ) + a\right )}}{b}\right )}{b c^{4} d^{2} f} \] Input:

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

Output:

-4*(b*log(sqrt(sqrt(f*x + e)*d)*c) + a)^(p + 1)*e^(-4*a/b)*exp_integral_e( 
-p, -4*(b*log(sqrt(sqrt(f*x + e)*d)*c) + a)/b)/(b*c^4*d^2*f)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*(log(sqrt(d)*(e + f*x)**(1/4)*c)*b + a)**p*log(sqrt(d)*(e + f*x)**(1/4) 
*c)*b*e*p + 4*(log(sqrt(d)*(e + f*x)**(1/4)*c)*b + a)**p*a*e*p + 4*(log(sq 
rt(d)*(e + f*x)**(1/4)*c)*b + a)**p*a*f*p*x + 4*(log(sqrt(d)*(e + f*x)**(1 
/4)*c)*b + a)**p*a*f*x + 4*int(((log(sqrt(d)*(e + f*x)**(1/4)*c)*b + a)**p 
*log(sqrt(d)*(e + f*x)**(1/4)*c)*x)/(4*log(sqrt(d)*(e + f*x)**(1/4)*c)*a*b 
*e + 4*log(sqrt(d)*(e + f*x)**(1/4)*c)*a*b*f*x + log(sqrt(d)*(e + f*x)**(1 
/4)*c)*b**2*e*p + log(sqrt(d)*(e + f*x)**(1/4)*c)*b**2*f*p*x + 4*a**2*e + 
4*a**2*f*x + a*b*e*p + a*b*f*p*x),x)*a*b**2*f**2*p**2 + 4*int(((log(sqrt(d 
)*(e + f*x)**(1/4)*c)*b + a)**p*log(sqrt(d)*(e + f*x)**(1/4)*c)*x)/(4*log( 
sqrt(d)*(e + f*x)**(1/4)*c)*a*b*e + 4*log(sqrt(d)*(e + f*x)**(1/4)*c)*a*b* 
f*x + log(sqrt(d)*(e + f*x)**(1/4)*c)*b**2*e*p + log(sqrt(d)*(e + f*x)**(1 
/4)*c)*b**2*f*p*x + 4*a**2*e + 4*a**2*f*x + a*b*e*p + a*b*f*p*x),x)*a*b**2 
*f**2*p + int(((log(sqrt(d)*(e + f*x)**(1/4)*c)*b + a)**p*log(sqrt(d)*(e + 
 f*x)**(1/4)*c)*x)/(4*log(sqrt(d)*(e + f*x)**(1/4)*c)*a*b*e + 4*log(sqrt(d 
)*(e + f*x)**(1/4)*c)*a*b*f*x + log(sqrt(d)*(e + f*x)**(1/4)*c)*b**2*e*p + 
 log(sqrt(d)*(e + f*x)**(1/4)*c)*b**2*f*p*x + 4*a**2*e + 4*a**2*f*x + a*b* 
e*p + a*b*f*p*x),x)*b**3*f**2*p**3 + int(((log(sqrt(d)*(e + f*x)**(1/4)*c) 
*b + a)**p*log(sqrt(d)*(e + f*x)**(1/4)*c)*x)/(4*log(sqrt(d)*(e + f*x)**(1 
/4)*c)*a*b*e + 4*log(sqrt(d)*(e + f*x)**(1/4)*c)*a*b*f*x + log(sqrt(d)*(e 
+ f*x)**(1/4)*c)*b**2*e*p + log(sqrt(d)*(e + f*x)**(1/4)*c)*b**2*f*p*x ...