\(\int (a+b \log (c \sqrt {d \sqrt {e+f x}}))^p \, dx\) [419]

Optimal result

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

[Out]

GAMMA(p+1,-4*(a+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)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2436, 2336, 2212, 2495} \[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=\frac {4^{-p} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )}{b}\right )}{c^4 d^2 f} \]

[In]

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

[Out]

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

Rule 2212

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 2336

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[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 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[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 2495

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]]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (a+b \log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )\right )^p \, dx,c \sqrt {d} \sqrt [4]{e+f x},c \sqrt {d \sqrt {e+f x}}\right ) \\ & = \text {Subst}\left (\frac {\text {Subst}\left (\int \left (a+b \log \left (c \sqrt {d} \sqrt [4]{x}\right )\right )^p \, dx,x,e+f x\right )}{f},c \sqrt {d} \sqrt [4]{e+f x},c \sqrt {d \sqrt {e+f x}}\right ) \\ & = \text {Subst}\left (\frac {4 \text {Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )\right )}{c^4 d^2 f},c \sqrt {d} \sqrt [4]{e+f x},c \sqrt {d \sqrt {e+f x}}\right ) \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=\frac {2^{-2 p} e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )}{b}\right )^{-p}}{c^4 d^2 f} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

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

[In]

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

[Out]

-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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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