Integrand size = 24, antiderivative size = 109 \[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=\frac {4^{-p} e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )}{b}\right )^{-p}}{c^4 d^2 f} \]
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)
Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=\frac {2^{-2 p} e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )}{b}\right )^{-p}}{c^4 d^2 f} \]
(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)
Time = 0.42 (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 |
\(\displaystyle \frac {4^{-p} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \sqrt {d} \sqrt [4]{e+f x}\right )\right )}{b}\right )}{c^4 d^2 f}\) |
(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)
3.5.19.3.1 Defintions of rubi rules used
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]
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]
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]
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]]
\[\int \left (a +b \ln \left (c \sqrt {d \sqrt {f x +e}}\right )\right )^{p}d x\]
\[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=\int { {\left (b \log \left (\sqrt {\sqrt {f x + e} d} c\right ) + a\right )}^{p} \,d x } \]
\[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=\int \left (a + b \log {\left (c \sqrt {d \sqrt {e + f x}} \right )}\right )^{p}\, dx \]
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64 \[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=-\frac {4 \, {\left (b \log \left (\sqrt {\sqrt {f x + e} d} c\right ) + a\right )}^{p + 1} e^{\left (-\frac {4 \, a}{b}\right )} E_{-p}\left (-\frac {4 \, {\left (b \log \left (\sqrt {\sqrt {f x + e} d} c\right ) + a\right )}}{b}\right )}{b c^{4} d^{2} f} \]
-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)
\[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=\int { {\left (b \log \left (\sqrt {\sqrt {f x + e} d} c\right ) + a\right )}^{p} \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c \sqrt {d \sqrt {e+f x}}\right )\right )^p \, dx=\int {\left (a+b\,\ln \left (c\,\sqrt {d\,\sqrt {e+f\,x}}\right )\right )}^p \,d x \]