\(\int x (a+b \log (c \sqrt {x}))^p \, dx\) [185]

Optimal result

Integrand size = 16, antiderivative size = 75 \[ \int x \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\frac {2^{-1-2 p} e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {x}\right )}{b}\right )^{-p}}{c^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2347, 2212} \[ \int x \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\frac {2^{-2 p-1} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {x}\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )}{c^4} \]

[In]

Int[x*(a + b*Log[c*Sqrt[x]])^p,x]

[Out]

(2^(-1 - 2*p)*Gamma[1 + p, (-4*(a + b*Log[c*Sqrt[x]]))/b]*(a + b*Log[c*Sqrt[x]])^p)/(c^4*E^((4*a)/b)*(-((a + b
*Log[c*Sqrt[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 2347

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \sqrt {x}\right )\right )}{c^4} \\ & = \frac {2^{-1-2 p} e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {x}\right )}{b}\right )^{-p}}{c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\frac {2^{-1-2 p} e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {x}\right )}{b}\right )^{-p}}{c^4} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.64 \[ \int x \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=-\frac {2 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p + 1} e^{\left (-\frac {4 \, a}{b}\right )} E_{-p}\left (-\frac {4 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )}}{b}\right )}{b c^{4}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x*(a + b*log(c*x^(1/2)))^p,x)

[Out]

int(x*(a + b*log(c*x^(1/2)))^p, x)