\(\int (d x)^m (a+b \log (c \sqrt {x}))^p \, dx\) [183]

Optimal result

Integrand size = 20, antiderivative size = 107 \[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a (1+m)}{b}} \left (c \sqrt {x}\right )^{-2 (1+m)} (d x)^{1+m} \Gamma \left (1+p,-\frac {2 (1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p}}{d (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2347, 2212} \[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a (m+1)}{b}} \left (c \sqrt {x}\right )^{-2 (m+1)} (d x)^{m+1} \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 (m+1) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )}{d (m+1)} \]

[In]

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

[Out]

((d*x)^(1 + m)*Gamma[1 + p, (-2*(1 + m)*(a + b*Log[c*Sqrt[x]]))/b]*(a + b*Log[c*Sqrt[x]])^p)/(2^p*d*E^((2*a*(1
+ m))/b)*(1 + m)*(c*Sqrt[x])^(2*(1 + m))*(-(((1 + m)*(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 {\left (2 \left (c \sqrt {x}\right )^{-2 (1+m)} (d x)^{1+m}\right ) \text {Subst}\left (\int e^{2 (1+m) x} (a+b x)^p \, dx,x,\log \left (c \sqrt {x}\right )\right )}{d} \\ & = \frac {2^{-p} e^{-\frac {2 a (1+m)}{b}} \left (c \sqrt {x}\right )^{-2 (1+m)} (d x)^{1+m} \Gamma \left (1+p,-\frac {2 (1+m) \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 {(1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p}}{d (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96 \[ \int (d x)^m \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a (1+m)}{b}} \left (c \sqrt {x}\right )^{-2 m} (d x)^m \Gamma \left (1+p,-\frac {2 (1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 (1+m)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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