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

Optimal. Leaf size=107 \[ \frac{2^{-p} e^{-\frac{2 a (m+1)}{b}} \left (c \sqrt{x}\right )^{-2 (m+1)} (d x)^{m+1} \left (a+b \log \left (c \sqrt{x}\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 (m+1) \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right )}{d (m+1)} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0839944, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2310, 2181} \[ \frac{2^{-p} e^{-\frac{2 a (m+1)}{b}} \left (c \sqrt{x}\right )^{-2 (m+1)} (d x)^{m+1} \left (a+b \log \left (c \sqrt{x}\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 (m+1) \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right )}{d (m+1)} \]

Antiderivative was successfully verified.

[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 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2181

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

Rubi steps

\begin{align*} \int (d x)^m \left (a+b \log \left (c \sqrt{x}\right )\right )^p \, dx &=\frac{\left (2 \left (c \sqrt{x}\right )^{-2 (1+m)} (d x)^{1+m}\right ) \operatorname{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]  time = 0.187964, size = 103, normalized size = 0.96 \[ \frac{2^{-p} e^{-\frac{2 a (m+1)}{b}} \left (c \sqrt{x}\right )^{-2 m} (d x)^m \left (a+b \log \left (c \sqrt{x}\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 (m+1) \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right )}{c^2 (m+1)} \]

Antiderivative was successfully verified.

[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]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b\ln \left ( c\sqrt{x} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m}{\left (b \log \left (c \sqrt{x}\right ) + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d x\right )^{m}{\left (b \log \left (c \sqrt{x}\right ) + a\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m}{\left (b \log \left (c \sqrt{x}\right ) + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)