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

Optimal. Leaf size=88 \[ \frac{2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \sqrt{d+e x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d+e x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \sqrt{d+e x}\right )\right )}{b}\right )}{c^2 e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0675028, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2389, 2299, 2181} \[ \frac{2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \sqrt{d+e x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d+e x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \sqrt{d+e x}\right )\right )}{b}\right )}{c^2 e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*Sqrt[d + e*x]])^p,x]

[Out]

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

Rule 2389

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 2299

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 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 \left (a+b \log \left (c \sqrt{d+e x}\right )\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c \sqrt{x}\right )\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac{2 \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \sqrt{d+e x}\right )\right )}{c^2 e}\\ &=\frac{2^{-p} e^{-\frac{2 a}{b}} \Gamma \left (1+p,-\frac{2 \left (a+b \log \left (c \sqrt{d+e x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt{d+e x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d+e x}\right )}{b}\right )^{-p}}{c^2 e}\\ \end{align*}

Mathematica [A]  time = 0.0500074, size = 88, normalized size = 1. \[ \frac{2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \sqrt{d+e x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d+e x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \sqrt{d+e x}\right )\right )}{b}\right )}{c^2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.267, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c\sqrt{ex+d} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.30587, size = 80, normalized size = 0.91 \begin{align*} -\frac{2 \,{\left (b \log \left (\sqrt{e x + d} c\right ) + a\right )}^{p + 1} e^{\left (-\frac{2 \, a}{b}\right )} E_{-p}\left (-\frac{2 \,{\left (b \log \left (\sqrt{e x + d} c\right ) + a\right )}}{b}\right )}{b c^{2} e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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