3.184 \(\int x^2 (a+b \log (c \sqrt{x}))^p \, dx\)

Optimal. Leaf size=80 \[ \frac{2^{-p} 3^{-p-1} e^{-\frac{6 a}{b}} \left (a+b \log \left (c \sqrt{x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{6 \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right )}{c^6} \]

[Out]

(3^(-1 - p)*Gamma[1 + p, (-6*(a + b*Log[c*Sqrt[x]]))/b]*(a + b*Log[c*Sqrt[x]])^p)/(2^p*c^6*E^((6*a)/b)*(-((a +
b*Log[c*Sqrt[x]])/b))^p)

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Rubi [A]  time = 0.0620918, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2310, 2181} \[ \frac{2^{-p} 3^{-p-1} e^{-\frac{6 a}{b}} \left (a+b \log \left (c \sqrt{x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{6 \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right )}{c^6} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*Sqrt[x]])^p,x]

[Out]

(3^(-1 - p)*Gamma[1 + p, (-6*(a + b*Log[c*Sqrt[x]]))/b]*(a + b*Log[c*Sqrt[x]])^p)/(2^p*c^6*E^((6*a)/b)*(-((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 x^2 \left (a+b \log \left (c \sqrt{x}\right )\right )^p \, dx &=\frac{2 \operatorname{Subst}\left (\int e^{6 x} (a+b x)^p \, dx,x,\log \left (c \sqrt{x}\right )\right )}{c^6}\\ &=\frac{2^{-p} 3^{-1-p} e^{-\frac{6 a}{b}} \Gamma \left (1+p,-\frac{6 \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^6}\\ \end{align*}

Mathematica [A]  time = 0.0516996, size = 80, normalized size = 1. \[ \frac{2^{-p} 3^{-p-1} e^{-\frac{6 a}{b}} \left (a+b \log \left (c \sqrt{x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{6 \left (a+b \log \left (c \sqrt{x}\right )\right )}{b}\right )}{c^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*Sqrt[x]])^p,x]

[Out]

(3^(-1 - p)*Gamma[1 + p, (-6*(a + b*Log[c*Sqrt[x]]))/b]*(a + b*Log[c*Sqrt[x]])^p)/(2^p*c^6*E^((6*a)/b)*(-((a +
b*Log[c*Sqrt[x]])/b))^p)

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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(x^2*(a+b*ln(c*x^(1/2)))^p,x)

[Out]

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

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Maxima [A]  time = 1.28815, size = 65, normalized size = 0.81 \begin{align*} -\frac{2 \,{\left (b \log \left (c \sqrt{x}\right ) + a\right )}^{p + 1} e^{\left (-\frac{6 \, a}{b}\right )} E_{-p}\left (-\frac{6 \,{\left (b \log \left (c \sqrt{x}\right ) + a\right )}}{b}\right )}{b c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b*log(c*sqrt(x)) + a)^(p + 1)*e^(-6*a/b)*exp_integral_e(-p, -6*(b*log(c*sqrt(x)) + a)/b)/(b*c^6)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (c \sqrt{x}\right ) + a\right )}^{p} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*sqrt(x)) + a)^p*x^2, x)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c \sqrt{x}\right ) + a\right )}^{p} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*sqrt(x)) + a)^p*x^2, x)