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

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

[Out]

3^(-1-p)*GAMMA(1+p,-6*(a+b*ln(c*x^(1/2)))/b)*(a+b*ln(c*x^(1/2)))^p/(2^p)/c^6/exp(6*a/b)/(((-a-b*ln(c*x^(1/2)))
/b)^p)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.11, size = 80, normalized size = 1.00 \[ \frac {2^{-p} 3^{-p-1} e^{-\frac {6 a}{b}} \left (a+b \log \left (c \sqrt {x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {x}\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \sqrt {x}\right )\right )}{b}\right )}{c^6} \]

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)

________________________________________________________________________________________

fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p} x^{2}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p} x^{2}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int x^{2} \left (b \ln \left (c \sqrt {x}\right )+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.64, size = 48, normalized size = 0.60 \[ -\frac {2 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p + 1} e^{\left (-\frac {6 \, a}{b}\right )} E_{-p}\left (-\frac {6 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )}}{b}\right )}{b c^{6}} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\left (a+b\,\ln \left (c\,\sqrt {x}\right )\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*log(c*x^(1/2)))^p,x)

[Out]

int(x^2*(a + b*log(c*x^(1/2)))^p, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \log {\left (c \sqrt {x} \right )}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________