3.176 \(\int x^2 (a+b \log (c x))^p \, dx\)

Optimal. Leaf size=63 \[ \frac {3^{-p-1} e^{-\frac {3 a}{b}} (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 (a+b \log (c x))}{b}\right )}{c^3} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2309, 2181} \[ \frac {3^{-p-1} e^{-\frac {3 a}{b}} (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {3 (a+b \log (c x))}{b}\right )}{c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*x])^p,x]

[Out]

(3^(-1 - p)*Gamma[1 + p, (-3*(a + b*Log[c*x]))/b]*(a + b*Log[c*x])^p)/(c^3*E^((3*a)/b)*(-((a + b*Log[c*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 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int x^2 (a+b \log (c x))^p \, dx &=\frac {\operatorname {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log (c x)\right )}{c^3}\\ &=\frac {3^{-1-p} e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c^3}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 63, normalized size = 1.00 \[ \frac {3^{-p-1} e^{-\frac {3 a}{b}} (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 (a+b \log (c x))}{b}\right )}{c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*x])^p,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x) + a)^p*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x) + a)^p*x^2, x)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int x^{2} \left (b \ln \left (c x \right )+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.83, size = 44, normalized size = 0.70 \[ -\frac {{\left (b \log \left (c x\right ) + a\right )}^{p + 1} e^{\left (-\frac {3 \, a}{b}\right )} E_{-p}\left (-\frac {3 \, {\left (b \log \left (c x\right ) + a\right )}}{b}\right )}{b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*log(c*x) + a)^(p + 1)*e^(-3*a/b)*exp_integral_e(-p, -3*(b*log(c*x) + a)/b)/(b*c^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,{\left (a+b\,\ln \left (c\,x\right )\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*log(c*x))^p,x)

[Out]

int(x^2*(a + b*log(c*x))^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*log(c*x))**p, x)

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