3.66 \(\int \frac {x^2}{a+b \log (c x^n)} \, dx\)

Optimal. Leaf size=51 \[ \frac {x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]

[Out]

x^3*Ei(3*(a+b*ln(c*x^n))/b/n)/b/exp(3*a/b/n)/n/((c*x^n)^(3/n))

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Rubi [A]  time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2310, 2178} \[ \frac {x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Log[c*x^n]),x]

[Out]

(x^3*ExpIntegralEi[(3*(a + b*Log[c*x^n]))/(b*n)])/(b*E^((3*a)/(b*n))*n*(c*x^n)^(3/n))

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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 \frac {x^2}{a+b \log \left (c x^n\right )} \, dx &=\frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 51, normalized size = 1.00 \[ \frac {x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Log[c*x^n]),x]

[Out]

(x^3*ExpIntegralEi[(3*(a + b*Log[c*x^n]))/(b*n)])/(b*E^((3*a)/(b*n))*n*(c*x^n)^(3/n))

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fricas [A]  time = 0.41, size = 42, normalized size = 0.82 \[ \frac {e^{\left (-\frac {3 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left (x^{3} e^{\left (\frac {3 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )}\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-3*(b*log(c) + a)/(b*n))*log_integral(x^3*e^(3*(b*log(c) + a)/(b*n)))/(b*n)

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giac [A]  time = 0.26, size = 48, normalized size = 0.94 \[ \frac {{\rm Ei}\left (\frac {3 \, \log \relax (c)}{n} + \frac {3 \, a}{b n} + 3 \, \log \relax (x)\right ) e^{\left (-\frac {3 \, a}{b n}\right )}}{b c^{\frac {3}{n}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x))*e^(-3*a/(b*n))/(b*c^(3/n)*n)

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{b \ln \left (c \,x^{n}\right )+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/(b*log(c*x^n) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*log(c*x^n)),x)

[Out]

int(x^2/(a + b*log(c*x^n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(a + b*log(c*x**n)), x)

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