3.65 \(\int \frac{x^3}{a+b \log (c x^n)} \, dx\)

Optimal. Leaf size=51 \[ \frac{x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]

[Out]

(x^4*ExpIntegralEi[(4*(a + b*Log[c*x^n]))/(b*n)])/(b*E^((4*a)/(b*n))*n*(c*x^n)^(4/n))

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Rubi [A]  time = 0.0559981, antiderivative size = 51, normalized size of antiderivative = 1., 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} \[ \frac{x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Log[c*x^n]),x]

[Out]

(x^4*ExpIntegralEi[(4*(a + b*Log[c*x^n]))/(b*n)])/(b*E^((4*a)/(b*n))*n*(c*x^n)^(4/n))

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 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

Rubi steps

\begin{align*} \int \frac{x^3}{a+b \log \left (c x^n\right )} \, dx &=\frac{\left (x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{e^{-\frac{4 a}{b n}} x^4 \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n}\\ \end{align*}

Mathematica [A]  time = 0.0577181, size = 51, normalized size = 1. \[ \frac{x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Log[c*x^n]),x]

[Out]

(x^4*ExpIntegralEi[(4*(a + b*Log[c*x^n]))/(b*n)])/(b*E^((4*a)/(b*n))*n*(c*x^n)^(4/n))

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Maple [F]  time = 0.167, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{a+b\ln \left ( c{x}^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b*ln(c*x^n)),x)

[Out]

int(x^3/(a+b*ln(c*x^n)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

integrate(x^3/(b*log(c*x^n) + a), x)

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Fricas [A]  time = 0.904, size = 108, normalized size = 2.12 \begin{align*} \frac{e^{\left (-\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \logintegral \left (x^{4} e^{\left (\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

e^(-4*(b*log(c) + a)/(b*n))*log_integral(x^4*e^(4*(b*log(c) + a)/(b*n)))/(b*n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{a + b \log{\left (c x^{n} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+b*ln(c*x**n)),x)

[Out]

Integral(x**3/(a + b*log(c*x**n)), x)

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Giac [A]  time = 1.19458, size = 65, normalized size = 1.27 \begin{align*} \frac{{\rm Ei}\left (\frac{4 \, \log \left (c\right )}{n} + \frac{4 \, a}{b n} + 4 \, \log \left (x\right )\right ) e^{\left (-\frac{4 \, a}{b n}\right )}}{b c^{\frac{4}{n}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x))*e^(-4*a/(b*n))/(b*c^(4/n)*n)