3.153 \(\int \frac{(d x)^m}{a+b \log (c x^n)} \, dx\)

Optimal. Leaf size=66 \[ \frac{(d x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n} \]

[Out]

((d*x)^(1 + m)*ExpIntegralEi[((1 + m)*(a + b*Log[c*x^n]))/(b*n)])/(b*d*E^((a*(1 + m))/(b*n))*n*(c*x^n)^((1 + m
)/n))

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Rubi [A]  time = 0.0736044, antiderivative size = 66, 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, 2178} \[ \frac{(d x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m/(a + b*Log[c*x^n]),x]

[Out]

((d*x)^(1 + m)*ExpIntegralEi[((1 + m)*(a + b*Log[c*x^n]))/(b*n)])/(b*d*E^((a*(1 + m))/(b*n))*n*(c*x^n)^((1 + m
)/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{(d x)^m}{a+b \log \left (c x^n\right )} \, dx &=\frac{\left ((d x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1+m) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{d n}\\ &=\frac{e^{-\frac{a (1+m)}{b n}} (d x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \text{Ei}\left (\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n}\\ \end{align*}

Mathematica [A]  time = 0.105898, size = 67, normalized size = 1.02 \[ \frac{x^{-m} (d x)^m \exp \left (-\frac{(m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \text{Ei}\left (\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m/(a + b*Log[c*x^n]),x]

[Out]

((d*x)^m*ExpIntegralEi[((1 + m)*(a + b*Log[c*x^n]))/(b*n)])/(b*E^(((1 + m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))
/(b*n))*n*x^m)

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Maple [F]  time = 0.245, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{a+b\ln \left ( c{x}^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x)^m/(b*log(c*x^n) + a), x)

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Fricas [A]  time = 0.989485, size = 163, normalized size = 2.47 \begin{align*} \frac{{\rm Ei}\left (\frac{{\left (b m + b\right )} n \log \left (x\right ) + a m +{\left (b m + b\right )} \log \left (c\right ) + a}{b n}\right ) e^{\left (\frac{b m n \log \left (d\right ) - a m -{\left (b m + b\right )} \log \left (c\right ) - a}{b n}\right )}}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(((b*m + b)*n*log(x) + a*m + (b*m + b)*log(c) + a)/(b*n))*e^((b*m*n*log(d) - a*m - (b*m + 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{\left (d x\right )^{m}}{a + b \log{\left (c x^{n} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*x)**m/(a + b*log(c*x**n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x)^m/(b*log(c*x^n) + a), x)