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

Optimal. Leaf size=86 \[ \frac{(e x)^{q+1} e^{-\frac{a (q+1)}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac{q+1}{m n}} \text{Ei}\left (\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{b e m n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.184267, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2310, 2178, 2445} \[ \frac{(e x)^{q+1} e^{-\frac{a (q+1)}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac{q+1}{m n}} \text{Ei}\left (\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{b e m n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

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

Mathematica [A]  time = 0.155784, size = 85, normalized size = 0.99 \[ \frac{x^{-q} (e x)^q \exp \left (-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )-b m n \log (x)\right )}{b m n}\right ) \text{Ei}\left (\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{b m n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{q}}{a+b\ln \left ( c \left ( d{x}^{m} \right ) ^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.908281, size = 244, normalized size = 2.84 \begin{align*} \frac{{\rm Ei}\left (\frac{a q +{\left (b q + b\right )} \log \left (c\right ) +{\left (b n q + b n\right )} \log \left (d\right ) +{\left (b m n q + b m n\right )} \log \left (x\right ) + a}{b m n}\right ) e^{\left (\frac{b m n q \log \left (e\right ) - a q -{\left (b q + b\right )} \log \left (c\right ) -{\left (b n q + b n\right )} \log \left (d\right ) - a}{b m n}\right )}}{b m n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei((a*q + (b*q + b)*log(c) + (b*n*q + b*n)*log(d) + (b*m*n*q + b*m*n)*log(x) + a)/(b*m*n))*e^((b*m*n*q*log(e)
- a*q - (b*q + b)*log(c) - (b*n*q + b*n)*log(d) - a)/(b*m*n))/(b*m*n)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.31952, size = 189, normalized size = 2.2 \begin{align*} \frac{{\rm Ei}\left (q \log \left (x\right ) + \frac{q \log \left (d\right )}{m} + \frac{q \log \left (c\right )}{m n} + \frac{\log \left (d\right )}{m} + \frac{a q}{b m n} + \frac{\log \left (c\right )}{m n} + \frac{a}{b m n} + \log \left (x\right )\right ) e^{\left (q - \frac{a q}{b m n} - \frac{a}{b m n}\right )}}{b c^{\frac{q}{m n}} c^{\frac{1}{m n}} d^{\frac{q}{m}} d^{\left (\frac{1}{m}\right )} m n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(q*log(x) + q*log(d)/m + q*log(c)/(m*n) + log(d)/m + a*q/(b*m*n) + log(c)/(m*n) + a/(b*m*n) + log(x))*e^(q -
a*q/(b*m*n) - a/(b*m*n))/(b*c^(q/(m*n))*c^(1/(m*n))*d^(q/m)*d^(1/m)*m*n)