\(\int (e x)^m \operatorname {ExpIntegralEi}(d (a+b \log (c x^n))) \, dx\) [47]

Optimal result

Integrand size = 19, antiderivative size = 100 \[ \int (e x)^m \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m} \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {e^{-\frac {a (1+m)}{b n}} (e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.88 \[ \int (e x)^m \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^m \left (x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} x^{-m} \operatorname {ExpIntegralEi}\left (\frac {(1+m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )}{1+m} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {7048, 2747, 2609}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (e x)^m \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\)

\(\Big \downarrow \) 7048

\(\displaystyle \frac {(e x)^{m+1} \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b n e^{a d} \left (c x^n\right )^{b d} (e x)^{-b d n} \int \frac {(e x)^{m+b d n}}{a+b \log \left (c x^n\right )}dx}{m+1}\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {(e x)^{m+1} \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b e^{a d} (e x)^{m+1} \left (c x^n\right )^{b d-\frac {b d n+m+1}{n}} \int \frac {\left (c x^n\right )^{\frac {m+b d n+1}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{e (m+1)}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {(e x)^{m+1} \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {(e x)^{m+1} e^{a d-\frac {a (b d n+m+1)}{b n}} \left (c x^n\right )^{b d-\frac {b d n+m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m+b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{e (m+1)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2747

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

rule 7048

Int[ExpIntegralEi[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_)) 
^(m_.), x_Symbol] :> Simp[(e*x)^(m + 1)*(ExpIntegralEi[d*(a + b*Log[c*x^n]) 
]/(e*(m + 1))), x] - Simp[b*n*E^(a*d)*((c*x^n)^(b*d)/((m + 1)*(e*x)^(b*d*n) 
))   Int[(e*x)^(m + b*d*n)/(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, 
 e, m, n}, x] && NeQ[m, -1]
 

Maple [F]

\[\int \left (e x \right )^{m} \operatorname {expIntegral}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.17 \[ \int (e x)^m \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x {\rm Ei}\left (b d \log \left (c x^{n}\right ) + a d\right ) e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} - {\rm Ei}\left (\frac {a b d n + a m + {\left (b^{2} d n + b m + b\right )} \log \left (c\right ) + {\left (b^{2} d n^{2} + {\left (b m + b\right )} n\right )} \log \left (x\right ) + a}{b n}\right ) e^{\left (\frac {b m n \log \left (e\right ) - a m - {\left (b m + b\right )} \log \left (c\right ) - a}{b n}\right )}}{m + 1} \] Input:

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

Output:

(x*Ei(b*d*log(c*x^n) + a*d)*e^(m*log(e) + m*log(x)) - Ei((a*b*d*n + a*m + 
(b^2*d*n + b*m + b)*log(c) + (b^2*d*n^2 + (b*m + b)*n)*log(x) + a)/(b*n))* 
e^((b*m*n*log(e) - a*m - (b*m + b)*log(c) - a)/(b*n)))/(m + 1)
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^m \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int (e x)^m \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} {\rm Ei}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int (e x)^m \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} {\rm Ei}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {ei}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \] Input:

int(ei(d*(a + b*log(c*x^n)))*(e*x)^m,x)
 

Output:

int(ei(d*(a + b*log(c*x^n)))*(e*x)^m, x)
 

Reduce [F]

\[ \int (e x)^m \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=e^{m} \left (\int x^{m} \mathit {ei} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )d x \right ) \] Input:

int((e*x)^m*Ei(d*(a+b*log(c*x^n))),x)
 

Output:

e**m*int(x**m*ei(log(x**n*c)*b*d + a*d),x)