3.5.8 \(\int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx\) [408]

3.5.8.1 Optimal result

Integrand size = 19, antiderivative size = 240 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {b d^2 e^2 e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4} \]

output

1/2*d^2*exp(e/(d*x+c))/b/(-a*d+b*c)^2+1/2*d^2*e*exp(e/(d*x+c))/(-a*d+b*c)^ 
3-1/2*exp(e/(d*x+c))/b/(b*x+a)^2+1/2*d*e*exp(e/(d*x+c))/(-a*d+b*c)^2/(b*x+ 
a)+d^2*e*exp(b*e/(-a*d+b*c))*Ei(-d*e*(b*x+a)/(-a*d+b*c)/(d*x+c))/(-a*d+b*c 
)^3+1/2*b*d^2*e^2*exp(b*e/(-a*d+b*c))*Ei(-d*e*(b*x+a)/(-a*d+b*c)/(d*x+c))/ 
(-a*d+b*c)^4
 

3.5.8.2 Mathematica [F]

\[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx \]

input

Integrate[E^(e/(c + d*x))/(a + b*x)^3,x]
 

output

Integrate[E^(e/(c + d*x))/(a + b*x)^3, x]
 

3.5.8.3 Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2653, 7293, 2009}

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.

\(\displaystyle \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx\)

\(\Big \downarrow \) 2653

\(\displaystyle -\frac {d e \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2 (c+d x)^2}dx}{2 b}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {d e \int \left (-\frac {2 d e^{\frac {e}{c+d x}} b^2}{(b c-a d)^3 (a+b x)}+\frac {e^{\frac {e}{c+d x}} b^2}{(b c-a d)^2 (a+b x)^2}+\frac {2 d^2 e^{\frac {e}{c+d x}} b}{(b c-a d)^3 (c+d x)}+\frac {d^2 e^{\frac {e}{c+d x}}}{(b c-a d)^2 (c+d x)^2}\right )dx}{2 b}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d e \left (-\frac {b^2 d e e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^4}-\frac {2 b d e^{\frac {b e}{b c-a d}} \operatorname {ExpIntegralEi}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}-\frac {b d e^{\frac {e}{c+d x}}}{(b c-a d)^3}-\frac {b e^{\frac {e}{c+d x}}}{(a+b x) (b c-a d)^2}-\frac {d e^{\frac {e}{c+d x}}}{e (b c-a d)^2}\right )}{2 b}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}\)

input

Int[E^(e/(c + d*x))/(a + b*x)^3,x]
 

output

-1/2*E^(e/(c + d*x))/(b*(a + b*x)^2) - (d*e*(-((b*d*E^(e/(c + d*x)))/(b*c 
- a*d)^3) - (d*E^(e/(c + d*x)))/((b*c - a*d)^2*e) - (b*E^(e/(c + d*x)))/(( 
b*c - a*d)^2*(a + b*x)) - (2*b*d*E^((b*e)/(b*c - a*d))*ExpIntegralEi[-((d* 
e*(a + b*x))/((b*c - a*d)*(c + d*x)))])/(b*c - a*d)^3 - (b^2*d*e*E^((b*e)/ 
(b*c - a*d))*ExpIntegralEi[-((d*e*(a + b*x))/((b*c - a*d)*(c + d*x)))])/(b 
*c - a*d)^4))/(2*b)
 

3.5.8.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2653

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))*((e_.) + (f_.)*(x_))^(m_), x_ 
Symbol] :> Simp[(e + f*x)^(m + 1)*(F^(a + b/(c + d*x))/(f*(m + 1))), x] + S 
imp[b*d*(Log[F]/(f*(m + 1)))   Int[(e + f*x)^(m + 1)*(F^(a + b/(c + d*x))/( 
c + d*x)^2), x], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] 
 && ILtQ[m, -1]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.5.8.4 Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(-\frac {e \left (\frac {d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{\frac {e}{d x +c}+\frac {b e}{a d -c b}}-{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )\right )}{\left (a d -c b \right )^{3}}-\frac {b e \,d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )^{2}}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}-\frac {{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{2}\right )}{\left (a d -c b \right )^{4}}\right )}{d}\)	\(240\)
default	\(-\frac {e \left (\frac {d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{\frac {e}{d x +c}+\frac {b e}{a d -c b}}-{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )\right )}{\left (a d -c b \right )^{3}}-\frac {b e \,d^{3} \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )^{2}}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}-\frac {{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{2}\right )}{\left (a d -c b \right )^{4}}\right )}{d}\)	\(240\)
risch	\(\frac {e \,d^{2} {\mathrm e}^{\frac {e}{d x +c}}}{\left (a d -c b \right )^{3} \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}+\frac {e \,d^{2} {\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{\left (a d -c b \right )^{3}}-\frac {e^{2} d^{2} b \,{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (a d -c b \right )^{4} \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )^{2}}-\frac {e^{2} d^{2} b \,{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (a d -c b \right )^{4} \left (\frac {e}{d x +c}+\frac {b e}{a d -c b}\right )}-\frac {e^{2} d^{2} b \,{\mathrm e}^{-\frac {b e}{a d -c b}} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{2 \left (a d -c b \right )^{4}}\)	\(278\)

input

int(exp(e/(d*x+c))/(b*x+a)^3,x,method=_RETURNVERBOSE)
 

output

-1/d*e*(d^3/(a*d-b*c)^3*(-exp(e/(d*x+c))/(e/(d*x+c)+b*e/(a*d-b*c))-exp(-b* 
e/(a*d-b*c))*Ei(1,-e/(d*x+c)-b*e/(a*d-b*c)))-b*e/(a*d-b*c)^4*d^3*(-1/2*exp 
(e/(d*x+c))/(e/(d*x+c)+b*e/(a*d-b*c))^2-1/2*exp(e/(d*x+c))/(e/(d*x+c)+b*e/ 
(a*d-b*c))-1/2*exp(-b*e/(a*d-b*c))*Ei(1,-e/(d*x+c)-b*e/(a*d-b*c))))
 

3.5.8.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (224) = 448\).

Time = 0.29 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.15 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\frac {{\left (a^{2} b d^{2} e^{2} + {\left (b^{3} d^{2} e^{2} + 2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} + 2 \, {\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} e + 2 \, {\left (a b^{2} d^{2} e^{2} + 2 \, {\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} e\right )} x\right )} {\rm Ei}\left (-\frac {b d e x + a d e}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac {b e}{b c - a d}\right )} - {\left (b^{3} c^{4} - 4 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3} - {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} - {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e - {\left (2 \, a b^{2} c^{2} d^{2} - 4 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4} + {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{2 \, {\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4} + {\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 6 \, a^{3} b^{3} c^{2} d^{2} - 4 \, a^{4} b^{2} c d^{3} + a^{5} b d^{4}\right )} x\right )}} \]

input

integrate(exp(e/(d*x+c))/(b*x+a)^3,x, algorithm="fricas")
 

output

1/2*((a^2*b*d^2*e^2 + (b^3*d^2*e^2 + 2*(b^3*c*d^2 - a*b^2*d^3)*e)*x^2 + 2* 
(a^2*b*c*d^2 - a^3*d^3)*e + 2*(a*b^2*d^2*e^2 + 2*(a*b^2*c*d^2 - a^2*b*d^3) 
*e)*x)*Ei(-(b*d*e*x + a*d*e)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x))*e^(b*e/( 
b*c - a*d)) - (b^3*c^4 - 4*a*b^2*c^3*d + 5*a^2*b*c^2*d^2 - 2*a^3*c*d^3 - ( 
b^3*c^2*d^2 - 2*a*b^2*c*d^3 + a^2*b*d^4 + (b^3*c*d^2 - a*b^2*d^3)*e)*x^2 - 
 (a*b^2*c^2*d - a^2*b*c*d^2)*e - (2*a*b^2*c^2*d^2 - 4*a^2*b*c*d^3 + 2*a^3* 
d^4 + (b^3*c^2*d - a^2*b*d^3)*e)*x)*e^(e/(d*x + c)))/(a^2*b^4*c^4 - 4*a^3* 
b^3*c^3*d + 6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4 + (b^6*c^4 - 4*a*b 
^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^2 + 2*(a*b 
^5*c^4 - 4*a^2*b^4*c^3*d + 6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^4 
)*x)
 

3.5.8.6 Sympy [F]

\[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\int \frac {e^{\frac {e}{c + d x}}}{\left (a + b x\right )^{3}}\, dx \]

input

integrate(exp(e/(d*x+c))/(b*x+a)**3,x)
 

output

Integral(exp(e/(c + d*x))/(a + b*x)**3, x)
 

3.5.8.7 Maxima [F]

\[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\int { \frac {e^{\left (\frac {e}{d x + c}\right )}}{{\left (b x + a\right )}^{3}} \,d x } \]

input

integrate(exp(e/(d*x+c))/(b*x+a)^3,x, algorithm="maxima")
 

output

integrate(e^(e/(d*x + c))/(b*x + a)^3, x)
 

3.5.8.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1733 vs. \(2 (224) = 448\).

Time = 0.34 (sec) , antiderivative size = 1733, normalized size of antiderivative = 7.22 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\text {Too large to display} \]

input

integrate(exp(e/(d*x+c))/(b*x+a)^3,x, algorithm="giac")
 

output

1/2*(2*b^3*c*d*e^4*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a* 
d))*e^(b*e/(b*c - a*d)) - 4*b^3*c^2*d*e^4*Ei(-(b*e - b*c*e/(d*x + c) + a*d 
*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d))/(d*x + c) + 2*b^3*c^3*d*e^4 
*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - 
a*d))/(d*x + c)^2 - 2*a*b^2*d^2*e^4*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d* 
x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d)) + 8*a*b^2*c*d^2*e^4*Ei(-(b*e - b* 
c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d))/(d*x + c 
) - 6*a*b^2*c^2*d^2*e^4*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c 
 - a*d))*e^(b*e/(b*c - a*d))/(d*x + c)^2 - 4*a^2*b*d^3*e^4*Ei(-(b*e - b*c* 
e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d))/(d*x + c) 
+ 6*a^2*b*c*d^3*e^4*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a 
*d))*e^(b*e/(b*c - a*d))/(d*x + c)^2 - 2*a^3*d^4*e^4*Ei(-(b*e - b*c*e/(d*x 
 + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d))/(d*x + c)^2 + b^ 
3*d*e^5*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/ 
(b*c - a*d)) - 2*b^3*c*d*e^5*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c)) 
/(b*c - a*d))*e^(b*e/(b*c - a*d))/(d*x + c) + b^3*c^2*d*e^5*Ei(-(b*e - b*c 
*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d))/(d*x + c) 
^2 + 2*a*b^2*d^2*e^5*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - 
a*d))*e^(b*e/(b*c - a*d))/(d*x + c) - 2*a*b^2*c*d^2*e^5*Ei(-(b*e - b*c*e/( 
d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d))/(d*x + c)^...
 

3.5.8.9 Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx=\int \frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}}{{\left (a+b\,x\right )}^3} \,d x \]

input

int(exp(e/(c + d*x))/(a + b*x)^3,x)
 

output

int(exp(e/(c + d*x))/(a + b*x)^3, x)