\(\int \frac {e^{a+b x}}{x^2 (c+d x^2)} \, dx\) [31]

Optimal result

Integrand size = 20, antiderivative size = 145 \[ \int \frac {e^{a+b x}}{x^2 \left (c+d x^2\right )} \, dx=-\frac {e^{a+b x}}{c x}+\frac {b e^a \operatorname {ExpIntegralEi}(b x)}{c}+\frac {\sqrt {d} e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 (-c)^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.83 \[ \int \frac {e^{a+b x}}{x^2 \left (c+d x^2\right )} \, dx=\frac {-\frac {e^{a+b x}}{x}+b e^a \operatorname {ExpIntegralEi}(b x)}{c}+\frac {i \sqrt {d} e^{a-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (b \left (-\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (b \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )\right )}{2 c^{3/2}} \] Input:

Integrate[E^(a + b*x)/(x^2*(c + d*x^2)),x]
 

Output:

(-(E^(a + b*x)/x) + b*E^a*ExpIntegralEi[b*x])/c + ((I/2)*Sqrt[d]*E^(a - (I 
*b*Sqrt[c])/Sqrt[d])*(E^(((2*I)*b*Sqrt[c])/Sqrt[d])*ExpIntegralEi[b*(((-I) 
*Sqrt[c])/Sqrt[d] + x)] - ExpIntegralEi[b*((I*Sqrt[c])/Sqrt[d] + x)]))/c^( 
3/2)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2701, 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.

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

\(\Big \downarrow \) 2701

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

\(\Big \downarrow \) 2009

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

Input:

Int[E^(a + b*x)/(x^2*(c + d*x^2)),x]
 

Output:

-(E^(a + b*x)/(c*x)) + (b*E^a*ExpIntegralEi[b*x])/c + (Sqrt[d]*E^(a + (b*S 
qrt[-c])/Sqrt[d])*ExpIntegralEi[-((b*(Sqrt[-c] - Sqrt[d]*x))/Sqrt[d])])/(2 
*(-c)^(3/2)) - (Sqrt[d]*E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[(b*(Sqr 
t[-c] + Sqrt[d]*x))/Sqrt[d]])/(2*(-c)^(3/2))
 

Defintions of rubi rules used

rule 2009

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

rule 2701

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_) + (c_)*(x_)^ 
2), x_Symbol] :> Int[ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + c*x^2), x] 
, x] /; FreeQ[{F, a, c, d, e, g, n}, x] && PolynomialQ[u, x] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(b \left (-\frac {{\mathrm e}^{b x +a}}{c b x}+\frac {d \left ({\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right )-{\mathrm e}^{-\frac {b \sqrt {-c d}-d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right )\right )}{2 c b \sqrt {-c d}}-\frac {{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right )}{c}\right )\)	\(142\)
default	\(b \left (-\frac {{\mathrm e}^{b x +a}}{c b x}+\frac {d \left ({\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right )-{\mathrm e}^{-\frac {b \sqrt {-c d}-d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right )\right )}{2 c b \sqrt {-c d}}-\frac {{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right )}{c}\right )\)	\(142\)
risch	\(-\frac {{\mathrm e}^{b x +a}}{c x}+\frac {d \,{\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right )}{2 c \sqrt {-c d}}-\frac {d \,{\mathrm e}^{\frac {-b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right )}{2 c \sqrt {-c d}}-\frac {b \,{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right )}{c}\)	\(142\)

Input:

int(exp(b*x+a)/x^2/(d*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88 \[ \int \frac {e^{a+b x}}{x^2 \left (c+d x^2\right )} \, dx=\frac {2 \, b^{2} c x {\rm Ei}\left (b x\right ) e^{a} + \sqrt {-\frac {b^{2} c}{d}} d x {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} - \sqrt {-\frac {b^{2} c}{d}} d x {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )} - 2 \, b c e^{\left (b x + a\right )}}{2 \, b c^{2} x} \] Input:

integrate(exp(b*x+a)/x^2/(d*x^2+c),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {e^{a+b x}}{x^2 \left (c+d x^2\right )} \, dx=e^{a} \int \frac {e^{b x}}{c x^{2} + d x^{4}}\, dx \] Input:

integrate(exp(b*x+a)/x**2/(d*x**2+c),x)
 

Output:

exp(a)*Integral(exp(b*x)/(c*x**2 + d*x**4), x)
 

Maxima [F]

\[ \int \frac {e^{a+b x}}{x^2 \left (c+d x^2\right )} \, dx=\int { \frac {e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x^{2}} \,d x } \] Input:

integrate(exp(b*x+a)/x^2/(d*x^2+c),x, algorithm="maxima")
 

Output:

integrate(e^(b*x + a)/((d*x^2 + c)*x^2), x)
 

Giac [F]

\[ \int \frac {e^{a+b x}}{x^2 \left (c+d x^2\right )} \, dx=\int { \frac {e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x^{2}} \,d x } \] Input:

integrate(exp(b*x+a)/x^2/(d*x^2+c),x, algorithm="giac")
 

Output:

integrate(e^(b*x + a)/((d*x^2 + c)*x^2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{a+b x}}{x^2 \left (c+d x^2\right )} \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}}{x^2\,\left (d\,x^2+c\right )} \,d x \] Input:

int(exp(a + b*x)/(x^2*(c + d*x^2)),x)
 

Output:

int(exp(a + b*x)/(x^2*(c + d*x^2)), x)
 

Reduce [F]

\[ \int \frac {e^{a+b x}}{x^2 \left (c+d x^2\right )} \, dx=e^{a} \left (\int \frac {e^{b x}}{d \,x^{4}+c \,x^{2}}d x \right ) \] Input:

int(exp(b*x+a)/x^2/(d*x^2+c),x)
 

Output:

e**a*int(e**(b*x)/(c*x**2 + d*x**4),x)