\(\int \frac {e^{a+b x} x}{c+d x^2} \, dx\) [28]

Optimal result

Integrand size = 18, antiderivative size = 100 \[ \int \frac {e^{a+b x} x}{c+d x^2} \, dx=\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 d}+\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.83 \[ \int \frac {e^{a+b x} x}{c+d x^2} \, dx=\frac {e^{a-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (b \left (-\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )+\operatorname {ExpIntegralEi}\left (b \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )\right )}{2 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

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

\(\Big \downarrow \) 2701

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right )}{2 d}-\frac {{\mathrm e}^{\frac {-b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right )}{2 d}\)	\(100\)
derivativedivides	\(\frac {-\frac {b \left ({\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right ) \sqrt {-c d}\, b +{\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right ) a d +{\mathrm e}^{-\frac {b \sqrt {-c d}-d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right ) \sqrt {-c d}\, b -{\mathrm e}^{-\frac {b \sqrt {-c d}-d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right ) a d \right )}{2 d \sqrt {-c d}}+\frac {a b \left ({\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right )-{\mathrm e}^{-\frac {b \sqrt {-c d}-d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right )\right )}{2 \sqrt {-c d}}}{b^{2}}\)	\(323\)
default	\(\frac {-\frac {b \left ({\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right ) \sqrt {-c d}\, b +{\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right ) a d +{\mathrm e}^{-\frac {b \sqrt {-c d}-d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right ) \sqrt {-c d}\, b -{\mathrm e}^{-\frac {b \sqrt {-c d}-d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right ) a d \right )}{2 d \sqrt {-c d}}+\frac {a b \left ({\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+d a -d \left (b x +a \right )}{d}\right )-{\mathrm e}^{-\frac {b \sqrt {-c d}-d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-d a +d \left (b x +a \right )}{d}\right )\right )}{2 \sqrt {-c d}}}{b^{2}}\)	\(323\)

Input:

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

Output:

-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)- 
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)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.72 \[ \int \frac {e^{a+b x} x}{c+d x^2} \, dx=\frac {{\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} + {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )}}{2 \, d} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

x*e^(b*x + a)/(b*d*x^2 + b*c) + integrate((d*x^2*e^a - c*e^a)*e^(b*x)/(b*d 
^2*x^4 + 2*b*c*d*x^2 + b*c^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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