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

Optimal. Leaf size=100 \[ \frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d}+\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.128842, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2271, 2178} \[ \frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d}+\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(E^(a + b*x)*x)/(c + d*x^2),x]

[Out]

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

Rule 2271

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_) + (c_)*(x_)^2), x_Symbol] :> Int[ExpandIntegran
d[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] && Intege
rQ[m]

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

Rubi steps

\begin{align*} \int \frac{e^{a+b x} x}{c+d x^2} \, dx &=\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\\ &=-\frac{\int \frac{e^{a+b x}}{\sqrt{-c}-\sqrt{d} x} \, dx}{2 \sqrt{d}}+\frac{\int \frac{e^{a+b x}}{\sqrt{-c}+\sqrt{d} x} \, dx}{2 \sqrt{d}}\\ &=\frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d}+\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d}\\ \end{align*}

Mathematica [C]  time = 0.0542735, size = 83, normalized size = 0.83 \[ \frac{e^{a-\frac{i b \sqrt{c}}{\sqrt{d}}} \left (e^{\frac{2 i b \sqrt{c}}{\sqrt{d}}} \text{Ei}\left (b \left (x-\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+\text{Ei}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)]))/(2*d)

________________________________________________________________________________________

Maple [B]  time = 0.012, size = 323, normalized size = 3.2 \begin{align*}{\frac{1}{{b}^{2}} \left ( -{\frac{b}{2\,d} \left ( \sqrt{-cd}{{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}-d \left ( bx+a \right ) +ad \right ) } \right ) b+\sqrt{-cd}{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+d \left ( bx+a \right ) -ad \right ) } \right ) b+{{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}-d \left ( bx+a \right ) +ad \right ) } \right ) ad-{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+d \left ( bx+a \right ) -ad \right ) } \right ) ad \right ){\frac{1}{\sqrt{-cd}}}}+{\frac{ab}{2} \left ({{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}-d \left ( bx+a \right ) +ad \right ) } \right ) -{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+d \left ( bx+a \right ) -ad \right ) } \right ) \right ){\frac{1}{\sqrt{-cd}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x e^{\left (b x + a\right )}}{b d x^{2} + b c} + \int \frac{{\left (d x^{2} e^{a} - c e^{a}\right )} e^{\left (b x\right )}}{b d^{2} x^{4} + 2 \, b c d x^{2} + b c^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

________________________________________________________________________________________

Fricas [A]  time = 1.57449, size = 144, normalized size = 1.44 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} \int \frac{x e^{b x}}{c + d x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x e^{\left (b x + a\right )}}{d x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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