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

Optimal. Leaf size=111 \[ -\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \text {Ei}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 c}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \text {Ei}\left (\frac {b \left (\sqrt {d} x+\sqrt {-c}\right )}{\sqrt {d}}\right )}{2 c}+\frac {e^a \text {Ei}(b x)}{c} \]

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2271, 2178} \[ -\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \text {Ei}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 c}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \text {Ei}\left (\frac {b \left (\sqrt {d} x+\sqrt {-c}\right )}{\sqrt {d}}\right )}{2 c}+\frac {e^a \text {Ei}(b x)}{c} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x)/(x*(c + d*x^2)),x]

[Out]

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

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

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]

Rubi steps

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

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Mathematica [C]  time = 0.13, size = 93, normalized size = 0.84 \[ \frac {e^a \left (2 \text {Ei}(b x)-e^{-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \text {Ei}\left (b \left (x-\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )+\text {Ei}\left (b \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )\right )\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 80, normalized size = 0.72 \[ -\frac {{\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} + {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )} - 2 \, {\rm Ei}\left (b x\right ) e^{a}}{2 \, c} \]

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)) - 2*Ei
(b*x)*e^a)/c

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x}\,{d x} \]

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(e^(b*x + a)/((d*x^2 + c)*x), x)

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maple [A]  time = 0.03, size = 112, normalized size = 1.01 \[ -\frac {\Ei \left (1, -b x \right ) {\mathrm e}^{a}}{c}+\frac {\Ei \left (1, \frac {a d +\sqrt {-c d}\, b -\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{\frac {a d +\sqrt {-c d}\, b}{d}}+\Ei \left (1, -\frac {-a d +\sqrt {-c d}\, b +\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{-\frac {-a d +\sqrt {-c d}\, b}{d}}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x}\,{d x} \]

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]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{a+b\,x}}{x\,\left (d\,x^2+c\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + b*x)/(x*(c + d*x^2)),x)

[Out]

int(exp(a + b*x)/(x*(c + d*x^2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{a} \int \frac {e^{b x}}{c x + d x^{3}}\, dx \]

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(exp(b*x)/(c*x + d*x**3), x)

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