Optimal. Leaf size=111 \[ -\frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 c}-\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 c}+\frac{e^a \text{ExpIntegralEi}(b x)}{c} \]
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Rubi [A] time = 0.39909, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 c}-\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 c}+\frac{e^a \text{ExpIntegralEi}(b x)}{c} \]
Antiderivative was successfully verified.
[In] Int[E^(a + b*x)/(x*(c + d*x^2)),x]
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Rubi in Sympy [A] time = 43.1497, size = 94, normalized size = 0.85 \[ \frac{e^{a} \operatorname{Ei}{\left (b x \right )}}{c} - \frac{e^{a - \frac{b \sqrt{- c}}{\sqrt{d}}} \operatorname{Ei}{\left (\frac{b \left (\sqrt{d} x + \sqrt{- c}\right )}{\sqrt{d}} \right )}}{2 c} - \frac{e^{a + \frac{b \sqrt{- c}}{\sqrt{d}}} \operatorname{Ei}{\left (\frac{b \left (\sqrt{d} x - \sqrt{- c}\right )}{\sqrt{d}} \right )}}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(b*x+a)/x/(d*x**2+c),x)
[Out]
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Mathematica [C] time = 0.0944766, size = 93, normalized size = 0.84 \[ \frac{e^a \left (2 \text{ExpIntegralEi}(b x)-e^{-\frac{i b \sqrt{c}}{\sqrt{d}}} \left (e^{\frac{2 i b \sqrt{c}}{\sqrt{d}}} \text{ExpIntegralEi}\left (b \left (x-\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+\text{ExpIntegralEi}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )\right )\right )}{2 c} \]
Antiderivative was successfully verified.
[In] Integrate[E^(a + b*x)/(x*(c + d*x^2)),x]
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Maple [A] time = 0.026, size = 112, normalized size = 1. \[ -{\frac{{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{c}}+{\frac{1}{2\,c} \left ({{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}-d \left ( bx+a \right ) +ad \right ) } \right ) +{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+d \left ( bx+a \right ) -ad \right ) } \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(b*x+a)/x/(d*x^2+c),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(b*x + a)/((d*x^2 + c)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.268467, size = 108, normalized size = 0.97 \[ -\frac{{\rm Ei}\left (b x - \sqrt{-\frac{b^{2} c}{d}}\right ) e^{\left (a + \sqrt{-\frac{b^{2} c}{d}}\right )} +{\rm Ei}\left (b x + \sqrt{-\frac{b^{2} c}{d}}\right ) e^{\left (a - \sqrt{-\frac{b^{2} c}{d}}\right )} - 2 \,{\rm Ei}\left (b x\right ) e^{a}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(b*x + a)/((d*x^2 + c)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ e^{a} \int \frac{e^{b x}}{c x + d x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(b*x+a)/x/(d*x**2+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(b*x + a)/((d*x^2 + c)*x),x, algorithm="giac")
[Out]