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

Optimal. Leaf size=132 \[ \frac{\sqrt{-c} e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d^{3/2}}-\frac{\sqrt{-c} e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 d^{3/2}}+\frac{e^{a+b x}}{b d} \]

[Out]

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

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Rubi [A]  time = 0.23545, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2271, 2194, 2269, 2178} \[ \frac{\sqrt{-c} e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d^{3/2}}-\frac{\sqrt{-c} e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 d^{3/2}}+\frac{e^{a+b x}}{b d} \]

Antiderivative was successfully verified.

[In]

Int[(E^(a + b*x)*x^2)/(c + d*x^2),x]

[Out]

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

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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2269

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[F^(g*(d +
e*x)^n), 1/(a + c*x^2), x], x] /; FreeQ[{F, a, c, d, e, g, n}, x]

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^2}{c+d x^2} \, dx &=\int \left (\frac{e^{a+b x}}{d}-\frac{c e^{a+b x}}{d \left (c+d x^2\right )}\right ) \, dx\\ &=\frac{\int e^{a+b x} \, dx}{d}-\frac{c \int \frac{e^{a+b x}}{c+d x^2} \, dx}{d}\\ &=\frac{e^{a+b x}}{b d}-\frac{c \int \left (\frac{\sqrt{-c} e^{a+b x}}{2 c \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\sqrt{-c} e^{a+b x}}{2 c \left (\sqrt{-c}+\sqrt{d} x\right )}\right ) \, dx}{d}\\ &=\frac{e^{a+b x}}{b d}-\frac{\sqrt{-c} \int \frac{e^{a+b x}}{\sqrt{-c}-\sqrt{d} x} \, dx}{2 d}-\frac{\sqrt{-c} \int \frac{e^{a+b x}}{\sqrt{-c}+\sqrt{d} x} \, dx}{2 d}\\ &=\frac{e^{a+b x}}{b d}+\frac{\sqrt{-c} 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^{3/2}}-\frac{\sqrt{-c} 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^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.122203, size = 120, normalized size = 0.91 \[ \frac{e^a \left (i b \sqrt{c} e^{\frac{i b \sqrt{c}}{\sqrt{d}}} \text{Ei}\left (b \left (x-\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )-i b \sqrt{c} e^{-\frac{i b \sqrt{c}}{\sqrt{d}}} \text{Ei}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+2 \sqrt{d} e^{b x}\right )}{2 b d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.017, size = 660, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]  time = 1.56955, size = 212, normalized size = 1.61 \begin{align*} \frac{\sqrt{-\frac{b^{2} c}{d}}{\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}}{\rm Ei}\left (b x + \sqrt{-\frac{b^{2} c}{d}}\right ) e^{\left (a - \sqrt{-\frac{b^{2} c}{d}}\right )} + 2 \, e^{\left (b x + a\right )}}{2 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} 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^2/(d*x^2+c),x, algorithm="giac")

[Out]

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