Optimal. Leaf size=132 \[ \frac{\sqrt{-c} e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d^{3/2}}-\frac{\sqrt{-c} e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 d^{3/2}}+\frac{e^{a+b x}}{b d} \]
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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.
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Rule 2271
Rule 2194
Rule 2269
Rule 2178
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.
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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.
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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.
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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.
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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.
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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.
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