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 = 0.24, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2178
Rule 2194
Rule 2269
Rule 2271
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*}
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Mathematica [C] time = 0.13, 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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fricas [A] time = 0.41, size = 106, normalized size = 0.80 \[ \frac {\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )} + 2 \, e^{\left (b x + a\right )}}{2 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} e^{\left (b x + a\right )}}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 660, normalized size = 5.00 \[ \frac {-\frac {\left (\Ei \left (1, \frac {a d +\sqrt {-c d}\, b -\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{\frac {a d +\sqrt {-c d}\, b}{d}}-\Ei \left (1, -\frac {-a d +\sqrt {-c d}\, b +\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{-\frac {-a d +\sqrt {-c d}\, b}{d}}\right ) a^{2} b}{2 \sqrt {-c d}}+\frac {b^{2} {\mathrm e}^{b x +a}}{d}+\frac {\left (a d \Ei \left (1, \frac {a d +\sqrt {-c d}\, b -\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{\frac {a d +\sqrt {-c d}\, b}{d}}-a d \Ei \left (1, -\frac {-a d +\sqrt {-c d}\, b +\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{-\frac {-a d +\sqrt {-c d}\, b}{d}}+\sqrt {-c d}\, b \Ei \left (1, \frac {a d +\sqrt {-c d}\, b -\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{\frac {a d +\sqrt {-c d}\, b}{d}}+\sqrt {-c d}\, b \Ei \left (1, -\frac {-a d +\sqrt {-c d}\, b +\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{-\frac {-a d +\sqrt {-c d}\, b}{d}}\right ) a b}{\sqrt {-c d}\, d}-\frac {\left (a^{2} d \Ei \left (1, \frac {a d +\sqrt {-c d}\, b -\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{\frac {a d +\sqrt {-c d}\, b}{d}}-a^{2} d \Ei \left (1, -\frac {-a d +\sqrt {-c d}\, b +\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{-\frac {-a d +\sqrt {-c d}\, b}{d}}-b^{2} c \Ei \left (1, \frac {a d +\sqrt {-c d}\, b -\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{\frac {a d +\sqrt {-c d}\, b}{d}}+b^{2} c \Ei \left (1, -\frac {-a d +\sqrt {-c d}\, b +\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{-\frac {-a d +\sqrt {-c d}\, b}{d}}+2 \sqrt {-c d}\, a b \Ei \left (1, \frac {a d +\sqrt {-c d}\, b -\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{\frac {a d +\sqrt {-c d}\, b}{d}}+2 \sqrt {-c d}\, a b \Ei \left (1, -\frac {-a d +\sqrt {-c d}\, b +\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{-\frac {-a d +\sqrt {-c d}\, b}{d}}\right ) b}{2 \sqrt {-c d}\, d}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x^{2} e^{\left (b x + a\right )}}{b d x^{2} + b c} - 2 \, c \int \frac {x e^{\left (b x + a\right )}}{b d^{2} x^{4} + 2 \, b c d x^{2} + b c^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\mathrm {e}}^{a+b\,x}}{d\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{a} \int \frac {x^{2} e^{b x}}{c + d x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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