Optimal. Leaf size=111 \[ \frac {\sqrt {\pi } \sqrt {e} (b c-a d) \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^2}-\frac {(c+d x) (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^2}-\frac {b e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^2}+\frac {b (c+d x)^2 e^{\frac {e}{(c+d x)^2}}}{2 d^2} \]
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Rubi [A] time = 0.13, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2226, 2206, 2211, 2204, 2214, 2210} \[ \frac {\sqrt {\pi } \sqrt {e} (b c-a d) \text {Erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^2}-\frac {(c+d x) (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^2}-\frac {b e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^2}+\frac {b (c+d x)^2 e^{\frac {e}{(c+d x)^2}}}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2206
Rule 2210
Rule 2211
Rule 2214
Rule 2226
Rubi steps
\begin {align*} \int e^{\frac {e}{(c+d x)^2}} (a+b x) \, dx &=\int \left (\frac {(-b c+a d) e^{\frac {e}{(c+d x)^2}}}{d}+\frac {b e^{\frac {e}{(c+d x)^2}} (c+d x)}{d}\right ) \, dx\\ &=\frac {b \int e^{\frac {e}{(c+d x)^2}} (c+d x) \, dx}{d}+\frac {(-b c+a d) \int e^{\frac {e}{(c+d x)^2}} \, dx}{d}\\ &=-\frac {(b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^2}+\frac {b e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{2 d^2}+\frac {(b e) \int \frac {e^{\frac {e}{(c+d x)^2}}}{c+d x} \, dx}{d}+\frac {(2 (-b c+a d) e) \int \frac {e^{\frac {e}{(c+d x)^2}}}{(c+d x)^2} \, dx}{d}\\ &=-\frac {(b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^2}+\frac {b e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{2 d^2}-\frac {b e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^2}+\frac {(2 (b c-a d) e) \operatorname {Subst}\left (\int e^{e x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=-\frac {(b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^2}+\frac {b e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{2 d^2}+\frac {(b c-a d) \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^2}-\frac {b e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 85, normalized size = 0.77 \[ -\frac {2 \sqrt {\pi } \sqrt {e} (a d-b c) \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )+(c+d x) e^{\frac {e}{(c+d x)^2}} (-2 a d+b c-b d x)+b e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 122, normalized size = 1.10 \[ -\frac {b e {\rm Ei}\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, \sqrt {\pi } {\left (b c d - a d^{2}\right )} \sqrt {-\frac {e}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {e}{d^{2}}}}{d x + c}\right ) - {\left (b d^{2} x^{2} + 2 \, a d^{2} x - b c^{2} + 2 \, a c d\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{2 \, d^{2}} \]
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 \[ \int {\left (b x + a\right )} e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 140, normalized size = 1.26 \[ -\frac {\left (\frac {\sqrt {\pi }\, e \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}\right ) a -\frac {\left (\frac {\sqrt {\pi }\, e \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}\right ) b c}{d}+\frac {\left (-\frac {e \Ei \left (1, -\frac {e}{\left (d x +c \right )^{2}}\right )}{2}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{2}\right ) b}{d}}{d} \]
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 \[ \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )} + \int \frac {{\left (b d e x^{2} + 2 \, a d e x\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} \]
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 \[ \int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,\left (a+b\,x\right ) \,d x \]
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 \[ \int \left (a + b x\right ) e^{\frac {e}{c^{2} + 2 c d x + d^{2} x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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