Optimal. Leaf size=215 \[ -\frac {\sqrt {\pi } \sqrt {e} (b c-a d)^2 \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^3}+\frac {b e (b c-a d) \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{d^3}-\frac {b (c+d x)^2 (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^3}+\frac {(c+d x) (b c-a d)^2 e^{\frac {e}{(c+d x)^2}}}{d^3}-\frac {2 \sqrt {\pi } b^2 e^{3/2} \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{3 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{(c+d x)^2}}}{3 d^3}+\frac {2 b^2 e (c+d x) e^{\frac {e}{(c+d x)^2}}}{3 d^3} \]
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Rubi [A] time = 0.23, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2226, 2206, 2211, 2204, 2214, 2210} \[ -\frac {\sqrt {\pi } \sqrt {e} (b c-a d)^2 \text {Erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^3}+\frac {b e (b c-a d) \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{d^3}-\frac {b (c+d x)^2 (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^3}+\frac {(c+d x) (b c-a d)^2 e^{\frac {e}{(c+d x)^2}}}{d^3}-\frac {2 \sqrt {\pi } b^2 e^{3/2} \text {Erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{3 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{(c+d x)^2}}}{3 d^3}+\frac {2 b^2 e (c+d x) e^{\frac {e}{(c+d x)^2}}}{3 d^3} \]
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)^2 \, dx &=\int \left (\frac {(-b c+a d)^2 e^{\frac {e}{(c+d x)^2}}}{d^2}-\frac {2 b (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^2}+\frac {b^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{d^2}\right ) \, dx\\ &=\frac {b^2 \int e^{\frac {e}{(c+d x)^2}} (c+d x)^2 \, dx}{d^2}-\frac {(2 b (b c-a d)) \int e^{\frac {e}{(c+d x)^2}} (c+d x) \, dx}{d^2}+\frac {(b c-a d)^2 \int e^{\frac {e}{(c+d x)^2}} \, dx}{d^2}\\ &=\frac {(b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^3}-\frac {b (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{d^3}+\frac {b^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{3 d^3}+\frac {\left (2 b^2 e\right ) \int e^{\frac {e}{(c+d x)^2}} \, dx}{3 d^2}-\frac {(2 b (b c-a d) e) \int \frac {e^{\frac {e}{(c+d x)^2}}}{c+d x} \, dx}{d^2}+\frac {\left (2 (b c-a d)^2 e\right ) \int \frac {e^{\frac {e}{(c+d x)^2}}}{(c+d x)^2} \, dx}{d^2}\\ &=\frac {(b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^3}+\frac {2 b^2 e e^{\frac {e}{(c+d x)^2}} (c+d x)}{3 d^3}-\frac {b (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{d^3}+\frac {b^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{3 d^3}+\frac {b (b c-a d) e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{d^3}-\frac {\left (2 (b c-a d)^2 e\right ) \operatorname {Subst}\left (\int e^{e x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^3}+\frac {\left (4 b^2 e^2\right ) \int \frac {e^{\frac {e}{(c+d x)^2}}}{(c+d x)^2} \, dx}{3 d^2}\\ &=\frac {(b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^3}+\frac {2 b^2 e e^{\frac {e}{(c+d x)^2}} (c+d x)}{3 d^3}-\frac {b (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{d^3}+\frac {b^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^3}+\frac {b (b c-a d) e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{d^3}-\frac {\left (4 b^2 e^2\right ) \operatorname {Subst}\left (\int e^{e x^2} \, dx,x,\frac {1}{c+d x}\right )}{3 d^3}\\ &=\frac {(b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^3}+\frac {2 b^2 e e^{\frac {e}{(c+d x)^2}} (c+d x)}{3 d^3}-\frac {b (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{d^3}+\frac {b^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^3}-\frac {2 b^2 e^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{3 d^3}+\frac {b (b c-a d) e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 176, normalized size = 0.82 \[ \frac {-\sqrt {\pi } \sqrt {e} \left (3 a^2 d^2-6 a b c d+b^2 \left (3 c^2+2 e\right )\right ) \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )+d x e^{\frac {e}{(c+d x)^2}} \left (3 a^2 d^2+3 a b d^2 x+b^2 \left (d^2 x^2+2 e\right )\right )+3 b e (b c-a d) \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{3 d^3}+\frac {c e^{\frac {e}{(c+d x)^2}} \left (3 a^2 d^2-3 a b c d+b^2 \left (c^2+2 e\right )\right )}{3 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 196, normalized size = 0.91 \[ \frac {3 \, {\left (b^{2} c - a b d\right )} e {\rm Ei}\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \sqrt {\pi } {\left (3 \, b^{2} c^{2} d - 6 \, a b c d^{2} + 3 \, a^{2} d^{3} + 2 \, b^{2} d e\right )} \sqrt {-\frac {e}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {e}{d^{2}}}}{d x + c}\right ) + {\left (b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2} + 2 \, b^{2} c e + {\left (3 \, a^{2} d^{3} + 2 \, b^{2} d e\right )} x\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{3 \, d^{3}} \]
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 \[ \int {\left (b x + a\right )}^{2} e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 313, normalized size = 1.46 \[ -\frac {\left (\frac {\sqrt {\pi }\, e \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}\right ) a^{2}-\frac {2 \left (\frac {\sqrt {\pi }\, e \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}\right ) a b c}{d}+\frac {\left (\frac {\sqrt {\pi }\, e \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}\right ) b^{2} c^{2}}{d^{2}}+\frac {2 \left (-\frac {e \Ei \left (1, -\frac {e}{\left (d x +c \right )^{2}}\right )}{2}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{2}\right ) a b}{d}-\frac {2 \left (-\frac {e \Ei \left (1, -\frac {e}{\left (d x +c \right )^{2}}\right )}{2}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{2}\right ) b^{2} c}{d^{2}}+\frac {\left (\frac {2 \left (\frac {\sqrt {\pi }\, e \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}\right ) e}{3}-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{3}\right ) b^{2}}{d^{2}}}{d} \]
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 \[ \frac {{\left (b^{2} d^{2} x^{3} + 3 \, a b d^{2} x^{2} + {\left (3 \, a^{2} d^{2} + 2 \, b^{2} e\right )} x\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{3 \, d^{2}} + \int -\frac {2 \, {\left (b^{2} c^{3} e + 3 \, {\left (b^{2} c d^{2} e - a b d^{3} e\right )} x^{2} - {\left (3 \, a^{2} d^{3} e - {\left (3 \, c^{2} d e - 2 \, d e^{2}\right )} b^{2}\right )} x\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{3 \, {\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,{\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int \left (a + b x\right )^{2} e^{\frac {e}{c^{2} + 2 c d x + d^{2} x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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