Optimal. Leaf size=215 \[ \frac {(b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^3}+\frac {2 b^2 e e^{\frac {e}{(c+d x)^2}} (c+d x)}{3 d^3}-\frac {b (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{d^3}+\frac {b^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^3}-\frac {2 b^2 e^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{3 d^3}+\frac {b (b c-a d) e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{d^3} \]
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Rubi [A]
time = 0.17, 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 = {2258, 2237,
2242, 2235, 2245, 2241} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2237
Rule 2241
Rule 2242
Rule 2245
Rule 2258
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 ) \text {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 ) \text {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 = 0.15, size = 176, normalized size = 0.82 \begin {gather*} \frac {c \left (-3 a b c d+3 a^2 d^2+b^2 \left (c^2+2 e\right )\right ) e^{\frac {e}{(c+d x)^2}}}{3 d^3}+\frac {d e^{\frac {e}{(c+d x)^2}} x \left (3 a^2 d^2+3 a b d^2 x+b^2 \left (2 e+d^2 x^2\right )\right )-\sqrt {e} \left (-6 a b c d+3 a^2 d^2+b^2 \left (3 c^2+2 e\right )\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )+3 b (b c-a d) e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 313, normalized size = 1.46
method | result | size |
derivativedivides | \(-\frac {a^{2} \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )+\frac {b^{2} \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{3}+\frac {2 e \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{3}\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{d^{2}}+\frac {2 b a \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \expIntegral \left (1, -\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d}-\frac {2 b^{2} c \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \expIntegral \left (1, -\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d^{2}}-\frac {2 b c a \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{d}}{d}\) | \(313\) |
default | \(-\frac {a^{2} \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )+\frac {b^{2} \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{3}+\frac {2 e \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{3}\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{d^{2}}+\frac {2 b a \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \expIntegral \left (1, -\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d}-\frac {2 b^{2} c \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \expIntegral \left (1, -\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d^{2}}-\frac {2 b c a \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{d}}{d}\) | \(313\) |
risch | \(a^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x +\frac {a^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{d}-\frac {a^{2} e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{d \sqrt {-e}}+\frac {b^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x^{3}}{3}+\frac {b^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c^{3}}{3 d^{3}}+\frac {2 b^{2} e \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x}{3 d^{2}}+\frac {2 b^{2} e \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{3 d^{3}}-\frac {2 b^{2} e^{2} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{3 d^{3} \sqrt {-e}}-\frac {c^{2} b^{2} e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{d^{3} \sqrt {-e}}+a b \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x^{2}-\frac {a b \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c^{2}}{d^{2}}+\frac {a b e \expIntegral \left (1, -\frac {e}{\left (d x +c \right )^{2}}\right )}{d^{2}}-\frac {b^{2} c e \expIntegral \left (1, -\frac {e}{\left (d x +c \right )^{2}}\right )}{d^{3}}+\frac {2 a b c e \sqrt {\pi }\, \erf \left (\frac {\sqrt {-e}}{d x +c}\right )}{d^{2} \sqrt {-e}}\) | \(320\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 199, normalized size = 0.93 \begin {gather*} -\frac {\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 {1}{d^{2}}} \operatorname {erfi}\left (\frac {d \sqrt {\frac {1}{d^{2}}} e^{\frac {1}{2}}}{d x + c}\right ) e^{\frac {1}{2}} - 3 \, {\left (b^{2} c - a b d\right )} {\rm Ei}\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) e - {\left (b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + 3 \, a^{2} d^{3} x + b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2} + 2 \, {\left (b^{2} d x + b^{2} c\right )} e\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{3 \, d^{3}} \end {gather*}
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 \begin {gather*} \int \left (a + b x\right )^{2} e^{\frac {e}{c^{2} + 2 c d x + d^{2} x^{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,{\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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