Optimal. Leaf size=322 \[ \frac {2 \sqrt {\pi } b^2 e^{3/2} (b c-a d) \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}-\frac {b^2 (c+d x)^3 (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^4}-\frac {2 b^2 e (c+d x) (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^4}+\frac {\sqrt {\pi } \sqrt {e} (b c-a d)^3 \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}-\frac {3 b e (b c-a d)^2 \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}+\frac {3 b (c+d x)^2 (b c-a d)^2 e^{\frac {e}{(c+d x)^2}}}{2 d^4}-\frac {(c+d x) (b c-a d)^3 e^{\frac {e}{(c+d x)^2}}}{d^4}-\frac {b^3 e^2 \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4}+\frac {b^3 (c+d x)^4 e^{\frac {e}{(c+d x)^2}}}{4 d^4}+\frac {b^3 e (c+d x)^2 e^{\frac {e}{(c+d x)^2}}}{4 d^4} \]
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Rubi [A] time = 0.34, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ \frac {2 \sqrt {\pi } b^2 e^{3/2} (b c-a d) \text {Erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}-\frac {b^2 (c+d x)^3 (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^4}-\frac {2 b^2 e (c+d x) (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^4}+\frac {\sqrt {\pi } \sqrt {e} (b c-a d)^3 \text {Erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}-\frac {3 b e (b c-a d)^2 \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}+\frac {3 b (c+d x)^2 (b c-a d)^2 e^{\frac {e}{(c+d x)^2}}}{2 d^4}-\frac {(c+d x) (b c-a d)^3 e^{\frac {e}{(c+d x)^2}}}{d^4}-\frac {b^3 e^2 \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4}+\frac {b^3 (c+d x)^4 e^{\frac {e}{(c+d x)^2}}}{4 d^4}+\frac {b^3 e (c+d x)^2 e^{\frac {e}{(c+d x)^2}}}{4 d^4} \]
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)^3 \, dx &=\int \left (\frac {(-b c+a d)^3 e^{\frac {e}{(c+d x)^2}}}{d^3}+\frac {3 b (b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^3}-\frac {3 b^2 (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{d^3}+\frac {b^3 e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{d^3}\right ) \, dx\\ &=\frac {b^3 \int e^{\frac {e}{(c+d x)^2}} (c+d x)^3 \, dx}{d^3}-\frac {\left (3 b^2 (b c-a d)\right ) \int e^{\frac {e}{(c+d x)^2}} (c+d x)^2 \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2\right ) \int e^{\frac {e}{(c+d x)^2}} (c+d x) \, dx}{d^3}-\frac {(b c-a d)^3 \int e^{\frac {e}{(c+d x)^2}} \, dx}{d^3}\\ &=-\frac {(b c-a d)^3 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{d^4}+\frac {b^3 e^{\frac {e}{(c+d x)^2}} (c+d x)^4}{4 d^4}+\frac {\left (b^3 e\right ) \int e^{\frac {e}{(c+d x)^2}} (c+d x) \, dx}{2 d^3}-\frac {\left (2 b^2 (b c-a d) e\right ) \int e^{\frac {e}{(c+d x)^2}} \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2 e\right ) \int \frac {e^{\frac {e}{(c+d x)^2}}}{c+d x} \, dx}{d^3}-\frac {\left (2 (b c-a d)^3 e\right ) \int \frac {e^{\frac {e}{(c+d x)^2}}}{(c+d x)^2} \, dx}{d^3}\\ &=-\frac {(b c-a d)^3 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}-\frac {2 b^2 (b c-a d) e e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{2 d^4}+\frac {b^3 e e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{4 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{d^4}+\frac {b^3 e^{\frac {e}{(c+d x)^2}} (c+d x)^4}{4 d^4}-\frac {3 b (b c-a d)^2 e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}+\frac {\left (2 (b c-a d)^3 e\right ) \operatorname {Subst}\left (\int e^{e x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^4}+\frac {\left (b^3 e^2\right ) \int \frac {e^{\frac {e}{(c+d x)^2}}}{c+d x} \, dx}{2 d^3}-\frac {\left (4 b^2 (b c-a d) e^2\right ) \int \frac {e^{\frac {e}{(c+d x)^2}}}{(c+d x)^2} \, dx}{d^3}\\ &=-\frac {(b c-a d)^3 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}-\frac {2 b^2 (b c-a d) e e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{2 d^4}+\frac {b^3 e e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{4 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{d^4}+\frac {b^3 e^{\frac {e}{(c+d x)^2}} (c+d x)^4}{4 d^4}+\frac {(b c-a d)^3 \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}-\frac {b^3 e^2 \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4}+\frac {\left (4 b^2 (b c-a d) e^2\right ) \operatorname {Subst}\left (\int e^{e x^2} \, dx,x,\frac {1}{c+d x}\right )}{d^4}\\ &=-\frac {(b c-a d)^3 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}-\frac {2 b^2 (b c-a d) e e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{2 d^4}+\frac {b^3 e e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{4 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{d^4}+\frac {b^3 e^{\frac {e}{(c+d x)^2}} (c+d x)^4}{4 d^4}+\frac {(b c-a d)^3 \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}+\frac {2 b^2 (b c-a d) e^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{2 d^4}-\frac {b^3 e^2 \text {Ei}\left (\frac {e}{(c+d x)^2}\right )}{4 d^4}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 243, normalized size = 0.75 \[ \frac {4 \sqrt {\pi } \sqrt {e} (b c-a d) \left (a^2 d^2-2 a b c d+b^2 \left (c^2+2 e\right )\right ) \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )-b e \left (6 a^2 d^2-12 a b c d+b^2 \left (6 c^2+e\right )\right ) \text {Ei}\left (\frac {e}{(c+d x)^2}\right )+d x e^{\frac {e}{(c+d x)^2}} \left (4 a^3 d^3+6 a^2 b d^3 x+4 a b^2 d \left (d^2 x^2+2 e\right )+b^3 \left (-6 c e+d^3 x^3+d e x\right )\right )}{4 d^4}-\frac {c e^{\frac {e}{(c+d x)^2}} \left (-4 a^3 d^3+6 a^2 b c d^2-4 a b^2 d \left (c^2+2 e\right )+b^3 \left (c^3+7 c e\right )\right )}{4 d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 311, normalized size = 0.97 \[ -\frac {4 \, \sqrt {\pi } {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4} + 2 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} e\right )} \sqrt {-\frac {e}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {e}{d^{2}}}}{d x + c}\right ) + {\left (b^{3} e^{2} + 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} e\right )} {\rm Ei}\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - {\left (b^{3} d^{4} x^{4} + 4 \, a b^{2} d^{4} x^{3} - b^{3} c^{4} + 4 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} + 4 \, a^{3} c d^{3} + {\left (6 \, a^{2} b d^{4} + b^{3} d^{2} e\right )} x^{2} - {\left (7 \, b^{3} c^{2} - 8 \, a b^{2} c d\right )} e + 2 \, {\left (2 \, a^{3} d^{4} - {\left (3 \, b^{3} c d - 4 \, a b^{2} d^{2}\right )} e\right )} x\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{4 \, d^{4}} \]
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 \[ \int {\left (b x + a\right )}^{3} e^{\left (\frac {e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 560, normalized size = 1.74 \[ -\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^{3}-\frac {3 \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} b c}{d}+\frac {3 \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^{2} c^{2}}{d^{2}}-\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^{3} c^{3}}{d^{3}}+\frac {3 \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^{2} b}{d}-\frac {6 \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^{2} c}{d^{2}}+\frac {3 \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^{3} c^{2}}{d^{3}}+\frac {3 \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 ) a \,b^{2}}{d^{2}}-\frac {3 \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^{3} c}{d^{3}}+\frac {\left (\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 ) e}{2}-\frac {\left (d x +c \right )^{4} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{4}\right ) b^{3}}{d^{3}}}{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 \[ \frac {{\left (b^{3} d^{3} x^{4} + 4 \, a b^{2} d^{3} x^{3} + {\left (6 \, a^{2} b d^{3} + b^{3} d e\right )} x^{2} + 2 \, {\left (2 \, a^{3} d^{3} - 3 \, b^{3} c e + 4 \, a b^{2} d e\right )} x\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{4 \, d^{3}} + \int \frac {{\left (3 \, b^{3} c^{4} e - 4 \, a b^{2} c^{3} d e - {\left (12 \, a b^{2} c d^{3} e - 6 \, a^{2} b d^{4} e - {\left (6 \, c^{2} d^{2} e + d^{2} e^{2}\right )} b^{3}\right )} x^{2} + 2 \, {\left (2 \, a^{3} d^{4} e - 2 \, {\left (3 \, c^{2} d^{2} e - 2 \, d^{2} e^{2}\right )} a b^{2} + {\left (4 \, c^{3} d e - 3 \, c d e^{2}\right )} b^{3}\right )} x\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{2 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}}\,{d x} \]
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 \[ \int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,{\left (a+b\,x\right )}^3 \,d x \]
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 \[ \int \left (a + b x\right )^{3} e^{\frac {e}{c^{2} + 2 c d x + d^{2} x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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