Optimal. Leaf size=194 \[ \frac {(b c-a d)^3 \text {erf}(a+b x)}{3 b^3 d}+\frac {d (b c-a d) \text {erf}(a+b x)}{2 b^3}-\frac {e^{-(a+b x)^2} (b c-a d)^2}{\sqrt {\pi } b^3}-\frac {d e^{-(a+b x)^2} (a+b x) (b c-a d)}{\sqrt {\pi } b^3}-\frac {d^2 e^{-(a+b x)^2} (a+b x)^2}{3 \sqrt {\pi } b^3}-\frac {d^2 e^{-(a+b x)^2}}{3 \sqrt {\pi } b^3}+\frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d} \]
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Rubi [A] time = 0.18, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6362, 2226, 2205, 2209, 2212} \[ \frac {(b c-a d)^3 \text {Erf}(a+b x)}{3 b^3 d}+\frac {d (b c-a d) \text {Erf}(a+b x)}{2 b^3}-\frac {e^{-(a+b x)^2} (b c-a d)^2}{\sqrt {\pi } b^3}-\frac {d e^{-(a+b x)^2} (a+b x) (b c-a d)}{\sqrt {\pi } b^3}-\frac {d^2 e^{-(a+b x)^2} (a+b x)^2}{3 \sqrt {\pi } b^3}-\frac {d^2 e^{-(a+b x)^2}}{3 \sqrt {\pi } b^3}+\frac {(c+d x)^3 \text {Erfc}(a+b x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2209
Rule 2212
Rule 2226
Rule 6362
Rubi steps
\begin {align*} \int (c+d x)^2 \text {erfc}(a+b x) \, dx &=\frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d}+\frac {(2 b) \int e^{-(a+b x)^2} (c+d x)^3 \, dx}{3 d \sqrt {\pi }}\\ &=\frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d}+\frac {(2 b) \int \left (\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^3}+\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{b^3}+\frac {3 d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^3}+\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{b^3}\right ) \, dx}{3 d \sqrt {\pi }}\\ &=\frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d}+\frac {\left (2 d^2\right ) \int e^{-(a+b x)^2} (a+b x)^3 \, dx}{3 b^2 \sqrt {\pi }}+\frac {(2 d (b c-a d)) \int e^{-(a+b x)^2} (a+b x)^2 \, dx}{b^2 \sqrt {\pi }}+\frac {\left (2 (b c-a d)^2\right ) \int e^{-(a+b x)^2} (a+b x) \, dx}{b^2 \sqrt {\pi }}+\frac {\left (2 (b c-a d)^3\right ) \int e^{-(a+b x)^2} \, dx}{3 b^2 d \sqrt {\pi }}\\ &=-\frac {(b c-a d)^2 e^{-(a+b x)^2}}{b^3 \sqrt {\pi }}-\frac {d (b c-a d) e^{-(a+b x)^2} (a+b x)}{b^3 \sqrt {\pi }}-\frac {d^2 e^{-(a+b x)^2} (a+b x)^2}{3 b^3 \sqrt {\pi }}+\frac {(b c-a d)^3 \text {erf}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d}+\frac {\left (2 d^2\right ) \int e^{-(a+b x)^2} (a+b x) \, dx}{3 b^2 \sqrt {\pi }}+\frac {(d (b c-a d)) \int e^{-(a+b x)^2} \, dx}{b^2 \sqrt {\pi }}\\ &=-\frac {d^2 e^{-(a+b x)^2}}{3 b^3 \sqrt {\pi }}-\frac {(b c-a d)^2 e^{-(a+b x)^2}}{b^3 \sqrt {\pi }}-\frac {d (b c-a d) e^{-(a+b x)^2} (a+b x)}{b^3 \sqrt {\pi }}-\frac {d^2 e^{-(a+b x)^2} (a+b x)^2}{3 b^3 \sqrt {\pi }}+\frac {d (b c-a d) \text {erf}(a+b x)}{2 b^3}+\frac {(b c-a d)^3 \text {erf}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \text {erfc}(a+b x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 159, normalized size = 0.82 \[ \frac {\frac {2 e^{-(a+b x)^2} \left (-\left (a^2+1\right ) d^2+\sqrt {\pi } b^3 x e^{(a+b x)^2} \left (3 c^2+3 c d x+d^2 x^2\right ) \text {erfc}(a+b x)+a b d (3 c+d x)-b^2 \left (3 c^2+3 c d x+d^2 x^2\right )\right )}{\sqrt {\pi }}-\left (2 a^3 d^2-6 a^2 b c d+3 a \left (2 b^2 c^2+d^2\right )-3 b c d\right ) \text {erf}(a+b x)}{6 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 197, normalized size = 1.02 \[ \frac {2 \, \pi b^{3} d^{2} x^{3} + 6 \, \pi b^{3} c d x^{2} + 6 \, \pi b^{3} c^{2} x - 2 \, \sqrt {\pi } {\left (b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} - 3 \, a b c d + {\left (a^{2} + 1\right )} d^{2} + {\left (3 \, b^{2} c d - a b d^{2}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} - {\left (2 \, \pi b^{3} d^{2} x^{3} + 6 \, \pi b^{3} c d x^{2} + 6 \, \pi b^{3} c^{2} x + \pi {\left (6 \, a b^{2} c^{2} - 3 \, {\left (2 \, a^{2} + 1\right )} b c d + {\left (2 \, a^{3} + 3 \, a\right )} d^{2}\right )}\right )} \operatorname {erf}\left (b x + a\right )}{6 \, \pi b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.88, size = 280, normalized size = 1.44 \[ \frac {1}{3} \, d^{2} x^{3} + c d x^{2} - {\left (x \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }}\right )} c^{2} - \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {erf}\left (b x + a\right ) + \frac {\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b}\right )} c d - \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } {\left (2 \, a^{3} + 3 \, a\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {2 \, {\left (b^{2} {\left (x + \frac {a}{b}\right )}^{2} - 3 \, a b {\left (x + \frac {a}{b}\right )} + 3 \, a^{2} + 1\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b^{2}}\right )} d^{2} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 428, normalized size = 2.21 \[ \frac {\frac {d^{2} \mathrm {erfc}\left (b x +a \right ) \left (b x +a \right )^{3}}{3 b^{2}}-\frac {d^{2} \mathrm {erfc}\left (b x +a \right ) \left (b x +a \right )^{2} a}{b^{2}}+\frac {d \,\mathrm {erfc}\left (b x +a \right ) \left (b x +a \right )^{2} c}{b}+\frac {d^{2} \mathrm {erfc}\left (b x +a \right ) \left (b x +a \right ) a^{2}}{b^{2}}-\frac {2 d \,\mathrm {erfc}\left (b x +a \right ) \left (b x +a \right ) a c}{b}+\mathrm {erfc}\left (b x +a \right ) \left (b x +a \right ) c^{2}-\frac {d^{2} \mathrm {erfc}\left (b x +a \right ) a^{3}}{3 b^{2}}+\frac {d \,\mathrm {erfc}\left (b x +a \right ) a^{2} c}{b}-\mathrm {erfc}\left (b x +a \right ) a \,c^{2}+\frac {b \,\mathrm {erfc}\left (b x +a \right ) c^{3}}{3 d}+\frac {\frac {2 d^{3} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )}{3}+\frac {b^{3} c^{3} \sqrt {\pi }\, \erf \left (b x +a \right )}{3}-\frac {a^{3} d^{3} \sqrt {\pi }\, \erf \left (b x +a \right )}{3}-a^{2} d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}-2 a \,d^{3} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )-b^{2} c^{2} d \,{\mathrm e}^{-\left (b x +a \right )^{2}}+2 b c \,d^{2} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )-a \,b^{2} c^{2} d \sqrt {\pi }\, \erf \left (b x +a \right )+a^{2} b c \,d^{2} \sqrt {\pi }\, \erf \left (b x +a \right )+2 a b c \,d^{2} {\mathrm e}^{-\left (b x +a \right )^{2}}}{b^{2} \sqrt {\pi }\, d}}{b} \]
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 \[ \int {\left (d x + c\right )}^{2} \operatorname {erfc}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 220, normalized size = 1.13 \[ \frac {d^2\,x^3\,\mathrm {erfc}\left (a+b\,x\right )}{3}-\frac {{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (\frac {b^2\,c^2}{\sqrt {\pi }}-\frac {a\,d\,b\,c}{\sqrt {\pi }}+\frac {a^2\,d^2+d^2}{3\,\sqrt {\pi }}\right )}{b^3}+\frac {\mathrm {erfc}\left (a+b\,x\right )\,\left (\frac {a\,d^2}{2}-b\,\left (c\,d\,a^2+\frac {c\,d}{2}\right )+\frac {a^3\,d^2}{3}+a\,b^2\,c^2\right )}{b^3}+c^2\,x\,\mathrm {erfc}\left (a+b\,x\right )+c\,d\,x^2\,\mathrm {erfc}\left (a+b\,x\right )+\frac {x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (a\,d^2-3\,b\,c\,d\right )}{3\,b^2\,\sqrt {\pi }}-\frac {d^2\,x^2\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{3\,b\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.78, size = 398, normalized size = 2.05 \[ \begin {cases} \frac {a^{3} d^{2} \operatorname {erfc}{\left (a + b x \right )}}{3 b^{3}} - \frac {a^{2} c d \operatorname {erfc}{\left (a + b x \right )}}{b^{2}} - \frac {a^{2} d^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b^{3}} + \frac {a c^{2} \operatorname {erfc}{\left (a + b x \right )}}{b} + \frac {a c d e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b^{2}} + \frac {a d^{2} x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b^{2}} + \frac {a d^{2} \operatorname {erfc}{\left (a + b x \right )}}{2 b^{3}} + c^{2} x \operatorname {erfc}{\left (a + b x \right )} + c d x^{2} \operatorname {erfc}{\left (a + b x \right )} + \frac {d^{2} x^{3} \operatorname {erfc}{\left (a + b x \right )}}{3} - \frac {c^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} - \frac {c d x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} - \frac {d^{2} x^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b} - \frac {c d \operatorname {erfc}{\left (a + b x \right )}}{2 b^{2}} - \frac {d^{2} e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{3 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \operatorname {erfc}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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