Optimal. Leaf size=186 \[ -\frac {(b c-a d)^3 \text {erfi}(a+b x)}{3 b^3 d}+\frac {d (b c-a d) \text {erfi}(a+b x)}{2 b^3}-\frac {e^{(a+b x)^2} (b c-a d)^2}{\sqrt {\pi } b^3}-\frac {d e^{(a+b x)^2} (a+b x) (b c-a d)}{\sqrt {\pi } b^3}-\frac {d^2 e^{(a+b x)^2} (a+b x)^2}{3 \sqrt {\pi } b^3}+\frac {d^2 e^{(a+b x)^2}}{3 \sqrt {\pi } b^3}+\frac {(c+d x)^3 \text {erfi}(a+b x)}{3 d} \]
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Rubi [A] time = 0.17, antiderivative size = 186, 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 = {6363, 2226, 2204, 2209, 2212} \[ -\frac {(b c-a d)^3 \text {Erfi}(a+b x)}{3 b^3 d}+\frac {d (b c-a d) \text {Erfi}(a+b x)}{2 b^3}-\frac {e^{(a+b x)^2} (b c-a d)^2}{\sqrt {\pi } b^3}-\frac {d e^{(a+b x)^2} (a+b x) (b c-a d)}{\sqrt {\pi } b^3}-\frac {d^2 e^{(a+b x)^2} (a+b x)^2}{3 \sqrt {\pi } b^3}+\frac {d^2 e^{(a+b x)^2}}{3 \sqrt {\pi } b^3}+\frac {(c+d x)^3 \text {Erfi}(a+b x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2209
Rule 2212
Rule 2226
Rule 6363
Rubi steps
\begin {align*} \int (c+d x)^2 \text {erfi}(a+b x) \, dx &=\frac {(c+d x)^3 \text {erfi}(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 {erfi}(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 {erfi}(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 {erfi}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \text {erfi}(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 {erfi}(a+b x)}{2 b^3}-\frac {(b c-a d)^3 \text {erfi}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \text {erfi}(a+b x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 142, normalized size = 0.76 \[ \frac {\sqrt {\pi } \text {erfi}(a+b x) \left (2 a^3 d^2-6 a^2 b c d+a \left (6 b^2 c^2-3 d^2\right )+2 b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )+3 b c d\right )-2 e^{(a+b x)^2} \left (\left (a^2-1\right ) d^2-a b d (3 c+d x)+b^2 \left (3 c^2+3 c d x+d^2 x^2\right )\right )}{6 \sqrt {\pi } b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 161, normalized size = 0.87 \[ -\frac {2 \, \sqrt {\pi } {\left (b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} - 3 \, a b c d + {\left (a^{2} - 1\right )} d^{2} + {\left (3 \, b^{2} c d - a b d^{2}\right )} x\right )} e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - {\left (2 \, \pi b^{3} d^{2} x^{3} + 6 \, \pi b^{3} c d x^{2} + 6 \, \pi b^{3} c^{2} x + \pi {\left (6 \, a b^{2} c^{2} - 3 \, {\left (2 \, a^{2} - 1\right )} b c d + {\left (2 \, a^{3} - 3 \, a\right )} d^{2}\right )}\right )} \operatorname {erfi}\left (b x + a\right )}{6 \, \pi b^{3}} \]
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 \[ \int {\left (d x + c\right )}^{2} \operatorname {erfi}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 414, normalized size = 2.23 \[ \frac {\frac {d^{2} \erfi \left (b x +a \right ) \left (b x +a \right )^{3}}{3 b^{2}}-\frac {d^{2} \erfi \left (b x +a \right ) \left (b x +a \right )^{2} a}{b^{2}}+\frac {d \erfi \left (b x +a \right ) \left (b x +a \right )^{2} c}{b}+\frac {d^{2} \erfi \left (b x +a \right ) \left (b x +a \right ) a^{2}}{b^{2}}-\frac {2 d \erfi \left (b x +a \right ) \left (b x +a \right ) a c}{b}+\erfi \left (b x +a \right ) \left (b x +a \right ) c^{2}-\frac {d^{2} \erfi \left (b x +a \right ) a^{3}}{3 b^{2}}+\frac {d \erfi \left (b x +a \right ) a^{2} c}{b}-\erfi \left (b x +a \right ) a \,c^{2}+\frac {b \erfi \left (b x +a \right ) c^{3}}{3 d}-\frac {2 \left (d^{3} \left (\frac {\left (b x +a \right )^{2} {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{2}\right )+\frac {b^{3} c^{3} \sqrt {\pi }\, \erfi \left (b x +a \right )}{2}-\frac {a^{3} d^{3} \sqrt {\pi }\, \erfi \left (b x +a \right )}{2}+\frac {3 a^{2} d^{3} {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-3 a \,d^{3} \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \erfi \left (b x +a \right )}{4}\right )+\frac {3 b^{2} c^{2} d \,{\mathrm e}^{\left (b x +a \right )^{2}}}{2}+3 b c \,d^{2} \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \erfi \left (b x +a \right )}{4}\right )-\frac {3 a \,b^{2} c^{2} d \sqrt {\pi }\, \erfi \left (b x +a \right )}{2}+\frac {3 a^{2} b c \,d^{2} \sqrt {\pi }\, \erfi \left (b x +a \right )}{2}-3 a b c \,d^{2} {\mathrm e}^{\left (b x +a \right )^{2}}\right )}{3 b^{2} d \sqrt {\pi }}}{b} \]
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 \[ \int {\left (d x + c\right )}^{2} \operatorname {erfi}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 190, normalized size = 1.02 \[ \frac {\frac {{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}\,\left (-a^2\,d^2+3\,a\,b\,c\,d-3\,b^2\,c^2+d^2\right )}{b^3}+\frac {x\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a\,d^2-3\,b\,c\,d\right )}{b^2}-\frac {d^2\,x^2\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}}{b}}{3\,\sqrt {\pi }}+\mathrm {erfi}\left (a+b\,x\right )\,\left (c^2\,x+c\,d\,x^2+\frac {d^2\,x^3}{3}\right )+\frac {\mathrm {erfi}\left (a+b\,x\right )\,\left (2\,a^3\,d^2-6\,a^2\,b\,c\,d+6\,a\,b^2\,c^2-3\,a\,d^2+3\,b\,c\,d\right )}{6\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.09, size = 398, normalized size = 2.14 \[ \begin {cases} \frac {a^{3} d^{2} \operatorname {erfi}{\left (a + b x \right )}}{3 b^{3}} - \frac {a^{2} c d \operatorname {erfi}{\left (a + b x \right )}}{b^{2}} - \frac {a^{2} d^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{3 \sqrt {\pi } b^{3}} + \frac {a c^{2} \operatorname {erfi}{\left (a + b x \right )}}{b} + \frac {a c d e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b^{2}} + \frac {a d^{2} x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{3 \sqrt {\pi } b^{2}} - \frac {a d^{2} \operatorname {erfi}{\left (a + b x \right )}}{2 b^{3}} + c^{2} x \operatorname {erfi}{\left (a + b x \right )} + c d x^{2} \operatorname {erfi}{\left (a + b x \right )} + \frac {d^{2} x^{3} \operatorname {erfi}{\left (a + b x \right )}}{3} - \frac {c^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} - \frac {c d x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} - \frac {d^{2} x^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{3 \sqrt {\pi } b} + \frac {c d \operatorname {erfi}{\left (a + b x \right )}}{2 b^{2}} + \frac {d^{2} e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{3 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \operatorname {erfi}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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