Integrand size = 16, antiderivative size = 366 \[ \int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx=\frac {d (b c-a d) e^{2 (a+b x)^2}}{b^3 \pi }+\frac {d^2 e^{2 (a+b x)^2} (a+b x)}{3 b^3 \pi }+\frac {2 d^2 e^{(a+b x)^2} \text {erfi}(a+b x)}{3 b^3 \sqrt {\pi }}-\frac {2 (b c-a d)^2 e^{(a+b x)^2} \text {erfi}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d (b c-a d) e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d^2 e^{(a+b x)^2} (a+b x)^2 \text {erfi}(a+b x)}{3 b^3 \sqrt {\pi }}+\frac {d (b c-a d) \text {erfi}(a+b x)^2}{2 b^3}+\frac {(b c-a d)^2 (a+b x) \text {erfi}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \text {erfi}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \text {erfi}(a+b x)^2}{3 b^3}+\frac {(b c-a d)^2 \sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^3}-\frac {5 d^2 \text {erfi}\left (\sqrt {2} (a+b x)\right )}{6 b^3 \sqrt {2 \pi }} \]
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Time = 0.26 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6504, 6489, 6519, 2235, 6501, 6522, 6510, 30, 2240, 2243} \[ \int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx=\frac {d (a+b x)^2 (b c-a d) \text {erfi}(a+b x)^2}{b^3}+\frac {(a+b x) (b c-a d)^2 \text {erfi}(a+b x)^2}{b^3}-\frac {2 d e^{(a+b x)^2} (a+b x) (b c-a d) \text {erfi}(a+b x)}{\sqrt {\pi } b^3}+\frac {d (b c-a d) \text {erfi}(a+b x)^2}{2 b^3}-\frac {2 e^{(a+b x)^2} (b c-a d)^2 \text {erfi}(a+b x)}{\sqrt {\pi } b^3}+\frac {\sqrt {\frac {2}{\pi }} (b c-a d)^2 \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^3}+\frac {d e^{2 (a+b x)^2} (b c-a d)}{\pi b^3}+\frac {d^2 (a+b x)^3 \text {erfi}(a+b x)^2}{3 b^3}-\frac {2 d^2 e^{(a+b x)^2} (a+b x)^2 \text {erfi}(a+b x)}{3 \sqrt {\pi } b^3}+\frac {2 d^2 e^{(a+b x)^2} \text {erfi}(a+b x)}{3 \sqrt {\pi } b^3}-\frac {5 d^2 \text {erfi}\left (\sqrt {2} (a+b x)\right )}{6 \sqrt {2 \pi } b^3}+\frac {d^2 e^{2 (a+b x)^2} (a+b x)}{3 \pi b^3} \]
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Rule 30
Rule 2235
Rule 2240
Rule 2243
Rule 6489
Rule 6501
Rule 6504
Rule 6510
Rule 6519
Rule 6522
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (b^2 c^2 \left (1+\frac {a d (-2 b c+a d)}{b^2 c^2}\right ) \text {erfi}(x)^2+2 b c d \left (1-\frac {a d}{b c}\right ) x \text {erfi}(x)^2+d^2 x^2 \text {erfi}(x)^2\right ) \, dx,x,a+b x\right )}{b^3} \\ & = \frac {d^2 \text {Subst}\left (\int x^2 \text {erfi}(x)^2 \, dx,x,a+b x\right )}{b^3}+\frac {(2 d (b c-a d)) \text {Subst}\left (\int x \text {erfi}(x)^2 \, dx,x,a+b x\right )}{b^3}+\frac {(b c-a d)^2 \text {Subst}\left (\int \text {erfi}(x)^2 \, dx,x,a+b x\right )}{b^3} \\ & = \frac {(b c-a d)^2 (a+b x) \text {erfi}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \text {erfi}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \text {erfi}(a+b x)^2}{3 b^3}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int e^{x^2} x^3 \text {erfi}(x) \, dx,x,a+b x\right )}{3 b^3 \sqrt {\pi }}-\frac {(4 d (b c-a d)) \text {Subst}\left (\int e^{x^2} x^2 \text {erfi}(x) \, dx,x,a+b x\right )}{b^3 \sqrt {\pi }}-\frac {\left (4 (b c-a d)^2\right ) \text {Subst}\left (\int e^{x^2} x \text {erfi}(x) \, dx,x,a+b x\right )}{b^3 \sqrt {\pi }} \\ & = -\frac {2 (b c-a d)^2 e^{(a+b x)^2} \text {erfi}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d (b c-a d) e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d^2 e^{(a+b x)^2} (a+b x)^2 \text {erfi}(a+b x)}{3 b^3 \sqrt {\pi }}+\frac {(b c-a d)^2 (a+b x) \text {erfi}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \text {erfi}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \text {erfi}(a+b x)^2}{3 b^3}+\frac {\left (4 d^2\right ) \text {Subst}\left (\int e^{2 x^2} x^2 \, dx,x,a+b x\right )}{3 b^3 \pi }+\frac {(4 d (b c-a d)) \text {Subst}\left (\int e^{2 x^2} x \, dx,x,a+b x\right )}{b^3 \pi }+\frac {\left (4 (b c-a d)^2\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,a+b x\right )}{b^3 \pi }+\frac {\left (4 d^2\right ) \text {Subst}\left (\int e^{x^2} x \text {erfi}(x) \, dx,x,a+b x\right )}{3 b^3 \sqrt {\pi }}+\frac {(2 d (b c-a d)) \text {Subst}\left (\int e^{x^2} \text {erfi}(x) \, dx,x,a+b x\right )}{b^3 \sqrt {\pi }} \\ & = \frac {d (b c-a d) e^{2 (a+b x)^2}}{b^3 \pi }+\frac {d^2 e^{2 (a+b x)^2} (a+b x)}{3 b^3 \pi }+\frac {2 d^2 e^{(a+b x)^2} \text {erfi}(a+b x)}{3 b^3 \sqrt {\pi }}-\frac {2 (b c-a d)^2 e^{(a+b x)^2} \text {erfi}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d (b c-a d) e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d^2 e^{(a+b x)^2} (a+b x)^2 \text {erfi}(a+b x)}{3 b^3 \sqrt {\pi }}+\frac {(b c-a d)^2 (a+b x) \text {erfi}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \text {erfi}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \text {erfi}(a+b x)^2}{3 b^3}+\frac {(b c-a d)^2 \sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^3}+\frac {(d (b c-a d)) \text {Subst}(\int x \, dx,x,\text {erfi}(a+b x))}{b^3}-\frac {d^2 \text {Subst}\left (\int e^{2 x^2} \, dx,x,a+b x\right )}{3 b^3 \pi }-\frac {\left (4 d^2\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,a+b x\right )}{3 b^3 \pi } \\ & = \frac {d (b c-a d) e^{2 (a+b x)^2}}{b^3 \pi }+\frac {d^2 e^{2 (a+b x)^2} (a+b x)}{3 b^3 \pi }+\frac {2 d^2 e^{(a+b x)^2} \text {erfi}(a+b x)}{3 b^3 \sqrt {\pi }}-\frac {2 (b c-a d)^2 e^{(a+b x)^2} \text {erfi}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d (b c-a d) e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{b^3 \sqrt {\pi }}-\frac {2 d^2 e^{(a+b x)^2} (a+b x)^2 \text {erfi}(a+b x)}{3 b^3 \sqrt {\pi }}+\frac {d (b c-a d) \text {erfi}(a+b x)^2}{2 b^3}+\frac {(b c-a d)^2 (a+b x) \text {erfi}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \text {erfi}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \text {erfi}(a+b x)^2}{3 b^3}-\frac {d^2 \sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{3 b^3}+\frac {(b c-a d)^2 \sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^3}-\frac {d^2 \text {erfi}\left (\sqrt {2} (a+b x)\right )}{6 b^3 \sqrt {2 \pi }} \\ \end{align*}
\[ \int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx=\int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx \]
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\[\int \left (d x +c \right )^{2} \operatorname {erfi}\left (b x +a \right )^{2}d x\]
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Time = 0.26 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.77 \[ \int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx=-\frac {\sqrt {2} \sqrt {\pi } {\left (12 \, b^{2} c^{2} - 24 \, a b c d + {\left (12 \, a^{2} - 5\right )} d^{2}\right )} \sqrt {-b^{2}} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {-b^{2}} {\left (b x + a\right )}}{b}\right ) + 8 \, \sqrt {\pi } {\left (b^{3} d^{2} x^{2} + 3 \, b^{3} c^{2} - 3 \, a b^{2} c d + {\left (a^{2} - 1\right )} b d^{2} + {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x\right )} \operatorname {erfi}\left (b x + a\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - 2 \, {\left (2 \, \pi b^{4} d^{2} x^{3} + 6 \, \pi b^{4} c d x^{2} + 6 \, \pi b^{4} c^{2} x + \pi {\left (6 \, a b^{3} c^{2} - 3 \, {\left (2 \, a^{2} - 1\right )} b^{2} c d + {\left (2 \, a^{3} - 3 \, a\right )} b d^{2}\right )}\right )} \operatorname {erfi}\left (b x + a\right )^{2} - 4 \, {\left (b^{2} d^{2} x + 3 \, b^{2} c d - 2 \, a b d^{2}\right )} e^{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2}\right )}}{12 \, \pi b^{4}} \]
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\[ \int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx=\int \left (c + d x\right )^{2} \operatorname {erfi}^{2}{\left (a + b x \right )}\, dx \]
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\[ \int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {erfi}\left (b x + a\right )^{2} \,d x } \]
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\[ \int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {erfi}\left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx=\int {\mathrm {erfi}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \]
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