Optimal. Leaf size=366 \[ \frac {d (a+b x)^2 (b c-a d) \text {erfi}(a+b x)^2}{b^3}+\frac {(a+b x) (b c-a d)^2 \text {erfi}(a+b x)^2}{b^3}-\frac {2 d e^{(a+b x)^2} (a+b x) (b c-a d) \text {erfi}(a+b x)}{\sqrt {\pi } b^3}+\frac {d (b c-a d) \text {erfi}(a+b x)^2}{2 b^3}-\frac {2 e^{(a+b x)^2} (b c-a d)^2 \text {erfi}(a+b x)}{\sqrt {\pi } b^3}+\frac {\sqrt {\frac {2}{\pi }} (b c-a d)^2 \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^3}+\frac {d e^{2 (a+b x)^2} (b c-a d)}{\pi b^3}+\frac {d^2 (a+b x)^3 \text {erfi}(a+b x)^2}{3 b^3}-\frac {2 d^2 e^{(a+b x)^2} (a+b x)^2 \text {erfi}(a+b x)}{3 \sqrt {\pi } b^3}+\frac {2 d^2 e^{(a+b x)^2} \text {erfi}(a+b x)}{3 \sqrt {\pi } b^3}-\frac {5 d^2 \text {erfi}\left (\sqrt {2} (a+b x)\right )}{6 \sqrt {2 \pi } b^3}+\frac {d^2 e^{2 (a+b x)^2} (a+b x)}{3 \pi b^3} \]
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Rubi [A] time = 0.36, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6369, 6354, 6384, 2204, 6366, 6387, 6375, 30, 2209, 2212} \[ \frac {d (a+b x)^2 (b c-a d) \text {Erfi}(a+b x)^2}{b^3}+\frac {(a+b x) (b c-a d)^2 \text {Erfi}(a+b x)^2}{b^3}-\frac {2 d e^{(a+b x)^2} (a+b x) (b c-a d) \text {Erfi}(a+b x)}{\sqrt {\pi } b^3}+\frac {d (b c-a d) \text {Erfi}(a+b x)^2}{2 b^3}-\frac {2 e^{(a+b x)^2} (b c-a d)^2 \text {Erfi}(a+b x)}{\sqrt {\pi } b^3}+\frac {\sqrt {\frac {2}{\pi }} (b c-a d)^2 \text {Erfi}\left (\sqrt {2} (a+b x)\right )}{b^3}+\frac {d e^{2 (a+b x)^2} (b c-a d)}{\pi b^3}+\frac {d^2 (a+b x)^3 \text {Erfi}(a+b x)^2}{3 b^3}-\frac {2 d^2 e^{(a+b x)^2} (a+b x)^2 \text {Erfi}(a+b x)}{3 \sqrt {\pi } b^3}+\frac {2 d^2 e^{(a+b x)^2} \text {Erfi}(a+b x)}{3 \sqrt {\pi } b^3}-\frac {5 d^2 \text {Erfi}\left (\sqrt {2} (a+b x)\right )}{6 \sqrt {2 \pi } b^3}+\frac {d^2 e^{2 (a+b x)^2} (a+b x)}{3 \pi b^3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2204
Rule 2209
Rule 2212
Rule 6354
Rule 6366
Rule 6369
Rule 6375
Rule 6384
Rule 6387
Rubi steps
\begin {align*} \int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx &=\frac {\operatorname {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 \operatorname {Subst}\left (\int x^2 \text {erfi}(x)^2 \, dx,x,a+b x\right )}{b^3}+\frac {(2 d (b c-a d)) \operatorname {Subst}\left (\int x \text {erfi}(x)^2 \, dx,x,a+b x\right )}{b^3}+\frac {(b c-a d)^2 \operatorname {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 ) \operatorname {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)) \operatorname {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 ) \operatorname {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 ) \operatorname {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)) \operatorname {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 ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,a+b x\right )}{b^3 \pi }+\frac {\left (4 d^2\right ) \operatorname {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)) \operatorname {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)) \operatorname {Subst}(\int x \, dx,x,\text {erfi}(a+b x))}{b^3}-\frac {d^2 \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,a+b x\right )}{3 b^3 \pi }-\frac {\left (4 d^2\right ) \operatorname {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*}
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Mathematica [F] time = 0.86, size = 0, normalized size = 0.00 \[ \int (c+d x)^2 \text {erfi}(a+b x)^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.61, size = 278, normalized size = 0.76 \[ \frac {\sqrt {2} \sqrt {\pi } {\left (12 \, b^{2} c^{2} - 24 \, a b c d + {\left (12 \, a^{2} - 5\right )} d^{2}\right )} \sqrt {b^{2}} \operatorname {erfi}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 8 \, \sqrt {\pi } {\left (b^{3} d^{2} x^{2} + 3 \, b^{3} c^{2} - 3 \, a b^{2} c d + {\left (a^{2} - 1\right )} b d^{2} + {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x\right )} \operatorname {erfi}\left (b x + a\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} + 2 \, {\left (2 \, \pi b^{4} d^{2} x^{3} + 6 \, \pi b^{4} c d x^{2} + 6 \, \pi b^{4} c^{2} x + \pi {\left (6 \, a b^{3} c^{2} - 3 \, {\left (2 \, a^{2} - 1\right )} b^{2} c d + {\left (2 \, a^{3} - 3 \, a\right )} b d^{2}\right )}\right )} \operatorname {erfi}\left (b x + a\right )^{2} + 4 \, {\left (b^{2} d^{2} x + 3 \, b^{2} c d - 2 \, a b d^{2}\right )} e^{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2}\right )}}{12 \, \pi b^{4}} \]
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 \[ \int {\left (d x + c\right )}^{2} \operatorname {erfi}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{2} \erfi \left (b x +a \right )^{2}\, dx \]
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 \[ \int {\left (d x + c\right )}^{2} \operatorname {erfi}\left (b x + a\right )^{2}\,{d x} \]
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 \[ \int {\mathrm {erfi}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \]
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 \[ \int \left (c + d x\right )^{2} \operatorname {erfi}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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