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 = 0.361178, antiderivative size = 366, normalized size of antiderivative = 1., 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 6369
Rule 6354
Rule 6384
Rule 2204
Rule 6366
Rule 6387
Rule 6375
Rule 30
Rule 2209
Rule 2212
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*}
Mathematica [F] time = 0.736608, size = 0, normalized size = 0. \[ \int (c+d x)^2 \text{Erfi}(a+b x)^2 \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.205, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{2} \left ({\it erfi} \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2} \operatorname{erfi}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.55522, size = 645, normalized size = 1.76 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{2} \operatorname{erfi}^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2} \operatorname{erfi}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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