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 = 0.165187, antiderivative size = 186, normalized size of antiderivative = 1., 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 6363
Rule 2226
Rule 2204
Rule 2209
Rule 2212
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*}
Mathematica [A] time = 0.174973, size = 142, normalized size = 0.76 \[ \frac{\sqrt{\pi } \text{Erfi}(a+b x) \left (-6 a^2 b c d+2 a^3 d^2+a \left (6 b^2 c^2-3 d^2\right )+2 b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )+3 b c d\right )-2 e^{(a+b x)^2} \left (\left (a^2-1\right ) d^2-a b d (3 c+d x)+b^2 \left (3 c^2+3 c d x+d^2 x^2\right )\right )}{6 \sqrt{\pi } b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 414, normalized size = 2.2 \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{2}{\it erfi} \left ( bx+a \right ) \left ( bx+a \right ) ^{3}}{3\,{b}^{2}}}-{\frac{{d}^{2}{\it erfi} \left ( bx+a \right ) \left ( bx+a \right ) ^{2}a}{{b}^{2}}}+{\frac{d{\it erfi} \left ( bx+a \right ) \left ( bx+a \right ) ^{2}c}{b}}+{\frac{{d}^{2}{\it erfi} \left ( bx+a \right ) \left ( bx+a \right ){a}^{2}}{{b}^{2}}}-2\,{\frac{d{\it erfi} \left ( bx+a \right ) \left ( bx+a \right ) ac}{b}}+{\it erfi} \left ( bx+a \right ) \left ( bx+a \right ){c}^{2}-{\frac{{d}^{2}{\it erfi} \left ( bx+a \right ){a}^{3}}{3\,{b}^{2}}}+{\frac{d{\it erfi} \left ( bx+a \right ){a}^{2}c}{b}}-{\it erfi} \left ( bx+a \right ) a{c}^{2}+{\frac{b{\it erfi} \left ( bx+a \right ){c}^{3}}{3\,d}}-{\frac{2}{3\,{b}^{2}\sqrt{\pi }d} \left ({\frac{{b}^{3}{c}^{3}\sqrt{\pi }{\it erfi} \left ( bx+a \right ) }{2}}+{d}^{3} \left ({\frac{ \left ( bx+a \right ) ^{2}{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}{2}}-{\frac{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}{2}} \right ) -{\frac{{a}^{3}{d}^{3}\sqrt{\pi }{\it erfi} \left ( bx+a \right ) }{2}}+{\frac{3\,{a}^{2}{d}^{3}{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}{2}}-3\,a{d}^{3} \left ( 1/2\, \left ( bx+a \right ){{\rm e}^{ \left ( bx+a \right ) ^{2}}}-1/4\,\sqrt{\pi }{\it erfi} \left ( bx+a \right ) \right ) +{\frac{3\,{b}^{2}{c}^{2}d{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}{2}}+3\,bc{d}^{2} \left ( 1/2\, \left ( bx+a \right ){{\rm e}^{ \left ( bx+a \right ) ^{2}}}-1/4\,\sqrt{\pi }{\it erfi} \left ( bx+a \right ) \right ) -{\frac{3\,a{b}^{2}{c}^{2}d\sqrt{\pi }{\it erfi} \left ( bx+a \right ) }{2}}+{\frac{3\,{a}^{2}bc{d}^{2}\sqrt{\pi }{\it erfi} \left ( bx+a \right ) }{2}}-3\,abc{d}^{2}{{\rm e}^{ \left ( bx+a \right ) ^{2}}} \right ) } \right ) } \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 )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31035, size = 362, normalized size = 1.95 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.38124, size = 398, normalized size = 2.14 \begin{align*} \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}{\left (a \right )} & \text{otherwise} \end{cases} \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 )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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