Optimal. Leaf size=467 \[ \frac{3 \text{Unintegrable}\left (e^{c+d x^2} \text{Erfi}(a+b x),x\right )}{4 d^2}-\frac{3 a b^2 e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{4 d^2 \left (b^2+d\right )^{3/2}}+\frac{3 b e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{4 \sqrt{\pi } d^2 \left (b^2+d\right )}+\frac{a^3 b^4 e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{2 d \left (b^2+d\right )^{7/2}}-\frac{3 a b^2 e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{4 d \left (b^2+d\right )^{5/2}}-\frac{a^2 b^3 e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )^3}+\frac{a b^2 x e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )^2}-\frac{b x^2 e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )}+\frac{b e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )^2}-\frac{3 x e^{c+d x^2} \text{Erfi}(a+b x)}{4 d^2}+\frac{x^3 e^{c+d x^2} \text{Erfi}(a+b x)}{2 d} \]
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Rubi [A] time = 0.892451, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int e^{c+d x^2} x^4 \text{Erfi}(a+b x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int e^{c+d x^2} x^4 \text{erfi}(a+b x) \, dx &=\frac{e^{c+d x^2} x^3 \text{erfi}(a+b x)}{2 d}-\frac{3 \int e^{c+d x^2} x^2 \text{erfi}(a+b x) \, dx}{2 d}-\frac{b \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^3 \, dx}{d \sqrt{\pi }}\\ &=-\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^2}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{3 e^{c+d x^2} x \text{erfi}(a+b x)}{4 d^2}+\frac{e^{c+d x^2} x^3 \text{erfi}(a+b x)}{2 d}+\frac{3 \int e^{c+d x^2} \text{erfi}(a+b x) \, dx}{4 d^2}+\frac{(3 b) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x \, dx}{2 d^2 \sqrt{\pi }}+\frac{b \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x \, dx}{d \left (b^2+d\right ) \sqrt{\pi }}+\frac{\left (a b^2\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^2 \, dx}{d \left (b^2+d\right ) \sqrt{\pi }}\\ &=\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}+\frac{3 b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{4 d^2 \left (b^2+d\right ) \sqrt{\pi }}+\frac{a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^2}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{3 e^{c+d x^2} x \text{erfi}(a+b x)}{4 d^2}+\frac{e^{c+d x^2} x^3 \text{erfi}(a+b x)}{2 d}+\frac{3 \int e^{c+d x^2} \text{erfi}(a+b x) \, dx}{4 d^2}-\frac{\left (a b^2\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{\left (a b^2\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{\left (a^2 b^3\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x \, dx}{d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{\left (3 a b^2\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{2 d^2 \left (b^2+d\right ) \sqrt{\pi }}\\ &=-\frac{a^2 b^3 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^3 \sqrt{\pi }}+\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}+\frac{3 b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{4 d^2 \left (b^2+d\right ) \sqrt{\pi }}+\frac{a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^2}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{3 e^{c+d x^2} x \text{erfi}(a+b x)}{4 d^2}+\frac{e^{c+d x^2} x^3 \text{erfi}(a+b x)}{2 d}+\frac{3 \int e^{c+d x^2} \text{erfi}(a+b x) \, dx}{4 d^2}+\frac{\left (a^3 b^4\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{d \left (b^2+d\right )^3 \sqrt{\pi }}-\frac{\left (a b^2 e^{c+\frac{a^2 d}{b^2+d}}\right ) \int e^{\frac{\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{\left (a b^2 e^{c+\frac{a^2 d}{b^2+d}}\right ) \int e^{\frac{\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{\left (3 a b^2 e^{c+\frac{a^2 d}{b^2+d}}\right ) \int e^{\frac{\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{2 d^2 \left (b^2+d\right ) \sqrt{\pi }}\\ &=-\frac{a^2 b^3 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^3 \sqrt{\pi }}+\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}+\frac{3 b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{4 d^2 \left (b^2+d\right ) \sqrt{\pi }}+\frac{a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^2}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{3 e^{c+d x^2} x \text{erfi}(a+b x)}{4 d^2}+\frac{e^{c+d x^2} x^3 \text{erfi}(a+b x)}{2 d}-\frac{3 a b^2 e^{c+\frac{a^2 d}{b^2+d}} \text{erfi}\left (\frac{a b+\left (b^2+d\right ) x}{\sqrt{b^2+d}}\right )}{4 d \left (b^2+d\right )^{5/2}}-\frac{3 a b^2 e^{c+\frac{a^2 d}{b^2+d}} \text{erfi}\left (\frac{a b+\left (b^2+d\right ) x}{\sqrt{b^2+d}}\right )}{4 d^2 \left (b^2+d\right )^{3/2}}+\frac{3 \int e^{c+d x^2} \text{erfi}(a+b x) \, dx}{4 d^2}+\frac{\left (a^3 b^4 e^{c+\frac{a^2 d}{b^2+d}}\right ) \int e^{\frac{\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{d \left (b^2+d\right )^3 \sqrt{\pi }}\\ &=-\frac{a^2 b^3 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^3 \sqrt{\pi }}+\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}+\frac{3 b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{4 d^2 \left (b^2+d\right ) \sqrt{\pi }}+\frac{a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}-\frac{b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^2}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{3 e^{c+d x^2} x \text{erfi}(a+b x)}{4 d^2}+\frac{e^{c+d x^2} x^3 \text{erfi}(a+b x)}{2 d}+\frac{a^3 b^4 e^{c+\frac{a^2 d}{b^2+d}} \text{erfi}\left (\frac{a b+\left (b^2+d\right ) x}{\sqrt{b^2+d}}\right )}{2 d \left (b^2+d\right )^{7/2}}-\frac{3 a b^2 e^{c+\frac{a^2 d}{b^2+d}} \text{erfi}\left (\frac{a b+\left (b^2+d\right ) x}{\sqrt{b^2+d}}\right )}{4 d \left (b^2+d\right )^{5/2}}-\frac{3 a b^2 e^{c+\frac{a^2 d}{b^2+d}} \text{erfi}\left (\frac{a b+\left (b^2+d\right ) x}{\sqrt{b^2+d}}\right )}{4 d^2 \left (b^2+d\right )^{3/2}}+\frac{3 \int e^{c+d x^2} \text{erfi}(a+b x) \, dx}{4 d^2}\\ \end{align*}
Mathematica [A] time = 0.485338, size = 0, normalized size = 0. \[ \int e^{c+d x^2} x^4 \text{Erfi}(a+b x) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{4}{\it erfi} \left ( bx+a \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{4} \operatorname{erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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