Optimal. Leaf size=285 \[ \frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{2 d^2 \left (b^2-d\right )^{3/2}}-\frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{d^3 \sqrt{b^2-d}}-\frac{b x e^{c-x^2 \left (b^2-d\right )}}{\sqrt{\pi } d^2 \left (b^2-d\right )}-\frac{3 b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{8 d \left (b^2-d\right )^{5/2}}+\frac{b x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )}+\frac{3 b x e^{c-x^2 \left (b^2-d\right )}}{4 \sqrt{\pi } d \left (b^2-d\right )^2}-\frac{x^2 \text{Erf}(b x) e^{c+d x^2}}{d^2}+\frac{\text{Erf}(b x) e^{c+d x^2}}{d^3}+\frac{x^4 \text{Erf}(b x) e^{c+d x^2}}{2 d} \]
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Rubi [A] time = 0.435986, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6385, 6382, 2205, 2212} \[ \frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{2 d^2 \left (b^2-d\right )^{3/2}}-\frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{d^3 \sqrt{b^2-d}}-\frac{b x e^{c-x^2 \left (b^2-d\right )}}{\sqrt{\pi } d^2 \left (b^2-d\right )}-\frac{3 b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{8 d \left (b^2-d\right )^{5/2}}+\frac{b x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )}+\frac{3 b x e^{c-x^2 \left (b^2-d\right )}}{4 \sqrt{\pi } d \left (b^2-d\right )^2}-\frac{x^2 \text{Erf}(b x) e^{c+d x^2}}{d^2}+\frac{\text{Erf}(b x) e^{c+d x^2}}{d^3}+\frac{x^4 \text{Erf}(b x) e^{c+d x^2}}{2 d} \]
Antiderivative was successfully verified.
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Rule 6385
Rule 6382
Rule 2205
Rule 2212
Rubi steps
\begin{align*} \int e^{c+d x^2} x^5 \text{erf}(b x) \, dx &=\frac{e^{c+d x^2} x^4 \text{erf}(b x)}{2 d}-\frac{2 \int e^{c+d x^2} x^3 \text{erf}(b x) \, dx}{d}-\frac{b \int e^{c-\left (b^2-d\right ) x^2} x^4 \, dx}{d \sqrt{\pi }}\\ &=\frac{b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt{\pi }}-\frac{e^{c+d x^2} x^2 \text{erf}(b x)}{d^2}+\frac{e^{c+d x^2} x^4 \text{erf}(b x)}{2 d}+\frac{2 \int e^{c+d x^2} x \text{erf}(b x) \, dx}{d^2}+\frac{(2 b) \int e^{c-\left (b^2-d\right ) x^2} x^2 \, dx}{d^2 \sqrt{\pi }}-\frac{(3 b) \int e^{c+\left (-b^2+d\right ) x^2} x^2 \, dx}{2 \left (b^2-d\right ) d \sqrt{\pi }}\\ &=-\frac{b e^{c-\left (b^2-d\right ) x^2} x}{\left (b^2-d\right ) d^2 \sqrt{\pi }}+\frac{3 b e^{c-\left (b^2-d\right ) x^2} x}{4 \left (b^2-d\right )^2 d \sqrt{\pi }}+\frac{b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt{\pi }}+\frac{e^{c+d x^2} \text{erf}(b x)}{d^3}-\frac{e^{c+d x^2} x^2 \text{erf}(b x)}{d^2}+\frac{e^{c+d x^2} x^4 \text{erf}(b x)}{2 d}-\frac{(2 b) \int e^{c-\left (b^2-d\right ) x^2} \, dx}{d^3 \sqrt{\pi }}+\frac{b \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{\left (b^2-d\right ) d^2 \sqrt{\pi }}-\frac{(3 b) \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{4 \left (b^2-d\right )^2 d \sqrt{\pi }}\\ &=-\frac{b e^{c-\left (b^2-d\right ) x^2} x}{\left (b^2-d\right ) d^2 \sqrt{\pi }}+\frac{3 b e^{c-\left (b^2-d\right ) x^2} x}{4 \left (b^2-d\right )^2 d \sqrt{\pi }}+\frac{b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt{\pi }}+\frac{e^{c+d x^2} \text{erf}(b x)}{d^3}-\frac{e^{c+d x^2} x^2 \text{erf}(b x)}{d^2}+\frac{e^{c+d x^2} x^4 \text{erf}(b x)}{2 d}-\frac{b e^c \text{erf}\left (\sqrt{b^2-d} x\right )}{\sqrt{b^2-d} d^3}+\frac{b e^c \text{erf}\left (\sqrt{b^2-d} x\right )}{2 \left (b^2-d\right )^{3/2} d^2}-\frac{3 b e^c \text{erf}\left (\sqrt{b^2-d} x\right )}{8 \left (b^2-d\right )^{5/2} d}\\ \end{align*}
Mathematica [A] time = 0.413958, size = 138, normalized size = 0.48 \[ \frac{e^c \left (\frac{b \left (20 b^2 d-8 b^4-15 d^2\right ) \text{Erfi}\left (x \sqrt{d-b^2}\right )}{\left (d-b^2\right )^{5/2}}+\frac{2 b d x e^{x^2 \left (d-b^2\right )} \left (2 b^2 \left (d x^2-2\right )+d \left (7-2 d x^2\right )\right )}{\sqrt{\pi } \left (b^2-d\right )^2}+4 e^{d x^2} \left (d^2 x^4-2 d x^2+2\right ) \text{Erf}(b x)\right )}{8 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.268, size = 312, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Erf} \left ( bx \right ){{\rm e}^{c}}}{{b}^{5}} \left ({\frac{{{\rm e}^{d{x}^{2}}}{b}^{6}{x}^{4}}{2\,d}}-2\,{\frac{{b}^{2}}{d} \left ( 1/2\,{\frac{{b}^{4}{x}^{2}{{\rm e}^{d{x}^{2}}}}{d}}-1/2\,{\frac{{b}^{4}{{\rm e}^{d{x}^{2}}}}{{d}^{2}}} \right ) } \right ) }-{\frac{{{\rm e}^{c}}}{\sqrt{\pi }{b}^{5}} \left ({\frac{{b}^{2}}{d} \left ({\frac{{x}^{3}{b}^{3}}{2}{{\rm e}^{ \left ( -1+{\frac{d}{{b}^{2}}} \right ){b}^{2}{x}^{2}}} \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}}-{\frac{3}{2} \left ({\frac{bx}{2}{{\rm e}^{ \left ( -1+{\frac{d}{{b}^{2}}} \right ){b}^{2}{x}^{2}}} \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}}-{\frac{\sqrt{\pi }}{4}{\it Erf} \left ( \sqrt{1-{\frac{d}{{b}^{2}}}}bx \right ) \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}{\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}} \right ) \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}} \right ) }+{\frac{{b}^{6}\sqrt{\pi }}{{d}^{3}}{\it Erf} \left ( \sqrt{1-{\frac{d}{{b}^{2}}}}bx \right ){\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}}-2\,{\frac{{b}^{4}}{{d}^{2}} \left ( 1/2\,{bx{{\rm e}^{ \left ( -1+{\frac{d}{{b}^{2}}} \right ){b}^{2}{x}^{2}}} \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}}-1/4\,{\sqrt{\pi }{\it Erf} \left ( \sqrt{1-{\frac{d}{{b}^{2}}}}bx \right ) \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}{\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}} \right ) } \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*} \frac{{\left (d^{2} x^{4} e^{c} - 2 \, d x^{2} e^{c} + 2 \, e^{c}\right )} \operatorname{erf}\left (b x\right ) e^{\left (d x^{2}\right )}}{2 \, d^{3}} - \frac{-\frac{b d^{2} x^{5} e^{c} \Gamma \left (\frac{5}{2},{\left (b^{2} - d\right )} x^{2}\right )}{2 \, \left ({\left (b^{2} - d\right )} x^{2}\right )^{\frac{5}{2}}} + \frac{b d x^{3} e^{c} \Gamma \left (\frac{3}{2},{\left (b^{2} - d\right )} x^{2}\right )}{\left ({\left (b^{2} - d\right )} x^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{\pi } b \operatorname{erf}\left (\sqrt{b^{2} - d} x\right ) e^{c}}{\sqrt{b^{2} - d}}}{\sqrt{\pi } d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.24942, size = 545, normalized size = 1.91 \begin{align*} -\frac{\pi{\left (8 \, b^{5} - 20 \, b^{3} d + 15 \, b d^{2}\right )} \sqrt{b^{2} - d} \operatorname{erf}\left (\sqrt{b^{2} - d} x\right ) e^{c} - 4 \,{\left (\pi{\left (b^{6} d^{2} - 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} - d^{5}\right )} x^{4} - 2 \, \pi{\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} + 2 \, \pi{\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )}\right )} \operatorname{erf}\left (b x\right ) e^{\left (d x^{2} + c\right )} - 2 \, \sqrt{\pi }{\left (2 \,{\left (b^{5} d^{2} - 2 \, b^{3} d^{3} + b d^{4}\right )} x^{3} -{\left (4 \, b^{5} d - 11 \, b^{3} d^{2} + 7 \, b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} + d x^{2} + c\right )}}{8 \, \pi{\left (b^{6} d^{3} - 3 \, b^{4} d^{4} + 3 \, b^{2} d^{5} - d^{6}\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 = 1.28484, size = 347, normalized size = 1.22 \begin{align*} -\frac{{\left (2 \, d x^{2} -{\left (d x^{2} + c\right )}^{2} + 2 \,{\left (d x^{2} + c\right )} c - c^{2} - 2\right )} \operatorname{erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \, d^{3}} + \frac{\sqrt{\pi } b d^{2}{\left (\frac{2 \,{\left (2 \, b^{2} x^{3} - 2 \, d x^{3} + 3 \, x\right )} e^{\left (-b^{2} x^{2} + d x^{2} + c\right )}}{b^{4} - 2 \, b^{2} d + d^{2}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{b^{2} - d} x\right ) e^{c}}{{\left (b^{4} - 2 \, b^{2} d + d^{2}\right )} \sqrt{b^{2} - d}}\right )} - 4 \, \sqrt{\pi } b d{\left (\frac{2 \, x e^{\left (-b^{2} x^{2} + d x^{2} + c\right )}}{b^{2} - d} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{b^{2} - d} x\right ) e^{c}}{{\left (b^{2} - d\right )}^{\frac{3}{2}}}\right )} + \frac{8 \, \pi b \operatorname{erf}\left (-\sqrt{b^{2} - d} x\right ) e^{c}}{\sqrt{b^{2} - d}}}{8 \, \pi d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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