Optimal. Leaf size=144 \[ \frac{x^4 e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{x^2 e^{b^2 x^2+c} \text{Erfi}(b x)}{b^4}+\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{b^6}-\frac{43 e^c \text{Erfi}\left (\sqrt{2} b x\right )}{32 \sqrt{2} b^6}-\frac{x^3 e^{2 b^2 x^2+c}}{4 \sqrt{\pi } b^3}+\frac{11 x e^{2 b^2 x^2+c}}{16 \sqrt{\pi } b^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.228873, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6387, 6384, 2204, 2212} \[ \frac{x^4 e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{x^2 e^{b^2 x^2+c} \text{Erfi}(b x)}{b^4}+\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{b^6}-\frac{43 e^c \text{Erfi}\left (\sqrt{2} b x\right )}{32 \sqrt{2} b^6}-\frac{x^3 e^{2 b^2 x^2+c}}{4 \sqrt{\pi } b^3}+\frac{11 x e^{2 b^2 x^2+c}}{16 \sqrt{\pi } b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6387
Rule 6384
Rule 2204
Rule 2212
Rubi steps
\begin{align*} \int e^{c+b^2 x^2} x^5 \text{erfi}(b x) \, dx &=\frac{e^{c+b^2 x^2} x^4 \text{erfi}(b x)}{2 b^2}-\frac{2 \int e^{c+b^2 x^2} x^3 \text{erfi}(b x) \, dx}{b^2}-\frac{\int e^{c+2 b^2 x^2} x^4 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{c+2 b^2 x^2} x^3}{4 b^3 \sqrt{\pi }}-\frac{e^{c+b^2 x^2} x^2 \text{erfi}(b x)}{b^4}+\frac{e^{c+b^2 x^2} x^4 \text{erfi}(b x)}{2 b^2}+\frac{2 \int e^{c+b^2 x^2} x \text{erfi}(b x) \, dx}{b^4}+\frac{3 \int e^{c+2 b^2 x^2} x^2 \, dx}{4 b^3 \sqrt{\pi }}+\frac{2 \int e^{c+2 b^2 x^2} x^2 \, dx}{b^3 \sqrt{\pi }}\\ &=\frac{11 e^{c+2 b^2 x^2} x}{16 b^5 \sqrt{\pi }}-\frac{e^{c+2 b^2 x^2} x^3}{4 b^3 \sqrt{\pi }}+\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{b^6}-\frac{e^{c+b^2 x^2} x^2 \text{erfi}(b x)}{b^4}+\frac{e^{c+b^2 x^2} x^4 \text{erfi}(b x)}{2 b^2}-\frac{3 \int e^{c+2 b^2 x^2} \, dx}{16 b^5 \sqrt{\pi }}-\frac{\int e^{c+2 b^2 x^2} \, dx}{2 b^5 \sqrt{\pi }}-\frac{2 \int e^{c+2 b^2 x^2} \, dx}{b^5 \sqrt{\pi }}\\ &=\frac{11 e^{c+2 b^2 x^2} x}{16 b^5 \sqrt{\pi }}-\frac{e^{c+2 b^2 x^2} x^3}{4 b^3 \sqrt{\pi }}+\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{b^6}-\frac{e^{c+b^2 x^2} x^2 \text{erfi}(b x)}{b^4}+\frac{e^{c+b^2 x^2} x^4 \text{erfi}(b x)}{2 b^2}-\frac{43 e^c \text{erfi}\left (\sqrt{2} b x\right )}{32 \sqrt{2} b^6}\\ \end{align*}
Mathematica [A] time = 0.0630072, size = 95, normalized size = 0.66 \[ \frac{e^c \left (32 \sqrt{\pi } e^{b^2 x^2} \left (b^4 x^4-2 b^2 x^2+2\right ) \text{Erfi}(b x)-4 b x e^{2 b^2 x^2} \left (4 b^2 x^2-11\right )-43 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} b x\right )\right )}{64 \sqrt{\pi } b^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{b}^{2}{x}^{2}+c}}{x}^{5}{\it erfi} \left ( bx \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.70616, size = 262, normalized size = 1.82 \begin{align*} -\frac{43 \, \sqrt{2} \pi \sqrt{b^{2}} \operatorname{erfi}\left (\sqrt{2} \sqrt{b^{2}} x\right ) e^{c} - 32 \,{\left (\pi b^{5} x^{4} - 2 \, \pi b^{3} x^{2} + 2 \, \pi b\right )} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} + 4 \, \sqrt{\pi }{\left (4 \, b^{4} x^{3} - 11 \, b^{2} x\right )} e^{\left (2 \, b^{2} x^{2} + c\right )}}{64 \, \pi b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]