Optimal. Leaf size=138 \[ -\frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 \sqrt {\pi } b^3}+\frac {3 \sqrt {\pi } e^c \text {erfi}(b x)}{8 b^5}-\frac {3 e^c x^2}{4 \sqrt {\pi } b^3}+\frac {x^3 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}-\frac {3 x e^{b^2 x^2+c} \text {erfc}(b x)}{4 b^4}+\frac {e^c x^4}{4 \sqrt {\pi } b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6386, 6377, 2204, 6376, 12, 30} \[ -\frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 \sqrt {\pi } b^3}+\frac {x^3 e^{b^2 x^2+c} \text {Erfc}(b x)}{2 b^2}-\frac {3 x e^{b^2 x^2+c} \text {Erfc}(b x)}{4 b^4}+\frac {3 \sqrt {\pi } e^c \text {Erfi}(b x)}{8 b^5}-\frac {3 e^c x^2}{4 \sqrt {\pi } b^3}+\frac {e^c x^4}{4 \sqrt {\pi } b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 2204
Rule 6376
Rule 6377
Rule 6386
Rubi steps
\begin {align*} \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx &=\frac {e^{c+b^2 x^2} x^3 \text {erfc}(b x)}{2 b^2}-\frac {3 \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx}{2 b^2}+\frac {\int e^c x^3 \, dx}{b \sqrt {\pi }}\\ &=-\frac {3 e^{c+b^2 x^2} x \text {erfc}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfc}(b x)}{2 b^2}+\frac {3 \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx}{4 b^4}-\frac {3 \int e^c x \, dx}{2 b^3 \sqrt {\pi }}+\frac {e^c \int x^3 \, dx}{b \sqrt {\pi }}\\ &=\frac {e^c x^4}{4 b \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfc}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfc}(b x)}{2 b^2}+\frac {3 \int e^{c+b^2 x^2} \, dx}{4 b^4}-\frac {3 \int e^{c+b^2 x^2} \text {erf}(b x) \, dx}{4 b^4}-\frac {\left (3 e^c\right ) \int x \, dx}{2 b^3 \sqrt {\pi }}\\ &=-\frac {3 e^c x^2}{4 b^3 \sqrt {\pi }}+\frac {e^c x^4}{4 b \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfc}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfc}(b x)}{2 b^2}+\frac {3 e^c \sqrt {\pi } \text {erfi}(b x)}{8 b^5}-\frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 b^3 \sqrt {\pi }}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.30, size = 147, normalized size = 1.07 \[ -\frac {e^c \left (-6 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )-2 b^4 x^4+2 \sqrt {\pi } b x e^{b^2 x^2} \left (2 b^2 x^2-3\right ) \text {erf}(b x)+6 b^2 x^2+6 \sqrt {\pi } b x e^{b^2 x^2}-4 \sqrt {\pi } b^3 x^3 e^{b^2 x^2}+3 \pi \text {erf}(b x) \text {erfi}(b x)-3 \pi \text {erfi}(b x)\right )}{8 \sqrt {\pi } b^5} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (x^{4} \operatorname {erf}\left (b x\right ) - x^{4}\right )} e^{\left (b^{2} x^{2} + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b^{2} x^{2}+c} x^{4} \mathrm {erfc}\left (b x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________