Optimal. Leaf size=148 \[ \frac {15 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{8 \sqrt {\pi } b^5}+\frac {15 x^2}{8 \sqrt {\pi } b^5}+\frac {5 x^4}{8 \sqrt {\pi } b^3}-\frac {x^5 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {15 x e^{-b^2 x^2} \text {erfi}(b x)}{8 b^6}-\frac {5 x^3 e^{-b^2 x^2} \text {erfi}(b x)}{4 b^4}+\frac {x^6}{6 \sqrt {\pi } b} \]
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Rubi [A] time = 0.14, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6387, 6378, 30} \[ \frac {15 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{8 \sqrt {\pi } b^5}-\frac {x^5 e^{-b^2 x^2} \text {Erfi}(b x)}{2 b^2}-\frac {5 x^3 e^{-b^2 x^2} \text {Erfi}(b x)}{4 b^4}-\frac {15 x e^{-b^2 x^2} \text {Erfi}(b x)}{8 b^6}+\frac {5 x^4}{8 \sqrt {\pi } b^3}+\frac {15 x^2}{8 \sqrt {\pi } b^5}+\frac {x^6}{6 \sqrt {\pi } b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6378
Rule 6387
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x^6 \text {erfi}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^5 \text {erfi}(b x)}{2 b^2}+\frac {5 \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx}{2 b^2}+\frac {\int x^5 \, dx}{b \sqrt {\pi }}\\ &=\frac {x^6}{6 b \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3 \text {erfi}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^5 \text {erfi}(b x)}{2 b^2}+\frac {15 \int e^{-b^2 x^2} x^2 \text {erfi}(b x) \, dx}{4 b^4}+\frac {5 \int x^3 \, dx}{2 b^3 \sqrt {\pi }}\\ &=\frac {5 x^4}{8 b^3 \sqrt {\pi }}+\frac {x^6}{6 b \sqrt {\pi }}-\frac {15 e^{-b^2 x^2} x \text {erfi}(b x)}{8 b^6}-\frac {5 e^{-b^2 x^2} x^3 \text {erfi}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^5 \text {erfi}(b x)}{2 b^2}+\frac {15 \int e^{-b^2 x^2} \text {erfi}(b x) \, dx}{8 b^6}+\frac {15 \int x \, dx}{4 b^5 \sqrt {\pi }}\\ &=\frac {15 x^2}{8 b^5 \sqrt {\pi }}+\frac {5 x^4}{8 b^3 \sqrt {\pi }}+\frac {x^6}{6 b \sqrt {\pi }}-\frac {15 e^{-b^2 x^2} x \text {erfi}(b x)}{8 b^6}-\frac {5 e^{-b^2 x^2} x^3 \text {erfi}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^5 \text {erfi}(b x)}{2 b^2}+\frac {15 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{8 b^5 \sqrt {\pi }}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 0.35 \[ \frac {x^2 \left (-9 \, _2F_2\left (1,1;-\frac {3}{2},2;-b^2 x^2\right )+4 b^4 x^4+3 b^2 x^2+9\right )}{24 \sqrt {\pi } b^5} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{6} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{6} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int x^{6} \erfi \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{6} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^6\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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