Optimal. Leaf size=25 \[ -1+\left (e^{5+x^2 \log (7)}+x\right ) \left (5-\frac {1}{2 x}+x\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.34, antiderivative size = 129, normalized size of antiderivative = 5.16, number of steps
used = 11, number of rules used = 7, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.127, Rules used = {12, 14, 6874,
2245, 2235, 2240, 2243} \begin {gather*} \frac {1}{2} e^5 \sqrt {\pi \log (7)} \text {Erfi}\left (x \sqrt {\log (7)}\right )-\frac {1}{2} e^5 \sqrt {\frac {\pi }{\log (7)}} \text {Erfi}\left (x \sqrt {\log (7)}\right )+\frac {1}{2} e^5 (1-\log (7)) \sqrt {\frac {\pi }{\log (7)}} \text {Erfi}\left (x \sqrt {\log (7)}\right )+e^5 7^{x^2} x-\frac {e^5 7^{x^2}}{2 x}+5 e^5 7^{x^2}+\frac {1}{4} (2 x+5)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2235
Rule 2240
Rule 2243
Rule 2245
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} \left (1+2 x^2+\left (-2 x^2+20 x^3+4 x^4\right ) \log (7)\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \left (2 (5+2 x)+\frac {7^{x^2} e^5 \left (1+2 x^2 (1-\log (7))+20 x^3 \log (7)+4 x^4 \log (7)\right )}{x^2}\right ) \, dx\\ &=\frac {1}{4} (5+2 x)^2+\frac {1}{2} e^5 \int \frac {7^{x^2} \left (1+2 x^2 (1-\log (7))+20 x^3 \log (7)+4 x^4 \log (7)\right )}{x^2} \, dx\\ &=\frac {1}{4} (5+2 x)^2+\frac {1}{2} e^5 \int \left (\frac {7^{x^2}}{x^2}-2\ 7^{x^2} (-1+\log (7))+20\ 7^{x^2} x \log (7)+4\ 7^{x^2} x^2 \log (7)\right ) \, dx\\ &=\frac {1}{4} (5+2 x)^2+\frac {1}{2} e^5 \int \frac {7^{x^2}}{x^2} \, dx+\left (e^5 (1-\log (7))\right ) \int 7^{x^2} \, dx+\left (2 e^5 \log (7)\right ) \int 7^{x^2} x^2 \, dx+\left (10 e^5 \log (7)\right ) \int 7^{x^2} x \, dx\\ &=5\ 7^{x^2} e^5-\frac {7^{x^2} e^5}{2 x}+7^{x^2} e^5 x+\frac {1}{4} (5+2 x)^2+\frac {1}{2} e^5 \text {erfi}\left (x \sqrt {\log (7)}\right ) (1-\log (7)) \sqrt {\frac {\pi }{\log (7)}}-e^5 \int 7^{x^2} \, dx+\left (e^5 \log (7)\right ) \int 7^{x^2} \, dx\\ &=5\ 7^{x^2} e^5-\frac {7^{x^2} e^5}{2 x}+7^{x^2} e^5 x+\frac {1}{4} (5+2 x)^2-\frac {1}{2} e^5 \text {erfi}\left (x \sqrt {\log (7)}\right ) \sqrt {\frac {\pi }{\log (7)}}+\frac {1}{2} e^5 \text {erfi}\left (x \sqrt {\log (7)}\right ) (1-\log (7)) \sqrt {\frac {\pi }{\log (7)}}+\frac {1}{2} e^5 \text {erfi}\left (x \sqrt {\log (7)}\right ) \sqrt {\pi \log (7)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 29, normalized size = 1.16 \begin {gather*} 5 x+x^2+\frac {1}{2} 7^{x^2} e^5 \left (10-\frac {1}{x}+2 x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 44, normalized size = 1.76
method | result | size |
risch | \(x^{2}+5 x +\frac {\left (2 x^{2}+10 x -1\right ) 7^{x^{2}} {\mathrm e}^{5}}{2 x}\) | \(30\) |
default | \(x \,{\mathrm e}^{x^{2} \ln \left (7\right )+5}+5 \,{\mathrm e}^{x^{2} \ln \left (7\right )+5}-\frac {{\mathrm e}^{x^{2} \ln \left (7\right )+5}}{2 x}+x^{2}+5 x\) | \(44\) |
norman | \(\frac {x^{3}+{\mathrm e}^{x^{2} \ln \left (7\right )+5} x^{2}+5 x^{2}+5 x \,{\mathrm e}^{x^{2} \ln \left (7\right )+5}-\frac {{\mathrm e}^{x^{2} \ln \left (7\right )+5}}{2}}{x}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.52, size = 135, normalized size = 5.40 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {-\log \left (7\right )}\right ) e^{5} \log \left (7\right )}{2 \, \sqrt {-\log \left (7\right )}} + x^{2} + \frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {-\log \left (7\right )}\right ) e^{5}}{2 \, \sqrt {-\log \left (7\right )}} - \frac {1}{2} \, {\left (\frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {-\log \left (7\right )}\right ) e^{5}}{\sqrt {-\log \left (7\right )} \log \left (7\right )} - \frac {2 \, x e^{\left (x^{2} \log \left (7\right ) + 5\right )}}{\log \left (7\right )}\right )} \log \left (7\right ) - \frac {\sqrt {-x^{2} \log \left (7\right )} e^{5} \Gamma \left (-\frac {1}{2}, -x^{2} \log \left (7\right )\right )}{4 \, x} + 5 \, x + 5 \, e^{\left (x^{2} \log \left (7\right ) + 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 36, normalized size = 1.44 \begin {gather*} \frac {2 \, x^{3} + 10 \, x^{2} + {\left (2 \, x^{2} + 10 \, x - 1\right )} e^{\left (x^{2} \log \left (7\right ) + 5\right )}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 29, normalized size = 1.16 \begin {gather*} x^{2} + 5 x + \frac {\left (2 x^{2} + 10 x - 1\right ) e^{x^{2} \log {\left (7 \right )} + 5}}{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 47, normalized size = 1.88 \begin {gather*} \frac {2 \cdot 7^{\left (x^{2}\right )} x^{2} e^{5} + 2 \, x^{3} + 10 \cdot 7^{\left (x^{2}\right )} x e^{5} + 10 \, x^{2} - 7^{\left (x^{2}\right )} e^{5}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.08, size = 37, normalized size = 1.48 \begin {gather*} x\,\left (7^{x^2}\,{\mathrm {e}}^5+5\right )+x^2+5\,7^{x^2}\,{\mathrm {e}}^5-\frac {7^{x^2}\,{\mathrm {e}}^5}{2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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