Optimal. Leaf size=27 \[ -1+e^{x^2}+x^2+\frac {2 x}{3 \left (-5-x+2 x^2\right )} \]
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Rubi [B] time = 0.54, antiderivative size = 310, normalized size of antiderivative = 11.48, number of steps used = 29, number of rules used = 12, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6742, 2209, 614, 618, 206, 638, 722, 738, 773, 632, 31, 800} \begin {gather*} \frac {4 x^3}{41}-\frac {38 (x+10) x^2}{41 \left (-2 x^2+x+5\right )}+\frac {79 x^2}{41}+\frac {56 (x+10) x}{123 \left (-2 x^2+x+5\right )}+e^{x^2}+\frac {10 (1-4 x)}{123 \left (-2 x^2+x+5\right )}+\frac {50 (x+10)}{41 \left (-2 x^2+x+5\right )}+\frac {8 (x+10) x^4}{41 \left (-2 x^2+x+5\right )}-\frac {8 (x+10) x^3}{41 \left (-2 x^2+x+5\right )}-\frac {30 x}{41}-\frac {19 \left (1681-61 \sqrt {41}\right ) \log \left (-4 x-\sqrt {41}+1\right )}{6724}-\frac {\left (1681-661 \sqrt {41}\right ) \log \left (-4 x-\sqrt {41}+1\right )}{1681}+\frac {\left (38663-3203 \sqrt {41}\right ) \log \left (-4 x-\sqrt {41}+1\right )}{6724}+\frac {\left (38663+3203 \sqrt {41}\right ) \log \left (-4 x+\sqrt {41}+1\right )}{6724}-\frac {\left (1681+661 \sqrt {41}\right ) \log \left (-4 x+\sqrt {41}+1\right )}{1681}-\frac {19 \left (1681+61 \sqrt {41}\right ) \log \left (-4 x+\sqrt {41}+1\right )}{6724}+\frac {300 \tanh ^{-1}\left (\frac {1-4 x}{\sqrt {41}}\right )}{41 \sqrt {41}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 614
Rule 618
Rule 632
Rule 638
Rule 722
Rule 738
Rule 773
Rule 800
Rule 2209
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 e^{x^2} x-\frac {10}{3 \left (-5-x+2 x^2\right )^2}+\frac {50 x}{\left (-5-x+2 x^2\right )^2}+\frac {56 x^2}{3 \left (-5-x+2 x^2\right )^2}-\frac {38 x^3}{\left (-5-x+2 x^2\right )^2}-\frac {8 x^4}{\left (-5-x+2 x^2\right )^2}+\frac {8 x^5}{\left (-5-x+2 x^2\right )^2}\right ) \, dx\\ &=2 \int e^{x^2} x \, dx-\frac {10}{3} \int \frac {1}{\left (-5-x+2 x^2\right )^2} \, dx-8 \int \frac {x^4}{\left (-5-x+2 x^2\right )^2} \, dx+8 \int \frac {x^5}{\left (-5-x+2 x^2\right )^2} \, dx+\frac {56}{3} \int \frac {x^2}{\left (-5-x+2 x^2\right )^2} \, dx-38 \int \frac {x^3}{\left (-5-x+2 x^2\right )^2} \, dx+50 \int \frac {x}{\left (-5-x+2 x^2\right )^2} \, dx\\ &=e^{x^2}+\frac {10 (1-4 x)}{123 \left (5+x-2 x^2\right )}+\frac {50 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {56 x (10+x)}{123 \left (5+x-2 x^2\right )}-\frac {38 x^2 (10+x)}{41 \left (5+x-2 x^2\right )}-\frac {8 x^3 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8 x^4 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8}{41} \int \frac {(-30-2 x) x^2}{-5-x+2 x^2} \, dx-\frac {8}{41} \int \frac {(-40-3 x) x^3}{-5-x+2 x^2} \, dx+\frac {40}{123} \int \frac {1}{-5-x+2 x^2} \, dx+\frac {38}{41} \int \frac {(-20-x) x}{-5-x+2 x^2} \, dx-\frac {50}{41} \int \frac {1}{-5-x+2 x^2} \, dx+\frac {560}{123} \int \frac {1}{-5-x+2 x^2} \, dx\\ &=e^{x^2}-\frac {19 x}{41}+\frac {10 (1-4 x)}{123 \left (5+x-2 x^2\right )}+\frac {50 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {56 x (10+x)}{123 \left (5+x-2 x^2\right )}-\frac {38 x^2 (10+x)}{41 \left (5+x-2 x^2\right )}-\frac {8 x^3 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8 x^4 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8}{41} \int \left (-\frac {31}{2}-x-\frac {155+41 x}{2 \left (-5-x+2 x^2\right )}\right ) \, dx-\frac {8}{41} \int \left (-\frac {113}{8}-\frac {83 x}{4}-\frac {3 x^2}{2}-\frac {565+943 x}{8 \left (-5-x+2 x^2\right )}\right ) \, dx+\frac {19}{41} \int \frac {-5-41 x}{-5-x+2 x^2} \, dx-\frac {80}{123} \operatorname {Subst}\left (\int \frac {1}{41-x^2} \, dx,x,-1+4 x\right )+\frac {100}{41} \operatorname {Subst}\left (\int \frac {1}{41-x^2} \, dx,x,-1+4 x\right )-\frac {1120}{123} \operatorname {Subst}\left (\int \frac {1}{41-x^2} \, dx,x,-1+4 x\right )\\ &=e^{x^2}-\frac {30 x}{41}+\frac {79 x^2}{41}+\frac {4 x^3}{41}+\frac {10 (1-4 x)}{123 \left (5+x-2 x^2\right )}+\frac {50 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {56 x (10+x)}{123 \left (5+x-2 x^2\right )}-\frac {38 x^2 (10+x)}{41 \left (5+x-2 x^2\right )}-\frac {8 x^3 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8 x^4 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {300 \tanh ^{-1}\left (\frac {1-4 x}{\sqrt {41}}\right )}{41 \sqrt {41}}+\frac {1}{41} \int \frac {565+943 x}{-5-x+2 x^2} \, dx-\frac {4}{41} \int \frac {155+41 x}{-5-x+2 x^2} \, dx-\frac {\left (19 \left (1681-61 \sqrt {41}\right )\right ) \int \frac {1}{-\frac {1}{2}+\frac {\sqrt {41}}{2}+2 x} \, dx}{3362}-\frac {\left (19 \left (1681+61 \sqrt {41}\right )\right ) \int \frac {1}{-\frac {1}{2}-\frac {\sqrt {41}}{2}+2 x} \, dx}{3362}\\ &=e^{x^2}-\frac {30 x}{41}+\frac {79 x^2}{41}+\frac {4 x^3}{41}+\frac {10 (1-4 x)}{123 \left (5+x-2 x^2\right )}+\frac {50 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {56 x (10+x)}{123 \left (5+x-2 x^2\right )}-\frac {38 x^2 (10+x)}{41 \left (5+x-2 x^2\right )}-\frac {8 x^3 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8 x^4 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {300 \tanh ^{-1}\left (\frac {1-4 x}{\sqrt {41}}\right )}{41 \sqrt {41}}-\frac {19 \left (1681-61 \sqrt {41}\right ) \log \left (1-\sqrt {41}-4 x\right )}{6724}-\frac {19 \left (1681+61 \sqrt {41}\right ) \log \left (1+\sqrt {41}-4 x\right )}{6724}+\frac {\left (38663-3203 \sqrt {41}\right ) \int \frac {1}{-\frac {1}{2}+\frac {\sqrt {41}}{2}+2 x} \, dx}{3362}-\frac {\left (2 \left (1681-661 \sqrt {41}\right )\right ) \int \frac {1}{-\frac {1}{2}+\frac {\sqrt {41}}{2}+2 x} \, dx}{1681}-\frac {\left (2 \left (1681+661 \sqrt {41}\right )\right ) \int \frac {1}{-\frac {1}{2}-\frac {\sqrt {41}}{2}+2 x} \, dx}{1681}+\frac {\left (38663+3203 \sqrt {41}\right ) \int \frac {1}{-\frac {1}{2}-\frac {\sqrt {41}}{2}+2 x} \, dx}{3362}\\ &=e^{x^2}-\frac {30 x}{41}+\frac {79 x^2}{41}+\frac {4 x^3}{41}+\frac {10 (1-4 x)}{123 \left (5+x-2 x^2\right )}+\frac {50 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {56 x (10+x)}{123 \left (5+x-2 x^2\right )}-\frac {38 x^2 (10+x)}{41 \left (5+x-2 x^2\right )}-\frac {8 x^3 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8 x^4 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {300 \tanh ^{-1}\left (\frac {1-4 x}{\sqrt {41}}\right )}{41 \sqrt {41}}+\frac {\left (38663-3203 \sqrt {41}\right ) \log \left (1-\sqrt {41}-4 x\right )}{6724}-\frac {\left (1681-661 \sqrt {41}\right ) \log \left (1-\sqrt {41}-4 x\right )}{1681}-\frac {19 \left (1681-61 \sqrt {41}\right ) \log \left (1-\sqrt {41}-4 x\right )}{6724}-\frac {19 \left (1681+61 \sqrt {41}\right ) \log \left (1+\sqrt {41}-4 x\right )}{6724}-\frac {\left (1681+661 \sqrt {41}\right ) \log \left (1+\sqrt {41}-4 x\right )}{1681}+\frac {\left (38663+3203 \sqrt {41}\right ) \log \left (1+\sqrt {41}-4 x\right )}{6724}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.29, size = 35, normalized size = 1.30 \begin {gather*} \frac {2}{3} \left (\frac {3 e^{x^2}}{2}+\frac {3 x^2}{2}+\frac {x}{-5-x+2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 49, normalized size = 1.81 \begin {gather*} \frac {6 \, x^{4} - 3 \, x^{3} - 15 \, x^{2} + 3 \, {\left (2 \, x^{2} - x - 5\right )} e^{\left (x^{2}\right )} + 2 \, x}{3 \, {\left (2 \, x^{2} - x - 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 55, normalized size = 2.04 \begin {gather*} \frac {6 \, x^{4} - 3 \, x^{3} + 6 \, x^{2} e^{\left (x^{2}\right )} - 15 \, x^{2} - 3 \, x e^{\left (x^{2}\right )} + 2 \, x - 15 \, e^{\left (x^{2}\right )}}{3 \, {\left (2 \, x^{2} - x - 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 22, normalized size = 0.81
method | result | size |
risch | \(x^{2}+\frac {x}{3 x^{2}-\frac {3}{2} x -\frac {15}{2}}+{\mathrm e}^{x^{2}}\) | \(22\) |
norman | \(\frac {-\frac {11 x^{2}}{3}-x^{3}+2 x^{4}+2 x^{2} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}} x -5 \,{\mathrm e}^{x^{2}}-\frac {10}{3}}{2 x^{2}-x -5}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 120, normalized size = 4.44 \begin {gather*} x^{2} - \frac {551 \, x + 1205}{82 \, {\left (2 \, x^{2} - x - 5\right )}} + \frac {241 \, x + 155}{41 \, {\left (2 \, x^{2} - x - 5\right )}} + \frac {19 \, {\left (31 \, x + 105\right )}}{82 \, {\left (2 \, x^{2} - x - 5\right )}} - \frac {28 \, {\left (21 \, x + 5\right )}}{123 \, {\left (2 \, x^{2} - x - 5\right )}} + \frac {10 \, {\left (4 \, x - 1\right )}}{123 \, {\left (2 \, x^{2} - x - 5\right )}} - \frac {50 \, {\left (x + 10\right )}}{41 \, {\left (2 \, x^{2} - x - 5\right )}} + e^{\left (x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.50, size = 23, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^{x^2}-\frac {2\,x}{3\,\left (-2\,x^2+x+5\right )}+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 20, normalized size = 0.74 \begin {gather*} x^{2} + \frac {2 x}{6 x^{2} - 3 x - 15} + e^{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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