Optimal. Leaf size=124 \[ -\frac {289}{250} \left (5 x^2+2 x+3\right )^{3/2} x^2+\frac {2149 \left (5 x^2+2 x+3\right )^{3/2} x}{2500}+\frac {7819 \left (5 x^2+2 x+3\right )^{3/2}}{7500}-\frac {4633 (5 x+1) \sqrt {5 x^2+2 x+3}}{12500}-\frac {7}{30} \left (5 x^2+2 x+3\right )^{3/2} x^3-\frac {32431 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{6250 \sqrt {5}} \]
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Rubi [A] time = 0.12, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1661, 640, 612, 619, 215} \[ -\frac {7}{30} \left (5 x^2+2 x+3\right )^{3/2} x^3-\frac {289}{250} \left (5 x^2+2 x+3\right )^{3/2} x^2+\frac {2149 \left (5 x^2+2 x+3\right )^{3/2} x}{2500}+\frac {7819 \left (5 x^2+2 x+3\right )^{3/2}}{7500}-\frac {4633 (5 x+1) \sqrt {5 x^2+2 x+3}}{12500}-\frac {32431 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{6250 \sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx &=-\frac {7}{30} x^3 \left (3+2 x+5 x^2\right )^{3/2}+\frac {1}{30} \int \sqrt {3+2 x+5 x^2} \left (60+390 x+273 x^2-867 x^3\right ) \, dx\\ &=-\frac {289}{250} x^2 \left (3+2 x+5 x^2\right )^{3/2}-\frac {7}{30} x^3 \left (3+2 x+5 x^2\right )^{3/2}+\frac {1}{750} \int \sqrt {3+2 x+5 x^2} \left (1500+14952 x+12894 x^2\right ) \, dx\\ &=\frac {2149 x \left (3+2 x+5 x^2\right )^{3/2}}{2500}-\frac {289}{250} x^2 \left (3+2 x+5 x^2\right )^{3/2}-\frac {7}{30} x^3 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int (-8682+234570 x) \sqrt {3+2 x+5 x^2} \, dx}{15000}\\ &=\frac {7819 \left (3+2 x+5 x^2\right )^{3/2}}{7500}+\frac {2149 x \left (3+2 x+5 x^2\right )^{3/2}}{2500}-\frac {289}{250} x^2 \left (3+2 x+5 x^2\right )^{3/2}-\frac {7}{30} x^3 \left (3+2 x+5 x^2\right )^{3/2}-\frac {4633 \int \sqrt {3+2 x+5 x^2} \, dx}{1250}\\ &=-\frac {4633 (1+5 x) \sqrt {3+2 x+5 x^2}}{12500}+\frac {7819 \left (3+2 x+5 x^2\right )^{3/2}}{7500}+\frac {2149 x \left (3+2 x+5 x^2\right )^{3/2}}{2500}-\frac {289}{250} x^2 \left (3+2 x+5 x^2\right )^{3/2}-\frac {7}{30} x^3 \left (3+2 x+5 x^2\right )^{3/2}-\frac {32431 \int \frac {1}{\sqrt {3+2 x+5 x^2}} \, dx}{6250}\\ &=-\frac {4633 (1+5 x) \sqrt {3+2 x+5 x^2}}{12500}+\frac {7819 \left (3+2 x+5 x^2\right )^{3/2}}{7500}+\frac {2149 x \left (3+2 x+5 x^2\right )^{3/2}}{2500}-\frac {289}{250} x^2 \left (3+2 x+5 x^2\right )^{3/2}-\frac {7}{30} x^3 \left (3+2 x+5 x^2\right )^{3/2}-\frac {\left (4633 \sqrt {\frac {7}{10}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{56}}} \, dx,x,2+10 x\right )}{12500}\\ &=-\frac {4633 (1+5 x) \sqrt {3+2 x+5 x^2}}{12500}+\frac {7819 \left (3+2 x+5 x^2\right )^{3/2}}{7500}+\frac {2149 x \left (3+2 x+5 x^2\right )^{3/2}}{2500}-\frac {289}{250} x^2 \left (3+2 x+5 x^2\right )^{3/2}-\frac {7}{30} x^3 \left (3+2 x+5 x^2\right )^{3/2}-\frac {32431 \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{6250 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 65, normalized size = 0.52 \[ \frac {5 \sqrt {5 x^2+2 x+3} \left (-43750 x^5-234250 x^4+48225 x^3+129895 x^2+105400 x+103386\right )-194586 \sqrt {5} \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{187500} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 77, normalized size = 0.62 \[ -\frac {1}{37500} \, {\left (43750 \, x^{5} + 234250 \, x^{4} - 48225 \, x^{3} - 129895 \, x^{2} - 105400 \, x - 103386\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {32431}{62500} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 72, normalized size = 0.58 \[ -\frac {1}{37500} \, {\left (5 \, {\left ({\left (5 \, {\left (10 \, {\left (175 \, x + 937\right )} x - 1929\right )} x - 25979\right )} x - 21080\right )} x - 103386\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {32431}{31250} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 98, normalized size = 0.79 \[ -\frac {7 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{3}}{30}-\frac {289 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{2}}{250}+\frac {2149 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x}{2500}-\frac {32431 \sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{31250}+\frac {7819 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{7500}-\frac {4633 \left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{25000} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 109, normalized size = 0.88 \[ -\frac {7}{30} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {289}{250} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {2149}{2500} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x + \frac {7819}{7500} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {4633}{2500} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {32431}{31250} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {4633}{12500} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.38, size = 153, normalized size = 1.23 \[ \frac {7819\,\sqrt {5\,x^2+2\,x+3}\,\left (200\,x^2+20\,x+108\right )}{300000}-\frac {7\,x^3\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{30}-\frac {10129\,\sqrt {5}\,\ln \left (\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (5\,x+1\right )}{5}\right )}{62500}-\frac {1447\,\left (\frac {x}{2}+\frac {1}{10}\right )\,\sqrt {5\,x^2+2\,x+3}}{2500}-\frac {289\,x^2\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{250}+\frac {2149\,x\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{2500}-\frac {54733\,\sqrt {5}\,\ln \left (2\,\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (10\,x+2\right )}{5}\right )}{62500} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- 13 x \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 7 x^{2} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int 31 x^{3} \sqrt {5 x^{2} + 2 x + 3}\, dx - \int 7 x^{4} \sqrt {5 x^{2} + 2 x + 3}\, dx - \int \left (- 2 \sqrt {5 x^{2} + 2 x + 3}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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