Optimal. Leaf size=166 \[ -\frac {77509 \left (5 x^2+2 x+3\right )^{3/2} x^2}{25000}+\frac {1781669 \left (5 x^2+2 x+3\right )^{3/2} x}{250000}+\frac {198439 \left (5 x^2+2 x+3\right )^{3/2}}{750000}-\frac {2521723 (5 x+1) \sqrt {5 x^2+2 x+3}}{1250000}+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5+\frac {989}{200} \left (5 x^2+2 x+3\right )^{3/2} x^4-\frac {25277 \left (5 x^2+2 x+3\right )^{3/2} x^3}{3000}-\frac {17652061 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{625000 \sqrt {5}} \]
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Rubi [A] time = 0.20, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1661, 640, 612, 619, 215} \[ \frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5+\frac {989}{200} \left (5 x^2+2 x+3\right )^{3/2} x^4-\frac {25277 \left (5 x^2+2 x+3\right )^{3/2} x^3}{3000}-\frac {77509 \left (5 x^2+2 x+3\right )^{3/2} x^2}{25000}+\frac {1781669 \left (5 x^2+2 x+3\right )^{3/2} x}{250000}+\frac {198439 \left (5 x^2+2 x+3\right )^{3/2}}{750000}-\frac {2521723 (5 x+1) \sqrt {5 x^2+2 x+3}}{1250000}-\frac {17652061 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{625000 \sqrt {5}} \]
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 )^2 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx &=\frac {49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}+\frac {1}{40} \int \sqrt {3+2 x+5 x^2} \left (80+840 x+1800 x^2-3760 x^3-7935 x^4+6923 x^5\right ) \, dx\\ &=\frac {989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac {49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int \sqrt {3+2 x+5 x^2} \left (2800+29400 x+63000 x^2-214676 x^3-353878 x^4\right ) \, dx}{1400}\\ &=-\frac {25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac {989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac {49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int \sqrt {3+2 x+5 x^2} \left (84000+882000 x+5074902 x^2-3255378 x^3\right ) \, dx}{42000}\\ &=-\frac {77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac {25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac {989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac {49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int \sqrt {3+2 x+5 x^2} \left (2100000+41582268 x+149660196 x^2\right ) \, dx}{1050000}\\ &=\frac {1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac {77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac {25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac {989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac {49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int (-406980588+83344380 x) \sqrt {3+2 x+5 x^2} \, dx}{21000000}\\ &=\frac {198439 \left (3+2 x+5 x^2\right )^{3/2}}{750000}+\frac {1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac {77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac {25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac {989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac {49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}-\frac {2521723 \int \sqrt {3+2 x+5 x^2} \, dx}{125000}\\ &=-\frac {2521723 (1+5 x) \sqrt {3+2 x+5 x^2}}{1250000}+\frac {198439 \left (3+2 x+5 x^2\right )^{3/2}}{750000}+\frac {1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac {77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac {25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac {989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac {49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}-\frac {17652061 \int \frac {1}{\sqrt {3+2 x+5 x^2}} \, dx}{625000}\\ &=-\frac {2521723 (1+5 x) \sqrt {3+2 x+5 x^2}}{1250000}+\frac {198439 \left (3+2 x+5 x^2\right )^{3/2}}{750000}+\frac {1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac {77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac {25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac {989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac {49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}-\frac {\left (2521723 \sqrt {\frac {7}{10}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{56}}} \, dx,x,2+10 x\right )}{1250000}\\ &=-\frac {2521723 (1+5 x) \sqrt {3+2 x+5 x^2}}{1250000}+\frac {198439 \left (3+2 x+5 x^2\right )^{3/2}}{750000}+\frac {1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac {77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac {25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac {989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac {49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}-\frac {17652061 \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{625000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 75, normalized size = 0.45 \[ \frac {5 \sqrt {5 x^2+2 x+3} \left (22968750 x^7+101906250 x^6-107112500 x^5-65693000 x^4+15583725 x^3+23531995 x^2+44333650 x-4588584\right )-105912366 \sqrt {5} \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{18750000} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 87, normalized size = 0.52 \[ \frac {1}{3750000} \, {\left (22968750 \, x^{7} + 101906250 \, x^{6} - 107112500 \, x^{5} - 65693000 \, x^{4} + 15583725 \, x^{3} + 23531995 \, x^{2} + 44333650 \, x - 4588584\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {17652061}{6250000} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 82, normalized size = 0.49 \[ \frac {1}{3750000} \, {\left (5 \, {\left ({\left (5 \, {\left (10 \, {\left (25 \, {\left (15 \, {\left (245 \, x + 1087\right )} x - 17138\right )} x - 262772\right )} x + 623349\right )} x + 4706399\right )} x + 8866730\right )} x - 4588584\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {17652061}{3125000} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 132, normalized size = 0.80 \[ \frac {49 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{5}}{40}+\frac {989 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{4}}{200}-\frac {25277 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{3}}{3000}-\frac {77509 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{2}}{25000}+\frac {1781669 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x}{250000}-\frac {17652061 \sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{3125000}+\frac {198439 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{750000}-\frac {2521723 \left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{2500000} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 143, normalized size = 0.86 \[ \frac {49}{40} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{5} + \frac {989}{200} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {25277}{3000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {77509}{25000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {1781669}{250000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x + \frac {198439}{750000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {2521723}{250000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {17652061}{3125000} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {2521723}{1250000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.01, size = 187, normalized size = 1.13 \[ \frac {989\,x^4\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{200}-\frac {25277\,x^3\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{3000}-\frac {77509\,x^2\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{25000}+\frac {49\,x^5\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{40}-\frac {33915049\,\sqrt {5}\,\ln \left (\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (5\,x+1\right )}{5}\right )}{6250000}-\frac {4845007\,\left (\frac {x}{2}+\frac {1}{10}\right )\,\sqrt {5\,x^2+2\,x+3}}{250000}+\frac {198439\,\sqrt {5\,x^2+2\,x+3}\,\left (200\,x^2+20\,x+108\right )}{30000000}+\frac {1781669\,x\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{250000}-\frac {1389073\,\sqrt {5}\,\ln \left (2\,\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (10\,x+2\right )}{5}\right )}{6250000} \]
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 \[ \int \left (x^{2} + 5 x + 2\right ) \sqrt {5 x^{2} + 2 x + 3} \left (7 x^{2} - 4 x - 1\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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