Optimal. Leaf size=208 \[ \frac {98060877 \left (5 x^2+2 x+3\right )^{3/2} x^2}{4375000}+\frac {1045360143 \left (5 x^2+2 x+3\right )^{3/2} x}{43750000}-\frac {1968340667 \left (5 x^2+2 x+3\right )^{3/2}}{131250000}-\frac {77159983 (5 x+1) \sqrt {5 x^2+2 x+3}}{31250000}-\frac {343}{50} \left (5 x^2+2 x+3\right )^{3/2} x^7-\frac {50519 \left (5 x^2+2 x+3\right )^{3/2} x^6}{2250}+\frac {190939 \left (5 x^2+2 x+3\right )^{3/2} x^5}{3000}-\frac {888751 \left (5 x^2+2 x+3\right )^{3/2} x^4}{105000}-\frac {90960857 \left (5 x^2+2 x+3\right )^{3/2} x^3}{1575000}-\frac {540119881 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{15625000 \sqrt {5}} \]
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Rubi [A] time = 0.35, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ -\frac {343}{50} \left (5 x^2+2 x+3\right )^{3/2} x^7-\frac {50519 \left (5 x^2+2 x+3\right )^{3/2} x^6}{2250}+\frac {190939 \left (5 x^2+2 x+3\right )^{3/2} x^5}{3000}-\frac {888751 \left (5 x^2+2 x+3\right )^{3/2} x^4}{105000}-\frac {90960857 \left (5 x^2+2 x+3\right )^{3/2} x^3}{1575000}+\frac {98060877 \left (5 x^2+2 x+3\right )^{3/2} x^2}{4375000}+\frac {1045360143 \left (5 x^2+2 x+3\right )^{3/2} x}{43750000}-\frac {1968340667 \left (5 x^2+2 x+3\right )^{3/2}}{131250000}-\frac {77159983 (5 x+1) \sqrt {5 x^2+2 x+3}}{31250000}-\frac {540119881 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{15625000 \sqrt {5}} \]
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 )^3 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx &=-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac {1}{50} \int \sqrt {3+2 x+5 x^2} \left (100+1450 x+5750 x^2-3050 x^3-43550 x^4+6350 x^5+110453 x^6-50519 x^7\right ) \, dx\\ &=-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int \sqrt {3+2 x+5 x^2} \left (4500+65250 x+258750 x^2-137250 x^3-1959750 x^4+1195092 x^5+5728170 x^6\right ) \, dx}{2250}\\ &=\frac {190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int \sqrt {3+2 x+5 x^2} \left (180000+2610000 x+10350000 x^2-5490000 x^3-164312550 x^4-26662530 x^5\right ) \, dx}{90000}\\ &=-\frac {888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac {190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int \sqrt {3+2 x+5 x^2} \left (6300000+91350000 x+362250000 x^2+127800360 x^3-5457651420 x^4\right ) \, dx}{3150000}\\ &=-\frac {90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac {888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac {190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int \sqrt {3+2 x+5 x^2} \left (189000000+2740500000 x+59986362780 x^2+52952873580 x^3\right ) \, dx}{94500000}\\ &=\frac {98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac {90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac {888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac {190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int \sqrt {3+2 x+5 x^2} \left (4725000000-249204741480 x+1128988954440 x^2\right ) \, dx}{2362500000}\\ &=\frac {1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac {98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac {90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac {888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac {190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}+\frac {\int (-3292466863320-10629039601800 x) \sqrt {3+2 x+5 x^2} \, dx}{47250000000}\\ &=-\frac {1968340667 \left (3+2 x+5 x^2\right )^{3/2}}{131250000}+\frac {1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac {98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac {90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac {888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac {190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}-\frac {77159983 \int \sqrt {3+2 x+5 x^2} \, dx}{3125000}\\ &=-\frac {77159983 (1+5 x) \sqrt {3+2 x+5 x^2}}{31250000}-\frac {1968340667 \left (3+2 x+5 x^2\right )^{3/2}}{131250000}+\frac {1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac {98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac {90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac {888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac {190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}-\frac {540119881 \int \frac {1}{\sqrt {3+2 x+5 x^2}} \, dx}{15625000}\\ &=-\frac {77159983 (1+5 x) \sqrt {3+2 x+5 x^2}}{31250000}-\frac {1968340667 \left (3+2 x+5 x^2\right )^{3/2}}{131250000}+\frac {1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac {98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac {90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac {888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac {190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}-\frac {\left (77159983 \sqrt {\frac {7}{10}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{56}}} \, dx,x,2+10 x\right )}{31250000}\\ &=-\frac {77159983 (1+5 x) \sqrt {3+2 x+5 x^2}}{31250000}-\frac {1968340667 \left (3+2 x+5 x^2\right )^{3/2}}{131250000}+\frac {1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac {98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac {90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac {888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac {190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}-\frac {540119881 \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{15625000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 85, normalized size = 0.41 \[ \frac {-5 \sqrt {5 x^2+2 x+3} \left (67528125000 x^9+248031875000 x^8-497593468750 x^7-34674656250 x^6+225922362500 x^5+56757413000 x^4+17642392275 x^3-78839046795 x^2-57768004650 x+93436408944\right )-68055105006 \sqrt {5} \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{9843750000} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 97, normalized size = 0.47 \[ -\frac {1}{1968750000} \, {\left (67528125000 \, x^{9} + 248031875000 \, x^{8} - 497593468750 \, x^{7} - 34674656250 \, x^{6} + 225922362500 \, x^{5} + 56757413000 \, x^{4} + 17642392275 \, x^{3} - 78839046795 \, x^{2} - 57768004650 \, x + 93436408944\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {540119881}{156250000} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 92, normalized size = 0.44 \[ -\frac {1}{1968750000} \, {\left (5 \, {\left ({\left (5 \, {\left (10 \, {\left (25 \, {\left (5 \, {\left (49 \, {\left (140 \, {\left (315 \, x + 1157\right )} x - 324959\right )} x - 1109589\right )} x + 36147578\right )} x + 227029652\right )} x + 705695691\right )} x - 15767809359\right )} x - 11553600930\right )} x + 93436408944\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {540119881}{78125000} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 166, normalized size = 0.80 \[ -\frac {343 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{7}}{50}-\frac {50519 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{6}}{2250}+\frac {190939 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{5}}{3000}-\frac {888751 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{4}}{105000}-\frac {90960857 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{3}}{1575000}+\frac {98060877 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x^{2}}{4375000}+\frac {1045360143 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}} x}{43750000}-\frac {540119881 \sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{78125000}-\frac {1968340667 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{131250000}-\frac {77159983 \left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{62500000} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 177, normalized size = 0.85 \[ -\frac {343}{50} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{7} - \frac {50519}{2250} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{6} + \frac {190939}{3000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{5} - \frac {888751}{105000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {90960857}{1575000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {98060877}{4375000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {1045360143}{43750000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x - \frac {1968340667}{131250000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {77159983}{6250000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {540119881}{78125000} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {77159983}{31250000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.31, size = 221, normalized size = 1.06 \[ \frac {98060877\,x^2\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{4375000}-\frac {90960857\,x^3\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{1575000}-\frac {888751\,x^4\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{105000}+\frac {190939\,x^5\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{3000}-\frac {50519\,x^6\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{2250}-\frac {343\,x^7\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{50}-\frac {3048580429\,\sqrt {5}\,\ln \left (\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (5\,x+1\right )}{5}\right )}{156250000}-\frac {3048580429\,\left (\frac {x}{2}+\frac {1}{10}\right )\,\sqrt {5\,x^2+2\,x+3}}{43750000}-\frac {1968340667\,\sqrt {5\,x^2+2\,x+3}\,\left (200\,x^2+20\,x+108\right )}{5250000000}+\frac {1045360143\,x\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{43750000}+\frac {1968340667\,\sqrt {5}\,\ln \left (2\,\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (10\,x+2\right )}{5}\right )}{156250000} \]
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 \[ - \int \left (- 29 x \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 115 x^{2} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int 61 x^{3} \sqrt {5 x^{2} + 2 x + 3}\, dx - \int 871 x^{4} \sqrt {5 x^{2} + 2 x + 3}\, dx - \int \left (- 127 x^{5} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 2065 x^{6} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int 1127 x^{7} \sqrt {5 x^{2} + 2 x + 3}\, dx - \int 343 x^{8} \sqrt {5 x^{2} + 2 x + 3}\, dx - \int \left (- 2 \sqrt {5 x^{2} + 2 x + 3}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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