Optimal. Leaf size=208 \[ -\frac{343}{50} \left (5 x^2+2 x+3\right )^{3/2} x^7-\frac{50519 \left (5 x^2+2 x+3\right )^{3/2} x^6}{2250}+\frac{190939 \left (5 x^2+2 x+3\right )^{3/2} x^5}{3000}-\frac{888751 \left (5 x^2+2 x+3\right )^{3/2} x^4}{105000}-\frac{90960857 \left (5 x^2+2 x+3\right )^{3/2} x^3}{1575000}+\frac{98060877 \left (5 x^2+2 x+3\right )^{3/2} x^2}{4375000}+\frac{1045360143 \left (5 x^2+2 x+3\right )^{3/2} x}{43750000}-\frac{1968340667 \left (5 x^2+2 x+3\right )^{3/2}}{131250000}-\frac{77159983 (5 x+1) \sqrt{5 x^2+2 x+3}}{31250000}-\frac{540119881 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{15625000 \sqrt{5}} \]
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Rubi [A] time = 0.352348, antiderivative size = 208, normalized size of antiderivative = 1., 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 1661
Rule 640
Rule 612
Rule 619
Rule 215
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*}
Mathematica [A] time = 0.315134, 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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Maple [A] time = 0.075, size = 166, normalized size = 0.8 \begin{align*}{\frac{190939\,{x}^{5}}{3000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{888751\,{x}^{4}}{105000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{90960857\,{x}^{3}}{1575000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{98060877\,{x}^{2}}{4375000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{1045360143\,x}{43750000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{771599830\,x+154319966}{62500000}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{540119881\,\sqrt{5}}{78125000}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }-{\frac{1968340667}{131250000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{343\,{x}^{7}}{50} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{50519\,{x}^{6}}{2250} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49426, size = 239, normalized size = 1.15 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13303, size = 417, normalized size = 2. \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 29 x \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 115 x^{2} \sqrt{5 x^{2} + 2 x + 3}\, 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 - 127 x^{5} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 2065 x^{6} \sqrt{5 x^{2} + 2 x + 3}\, 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 - 2 \sqrt{5 x^{2} + 2 x + 3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28566, size = 124, normalized size = 0.6 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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