Optimal. Leaf size=231 \[ -\frac{343}{60} \left (5 x^2+2 x+3\right )^{5/2} x^7-\frac{61103 \left (5 x^2+2 x+3\right )^{5/2} x^6}{3300}+\frac{1031177 \left (5 x^2+2 x+3\right )^{5/2} x^5}{20625}-\frac{796559 \left (5 x^2+2 x+3\right )^{5/2} x^4}{123750}-\frac{190236913 \left (5 x^2+2 x+3\right )^{5/2} x^3}{4950000}+\frac{2173004363 \left (5 x^2+2 x+3\right )^{5/2} x^2}{173250000}+\frac{837379699 \left (5 x^2+2 x+3\right )^{5/2} x}{72187500}-\frac{6133820867 \left (5 x^2+2 x+3\right )^{5/2}}{1203125000}-\frac{22840599 (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}}{62500000}-\frac{479652579 (5 x+1) \sqrt{5 x^2+2 x+3}}{312500000}-\frac{3357568053 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{156250000 \sqrt{5}} \]
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Rubi [A] time = 0.363843, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ -\frac{343}{60} \left (5 x^2+2 x+3\right )^{5/2} x^7-\frac{61103 \left (5 x^2+2 x+3\right )^{5/2} x^6}{3300}+\frac{1031177 \left (5 x^2+2 x+3\right )^{5/2} x^5}{20625}-\frac{796559 \left (5 x^2+2 x+3\right )^{5/2} x^4}{123750}-\frac{190236913 \left (5 x^2+2 x+3\right )^{5/2} x^3}{4950000}+\frac{2173004363 \left (5 x^2+2 x+3\right )^{5/2} x^2}{173250000}+\frac{837379699 \left (5 x^2+2 x+3\right )^{5/2} x}{72187500}-\frac{6133820867 \left (5 x^2+2 x+3\right )^{5/2}}{1203125000}-\frac{22840599 (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}}{62500000}-\frac{479652579 (5 x+1) \sqrt{5 x^2+2 x+3}}{312500000}-\frac{3357568053 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{156250000 \sqrt{5}} \]
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 ) \left (3+2 x+5 x^2\right )^{3/2} \, dx &=-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac{1}{60} \int \left (3+2 x+5 x^2\right )^{3/2} \left (120+1740 x+6900 x^2-3660 x^3-52260 x^4+7620 x^5+131103 x^6-61103 x^7\right ) \, dx\\ &=-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int \left (3+2 x+5 x^2\right )^{3/2} \left (6600+95700 x+379500 x^2-201300 x^3-2874300 x^4+1518954 x^5+8249416 x^6\right ) \, dx}{3300}\\ &=\frac{1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int \left (3+2 x+5 x^2\right )^{3/2} \left (330000+4785000 x+18975000 x^2-10065000 x^3-267456240 x^4-47793540 x^5\right ) \, dx}{165000}\\ &=-\frac{796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac{1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int \left (3+2 x+5 x^2\right )^{3/2} \left (14850000+215325000 x+853875000 x^2+120597480 x^3-11414214780 x^4\right ) \, dx}{7425000}\\ &=-\frac{190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac{796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac{1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int \left (3+2 x+5 x^2\right )^{3/2} \left (594000000+8613000000 x+136882933020 x^2+130380261780 x^3\right ) \, dx}{297000000}\\ &=\frac{2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac{190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac{796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac{1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int \left (3+2 x+5 x^2\right )^{3/2} \left (20790000000-480826570680 x+3617480299680 x^2\right ) \, dx}{10395000000}\\ &=\frac{837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac{2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac{190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac{796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac{1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int (-10228740899040-39747159218160 x) \left (3+2 x+5 x^2\right )^{3/2} \, dx}{311850000000}\\ &=-\frac{6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac{837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac{2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac{190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac{796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac{1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac{22840599 \int \left (3+2 x+5 x^2\right )^{3/2} \, dx}{3125000}\\ &=-\frac{22840599 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{62500000}-\frac{6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac{837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac{2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac{190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac{796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac{1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac{479652579 \int \sqrt{3+2 x+5 x^2} \, dx}{31250000}\\ &=-\frac{479652579 (1+5 x) \sqrt{3+2 x+5 x^2}}{312500000}-\frac{22840599 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{62500000}-\frac{6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac{837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac{2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac{190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac{796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac{1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac{3357568053 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{156250000}\\ &=-\frac{479652579 (1+5 x) \sqrt{3+2 x+5 x^2}}{312500000}-\frac{22840599 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{62500000}-\frac{6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac{837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac{2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac{190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac{796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac{1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac{\left (479652579 \sqrt{\frac{7}{10}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{312500000}\\ &=-\frac{479652579 (1+5 x) \sqrt{3+2 x+5 x^2}}{312500000}-\frac{22840599 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{62500000}-\frac{6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac{837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac{2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac{190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac{796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac{1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac{61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac{343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac{3357568053 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{156250000 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.398603, size = 95, normalized size = 0.41 \[ \frac{-5 \sqrt{5 x^2+2 x+3} \left (30950390625000 x^{11}+125007421875000 x^{10}-148393743750000 x^9-30505457500000 x^8-72918247281250 x^7+52106830406250 x^6+85130334087500 x^5-2573089891000 x^4-19041688239675 x^3-15865844408685 x^2-6352777129950 x+10506617068392\right )-4653589321458 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1082812500000} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 185, normalized size = 0.8 \begin{align*}{\frac{1031177\,{x}^{5}}{20625} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{796559\,{x}^{4}}{123750} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{190236913\,{x}^{3}}{4950000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{2173004363\,{x}^{2}}{173250000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{4796525790\,x+959305158}{625000000}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{3357568053\,\sqrt{5}}{781250000}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }+{\frac{837379699\,x}{72187500} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{228405990\,x+45681198}{125000000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{6133820867}{1203125000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{61103\,{x}^{6}}{3300} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{343\,{x}^{7}}{60} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53308, size = 278, normalized size = 1.2 \begin{align*} -\frac{343}{60} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{7} - \frac{61103}{3300} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{6} + \frac{1031177}{20625} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{5} - \frac{796559}{123750} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{4} - \frac{190236913}{4950000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{2173004363}{173250000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{837379699}{72187500} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x - \frac{6133820867}{1203125000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} - \frac{22840599}{12500000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x - \frac{22840599}{62500000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} - \frac{479652579}{62500000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x - \frac{3357568053}{781250000} \, \sqrt{5} \operatorname{arsinh}\left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{479652579}{312500000} \, \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 = 1.48695, size = 518, normalized size = 2.24 \begin{align*} -\frac{1}{216562500000} \,{\left (30950390625000 \, x^{11} + 125007421875000 \, x^{10} - 148393743750000 \, x^{9} - 30505457500000 \, x^{8} - 72918247281250 \, x^{7} + 52106830406250 \, x^{6} + 85130334087500 \, x^{5} - 2573089891000 \, x^{4} - 19041688239675 \, x^{3} - 15865844408685 \, x^{2} - 6352777129950 \, x + 10506617068392\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{3357568053}{1562500000} \, \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 - 91 x \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 413 x^{2} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 192 x^{3} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 2160 x^{4} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 1666 x^{5} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 2094 x^{6} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 1384 x^{7} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 7042 x^{8} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 6321 x^{9} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 1715 x^{10} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 6 \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 = 1.2009, size = 138, normalized size = 0.6 \begin{align*} -\frac{1}{216562500000} \,{\left (5 \,{\left ({\left (5 \,{\left (10 \,{\left (25 \,{\left (5 \,{\left (7 \,{\left (20 \,{\left (105 \,{\left (875 \,{\left (77 \, x + 311\right )} x - 323034\right )} x - 6972676\right )} x - 333340559\right )} x + 1667418573\right )} x + 13620853454\right )} x - 10292359564\right )} x - 761667529587\right )} x - 3173168881737\right )} x - 1270555425990\right )} x + 10506617068392\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{3357568053}{781250000} \, \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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