Optimal. Leaf size=231 \[ -\frac {479652579 (1+5 x) \sqrt {3+2 x+5 x^2}}{312500000}-\frac {22840599 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{62500000}-\frac {6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac {837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac {2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac {190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac {3357568053 \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{156250000 \sqrt {5}} \]
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Rubi [A]
time = 0.23, antiderivative size = 231, normalized size of antiderivative = 1.00, 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 = {1675, 654,
626, 633, 221} \begin {gather*} \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 {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 {3357568053 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{156250000 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 626
Rule 633
Rule 654
Rule 1675
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 ) \text {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*}
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Mathematica [A]
time = 0.89, size = 109, normalized size = 0.47 \begin {gather*} \frac {\sqrt {3+2 x+5 x^2} \left (-10506617068392+6352777129950 x+15865844408685 x^2+19041688239675 x^3+2573089891000 x^4-85130334087500 x^5-52106830406250 x^6+72918247281250 x^7+30505457500000 x^8+148393743750000 x^9-125007421875000 x^{10}-30950390625000 x^{11}\right )}{216562500000}+\frac {3357568053 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{156250000 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 185, normalized size = 0.80
method | result | size |
risch | \(-\frac {\left (30950390625000 x^{11}+125007421875000 x^{10}-148393743750000 x^{9}-30505457500000 x^{8}-72918247281250 x^{7}+52106830406250 x^{6}+85130334087500 x^{5}-2573089891000 x^{4}-19041688239675 x^{3}-15865844408685 x^{2}-6352777129950 x +10506617068392\right ) \sqrt {5 x^{2}+2 x +3}}{216562500000}-\frac {3357568053 \sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{781250000}\) | \(85\) |
trager | \(\left (-\frac {1715}{12} x^{11}-\frac {76195}{132} x^{10}+\frac {376873}{550} x^{9}+\frac {1743169}{12375} x^{8}+\frac {333340559}{990000} x^{7}-\frac {555806191}{2310000} x^{6}-\frac {6810426727}{17325000} x^{5}+\frac {2573089891}{216562500} x^{4}+\frac {253889176529}{2887500000} x^{3}+\frac {352574320193}{4812500000} x^{2}+\frac {14117282511}{481250000} x -\frac {145925237061}{3007812500}\right ) \sqrt {5 x^{2}+2 x +3}-\frac {3357568053 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (5 \RootOf \left (\textit {\_Z}^{2}-5\right ) x +\RootOf \left (\textit {\_Z}^{2}-5\right )+5 \sqrt {5 x^{2}+2 x +3}\right )}{781250000}\) | \(109\) |
default | \(-\frac {3357568053 \sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{781250000}-\frac {479652579 \left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{625000000}-\frac {22840599 \left (10 x +2\right ) \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{125000000}-\frac {6133820867 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{1203125000}+\frac {837379699 x \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{72187500}+\frac {2173004363 x^{2} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{173250000}-\frac {190236913 x^{3} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{4950000}-\frac {796559 x^{4} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{123750}+\frac {1031177 x^{5} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{20625}-\frac {61103 x^{6} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{3300}-\frac {343 x^{7} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{60}\) | \(185\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 206, normalized size = 0.89 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 107, normalized size = 0.46 \begin {gather*} -\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 {gather*}
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 \begin {gather*} - \int \left (- 91 x \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 413 x^{2} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 192 x^{3} \sqrt {5 x^{2} + 2 x + 3}\right )\, 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 \left (- 2094 x^{6} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 1384 x^{7} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 7042 x^{8} \sqrt {5 x^{2} + 2 x + 3}\right )\, 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 \left (- 6 \sqrt {5 x^{2} + 2 x + 3}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.34, size = 102, normalized size = 0.44 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (x^2+5\,x+2\right )\,{\left (5\,x^2+2\,x+3\right )}^{3/2}\,{\left (-7\,x^2+4\,x+1\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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