Optimal. Leaf size=212 \[ -\frac {459555525 (1-4 x) \sqrt {3-x+2 x^2}}{1073741824}-\frac {6660225 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{67108864}-\frac {57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac {1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac {509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac {80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac {3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}-\frac {10569777075 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2147483648 \sqrt {2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1675, 654,
626, 633, 221} \begin {gather*} \frac {80483 \left (2 x^2-x+3\right )^{7/2} x^2}{9216}+\frac {509257 \left (2 x^2-x+3\right )^{7/2} x}{294912}-\frac {1696165 \left (2 x^2-x+3\right )^{7/2}}{2752512}-\frac {57915 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{2097152}-\frac {6660225 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{67108864}-\frac {459555525 (1-4 x) \sqrt {2 x^2-x+3}}{1073741824}+\frac {125}{24} \left (2 x^2-x+3\right )^{7/2} x^5+\frac {1175}{96} \left (2 x^2-x+3\right )^{7/2} x^4+\frac {3823}{256} \left (2 x^2-x+3\right )^{7/2} x^3-\frac {10569777075 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2147483648 \sqrt {2}} \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 (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3 \, dx &=\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac {1}{24} \int \left (3-x+2 x^2\right )^{5/2} \left (192+864 x+2736 x^2+4968 x^3+4965 x^4+\frac {12925 x^5}{2}\right ) \, dx\\ &=\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac {1}{528} \int \left (3-x+2 x^2\right )^{5/2} \left (4224+19008 x+60192 x^2+31746 x^3+\frac {630795 x^4}{4}\right ) \, dx\\ &=\frac {3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac {\int \left (3-x+2 x^2\right )^{5/2} \left (84480+380160 x-\frac {861795 x^2}{4}+\frac {13279695 x^3}{8}\right ) \, dx}{10560}\\ &=\frac {80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac {3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac {\int \left (3-x+2 x^2\right )^{5/2} \left (1520640-\frac {12467565 x}{4}+\frac {84027405 x^2}{16}\right ) \, dx}{190080}\\ &=\frac {509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac {80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac {3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac {\int \left (\frac {137201625}{16}-\frac {839601675 x}{32}\right ) \left (3-x+2 x^2\right )^{5/2} \, dx}{3041280}\\ &=-\frac {1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac {509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac {80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac {3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac {173745 \int \left (3-x+2 x^2\right )^{5/2} \, dx}{262144}\\ &=-\frac {57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac {1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac {509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac {80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac {3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac {6660225 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{4194304}\\ &=-\frac {6660225 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{67108864}-\frac {57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac {1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac {509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac {80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac {3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac {459555525 \int \sqrt {3-x+2 x^2} \, dx}{134217728}\\ &=-\frac {459555525 (1-4 x) \sqrt {3-x+2 x^2}}{1073741824}-\frac {6660225 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{67108864}-\frac {57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac {1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac {509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac {80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac {3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac {10569777075 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{2147483648}\\ &=-\frac {459555525 (1-4 x) \sqrt {3-x+2 x^2}}{1073741824}-\frac {6660225 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{67108864}-\frac {57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac {1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac {509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac {80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac {3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac {\left (459555525 \sqrt {\frac {23}{2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{2147483648}\\ &=-\frac {459555525 (1-4 x) \sqrt {3-x+2 x^2}}{1073741824}-\frac {6660225 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{67108864}-\frac {57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac {1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac {509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac {80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac {3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac {125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}-\frac {10569777075 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2147483648 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 1.02, size = 105, normalized size = 0.50 \begin {gather*} \frac {4 \sqrt {3-x+2 x^2} \left (-1191399152715+4560943728924 x+10060731582048 x^2+20384824684416 x^3+26186527209472 x^4+34378613923840 x^5+28347538538496 x^6+27835561148416 x^7+14341894045696 x^8+12943588589568 x^9+2395786444800 x^{10}+2818572288000 x^{11}\right )-665895955725 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{270582939648} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 170, normalized size = 0.80
method | result | size |
risch | \(\frac {\left (2818572288000 x^{11}+2395786444800 x^{10}+12943588589568 x^{9}+14341894045696 x^{8}+27835561148416 x^{7}+28347538538496 x^{6}+34378613923840 x^{5}+26186527209472 x^{4}+20384824684416 x^{3}+10060731582048 x^{2}+4560943728924 x -1191399152715\right ) \sqrt {2 x^{2}-x +3}}{67645734912}+\frac {10569777075 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4294967296}\) | \(85\) |
trager | \(\left (\frac {125}{3} x^{11}+\frac {425}{12} x^{10}+\frac {6123}{32} x^{9}+\frac {244241}{1152} x^{8}+\frac {15169177}{36864} x^{7}+\frac {144183037}{344064} x^{6}+\frac {4196608145}{8257536} x^{5}+\frac {1826627177}{4718592} x^{4}+\frac {53085480949}{176160768} x^{3}+\frac {4990442253}{33554432} x^{2}+\frac {126692881359}{1879048192} x -\frac {132377683635}{7516192768}\right ) \sqrt {2 x^{2}-x +3}+\frac {10569777075 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{4294967296}\) | \(111\) |
default | \(-\frac {1696165 \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{2752512}+\frac {57915 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{2097152}+\frac {459555525 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{1073741824}+\frac {10569777075 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4294967296}+\frac {6660225 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{67108864}+\frac {509257 x \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{294912}+\frac {80483 x^{2} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{9216}+\frac {3823 x^{3} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{256}+\frac {1175 x^{4} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{96}+\frac {125 x^{5} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{24}\) | \(170\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 201, normalized size = 0.95 \begin {gather*} \frac {125}{24} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{5} + \frac {1175}{96} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{4} + \frac {3823}{256} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{3} + \frac {80483}{9216} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{2} + \frac {509257}{294912} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x - \frac {1696165}{2752512} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {57915}{524288} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {57915}{2097152} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {6660225}{16777216} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {6660225}{67108864} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {459555525}{268435456} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {10569777075}{4294967296} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {459555525}{1073741824} \, \sqrt {2 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.74, size = 108, normalized size = 0.51 \begin {gather*} \frac {1}{67645734912} \, {\left (2818572288000 \, x^{11} + 2395786444800 \, x^{10} + 12943588589568 \, x^{9} + 14341894045696 \, x^{8} + 27835561148416 \, x^{7} + 28347538538496 \, x^{6} + 34378613923840 \, x^{5} + 26186527209472 \, x^{4} + 20384824684416 \, x^{3} + 10060731582048 \, x^{2} + 4560943728924 \, x - 1191399152715\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {10569777075}{8589934592} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\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 (2 x^{2} - x + 3\right )^{\frac {5}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.25, size = 103, normalized size = 0.49 \begin {gather*} \frac {1}{67645734912} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (8 \, {\left (28 \, {\left (32 \, {\left (12 \, {\left (200 \, {\left (20 \, x + 17\right )} x + 18369\right )} x + 244241\right )} x + 15169177\right )} x + 432549111\right )} x + 4196608145\right )} x + 12786390239\right )} x + 159256442847\right )} x + 314397861939\right )} x + 1140235932231\right )} x - 1191399152715\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {10569777075}{4294967296} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{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 (2\,x^2-x+3\right )}^{5/2}\,{\left (5\,x^2+3\,x+2\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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