Optimal. Leaf size=170 \[ \frac {305}{144} x^2 \left (2 x^2-x+3\right )^{7/2}+\frac {8467 x \left (2 x^2-x+3\right )^{7/2}}{4608}+\frac {23225 \left (2 x^2-x+3\right )^{7/2}}{43008}-\frac {1547 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{98304}-\frac {177905 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{3145728}-\frac {4091815 (1-4 x) \sqrt {2 x^2-x+3}}{16777216}+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}-\frac {94111745 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{33554432 \sqrt {2}} \]
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Rubi [A] time = 0.13, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1661, 640, 612, 619, 215} \begin {gather*} \frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}+\frac {305}{144} x^2 \left (2 x^2-x+3\right )^{7/2}+\frac {8467 x \left (2 x^2-x+3\right )^{7/2}}{4608}+\frac {23225 \left (2 x^2-x+3\right )^{7/2}}{43008}-\frac {1547 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{98304}-\frac {177905 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{3145728}-\frac {4091815 (1-4 x) \sqrt {2 x^2-x+3}}{16777216}-\frac {94111745 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{33554432 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2 \, dx &=\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1}{20} \int \left (3-x+2 x^2\right )^{5/2} \left (80+240 x+355 x^2+\frac {1525 x^3}{2}\right ) \, dx\\ &=\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1}{360} \int \left (3-x+2 x^2\right )^{5/2} \left (1440-255 x+\frac {42335 x^2}{4}\right ) \, dx\\ &=\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {\int \left (-\frac {34845}{4}+\frac {348375 x}{8}\right ) \left (3-x+2 x^2\right )^{5/2} \, dx}{5760}\\ &=\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1547 \int \left (3-x+2 x^2\right )^{5/2} \, dx}{4096}\\ &=-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {177905 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{196608}\\ &=-\frac {177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {4091815 \int \sqrt {3-x+2 x^2} \, dx}{2097152}\\ &=-\frac {4091815 (1-4 x) \sqrt {3-x+2 x^2}}{16777216}-\frac {177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {94111745 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{33554432}\\ &=-\frac {4091815 (1-4 x) \sqrt {3-x+2 x^2}}{16777216}-\frac {177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {\left (4091815 \sqrt {\frac {23}{2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{33554432}\\ &=-\frac {4091815 (1-4 x) \sqrt {3-x+2 x^2}}{16777216}-\frac {177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}-\frac {94111745 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{33554432 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 85, normalized size = 0.50 \begin {gather*} \frac {4 \sqrt {2 x^2-x+3} \left (10569646080 x^9+2055208960 x^8+44163137536 x^7+26401898496 x^6+75389820928 x^5+57147467776 x^4+77872272000 x^3+42992644128 x^2+39533249652 x+14824182519\right )-5929039935 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4227858432} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.05, size = 100, normalized size = 0.59 \begin {gather*} \frac {\sqrt {2 x^2-x+3} \left (10569646080 x^9+2055208960 x^8+44163137536 x^7+26401898496 x^6+75389820928 x^5+57147467776 x^4+77872272000 x^3+42992644128 x^2+39533249652 x+14824182519\right )}{1056964608}-\frac {94111745 \log \left (2 \sqrt {2} \sqrt {2 x^2-x+3}-4 x+1\right )}{33554432 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 98, normalized size = 0.58 \begin {gather*} \frac {1}{1056964608} \, {\left (10569646080 \, x^{9} + 2055208960 \, x^{8} + 44163137536 \, x^{7} + 26401898496 \, x^{6} + 75389820928 \, x^{5} + 57147467776 \, x^{4} + 77872272000 \, x^{3} + 42992644128 \, x^{2} + 39533249652 \, x + 14824182519\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {94111745}{134217728} \, \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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giac [A] time = 0.52, size = 93, normalized size = 0.55 \begin {gather*} \frac {1}{1056964608} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (8 \, {\left (28 \, {\left (160 \, {\left (36 \, x + 7\right )} x + 24067\right )} x + 402861\right )} x + 9202859\right )} x + 27904037\right )} x + 608377125\right )} x + 1343520129\right )} x + 9883312413\right )} x + 14824182519\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {94111745}{67108864} \, \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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maple [A] time = 0.01, size = 136, normalized size = 0.80 \begin {gather*} \frac {5 \left (2 x^{2}-x +3\right )^{\frac {7}{2}} x^{3}}{4}+\frac {305 \left (2 x^{2}-x +3\right )^{\frac {7}{2}} x^{2}}{144}+\frac {8467 \left (2 x^{2}-x +3\right )^{\frac {7}{2}} x}{4608}+\frac {94111745 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{67108864}+\frac {23225 \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{43008}+\frac {4091815 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{16777216}+\frac {1547 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{98304}+\frac {177905 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{3145728} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 167, normalized size = 0.98 \begin {gather*} \frac {5}{4} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{3} + \frac {305}{144} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{2} + \frac {8467}{4608} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x + \frac {23225}{43008} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {1547}{24576} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {1547}{98304} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {177905}{786432} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {177905}{3145728} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {4091815}{4194304} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {94111745}{67108864} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {4091815}{16777216} \, \sqrt {2 \, x^{2} - x + 3} \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 = -0.01 \begin {gather*} \int {\left (2\,x^2-x+3\right )}^{5/2}\,{\left (5\,x^2+3\,x+2\right )}^2 \,d x \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 )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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