Optimal. Leaf size=195 \[ \frac {(151764102421+27596573612 x) \sqrt {3-x+2 x^2}}{55037657088 (5+2 x)}-\frac {(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac {182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac {14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}+\frac {369 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}}-\frac {1903976002333 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{660451885056 \sqrt {2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1664, 824,
826, 857, 633, 221, 738, 212} \begin {gather*} -\frac {14087245 \left (2 x^2-x+3\right )^{5/2}}{71663616 (2 x+5)^4}+\frac {182165 \left (2 x^2-x+3\right )^{5/2}}{248832 (2 x+5)^5}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}-\frac {(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{13759414272 (2 x+5)^3}+\frac {(27596573612 x+151764102421) \sqrt {2 x^2-x+3}}{55037657088 (2 x+5)}-\frac {1903976002333 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{660451885056 \sqrt {2}}+\frac {369 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 633
Rule 738
Rule 824
Rule 826
Rule 857
Rule 1664
Rubi steps
\begin {align*} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^7} \, dx &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}-\frac {1}{432} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (\frac {68375}{16}-\frac {28085 x}{4}+2916 x^2-1080 x^3\right )}{(5+2 x)^6} \, dx\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac {182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}+\frac {\int \frac {\left (3-x+2 x^2\right )^{3/2} \left (\frac {10561025}{16}-1010880 x+194400 x^2\right )}{(5+2 x)^5} \, dx}{155520}\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac {182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac {14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac {\int \frac {\left (\frac {420053845}{16}-\frac {182410625 x}{4}\right ) \left (3-x+2 x^2\right )^{3/2}}{(5+2 x)^4} \, dx}{44789760}\\ &=-\frac {(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac {182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac {14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}+\frac {\int \frac {\left (-\frac {206718515415}{8}+\frac {103487151045 x}{2}\right ) \sqrt {3-x+2 x^2}}{(5+2 x)^2} \, dx}{51597803520}\\ &=\frac {(151764102421+27596573612 x) \sqrt {3-x+2 x^2}}{55037657088 (5+2 x)}-\frac {(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac {182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac {14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac {\int \frac {-\frac {4760153161395}{4}+2379948687360 x}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{412782428160}\\ &=\frac {(151764102421+27596573612 x) \sqrt {3-x+2 x^2}}{55037657088 (5+2 x)}-\frac {(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac {182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac {14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac {369}{128} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx+\frac {1903976002333 \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{110075314176}\\ &=\frac {(151764102421+27596573612 x) \sqrt {3-x+2 x^2}}{55037657088 (5+2 x)}-\frac {(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac {182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac {14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac {1903976002333 \text {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )}{55037657088}-\frac {369 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{128 \sqrt {46}}\\ &=\frac {(151764102421+27596573612 x) \sqrt {3-x+2 x^2}}{55037657088 (5+2 x)}-\frac {(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac {182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac {14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}+\frac {369 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}}-\frac {1903976002333 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{660451885056 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.97, size = 120, normalized size = 0.62 \begin {gather*} \frac {\frac {12 \sqrt {3-x+2 x^2} \left (458411625354581+1011372787716826 x+910256842473992 x^2+422554114856528 x^3+103803827945872 x^4+11854023276320 x^5+275188285440 x^6\right )}{(5+2 x)^6}+1903976002333 \sqrt {2} \tanh ^{-1}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )+951979474944 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{660451885056} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 246, normalized size = 1.26 \[-\frac {3667 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{221184 \left (x +\frac {5}{2}\right )^{6}}+\frac {182165 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{7962624 \left (x +\frac {5}{2}\right )^{5}}+\frac {149610673 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{41278242816 \left (x +\frac {5}{2}\right )^{3}}-\frac {3607708597 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{2972033482752 \left (x +\frac {5}{2}\right )^{2}}+\frac {125860542215 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{106993205379072 \left (x +\frac {5}{2}\right )}-\frac {125860542215 \left (4 x -1\right ) \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{213986410758144}-\frac {82772668391 \left (4 x -1\right ) \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{3962711310336}-\frac {14087245 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{1146617856 \left (x +\frac {5}{2}\right )^{4}}-\frac {1903976002333 \sqrt {2}\, \arctanh \left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{1320903770112}+\frac {1903976002333 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{3962711310336}-\frac {369 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{256}+\frac {1903976002333 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{213986410758144}\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 297, normalized size = 1.52 \begin {gather*} \frac {3607708597}{1486016741376} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3456 \, {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} + \frac {182165 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{248832 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} - \frac {14087245 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{71663616 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac {149610673 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{5159780352 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac {3607708597 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{743008370688 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac {82772668391}{990677827584} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {369}{256} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {1903976002333}{1320903770112} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {165562389227}{330225942528} \, \sqrt {2 \, x^{2} - x + 3} + \frac {125860542215 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2972033482752 \, {\left (2 \, x + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 229, normalized size = 1.17 \begin {gather*} \frac {1903958949888 \, \sqrt {2} {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 1903976002333 \, \sqrt {2} {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (275188285440 \, x^{6} + 11854023276320 \, x^{5} + 103803827945872 \, x^{4} + 422554114856528 \, x^{3} + 910256842473992 \, x^{2} + 1011372787716826 \, x + 458411625354581\right )} \sqrt {2 \, x^{2} - x + 3}}{2641807540224 \, {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\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 \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 452 vs.
\(2 (160) = 320\).
time = 3.17, size = 452, normalized size = 2.32 \begin {gather*} \frac {369}{256} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac {1903976002333}{1320903770112} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {1903976002333}{1320903770112} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {5}{64} \, \sqrt {2 \, x^{2} - x + 3} + \frac {\sqrt {2} {\left (159278433934432 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{11} + 6347903280912544 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{10} + 48544526840833424 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{9} + 305716670132783088 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{8} + 88313821135911024 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{7} - 2423668581998843376 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{6} - 397211131697032056 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} + 11708897232532299576 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} - 12803484860728491138 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} + 12593033197867577234 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} - 3042533760672408875 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 589526263249780195\right )}}{110075314176 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{6}} \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 \frac {{\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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