Optimal. Leaf size=281 \[ \frac {(11359-12920 x) \sqrt {3-x+2 x^2}}{48050}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}-\frac {4}{125} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {\sqrt {11 \left (1+4 \sqrt {2}\right )} \left (2937349+1978861 \sqrt {2}\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (3531015707557+2498852071250 \sqrt {2}\right )}} \left (3957722+2937349 \sqrt {2}+\left (9832420+6895071 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{29791000}-\frac {\left (2937349-1978861 \sqrt {2}\right ) \sqrt {11 \left (-1+4 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (-3531015707557+2498852071250 \sqrt {2}\right )}} \left (3957722-2937349 \sqrt {2}+\left (9832420-6895071 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{29791000} \]
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Rubi [A]
time = 0.42, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {985, 1068,
1080, 1090, 633, 221, 1049, 1043, 212, 210} \begin {gather*} \frac {\sqrt {11 \left (1+4 \sqrt {2}\right )} \left (2937349+1978861 \sqrt {2}\right ) \text {ArcTan}\left (\frac {\sqrt {\frac {11}{62 \left (3531015707557+2498852071250 \sqrt {2}\right )}} \left (\left (9832420+6895071 \sqrt {2}\right ) x+2937349 \sqrt {2}+3957722\right )}{\sqrt {2 x^2-x+3}}\right )}{29791000}+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac {(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{3844 \left (5 x^2+3 x+2\right )}+\frac {(11359-12920 x) \sqrt {2 x^2-x+3}}{48050}-\frac {\left (2937349-1978861 \sqrt {2}\right ) \sqrt {11 \left (4 \sqrt {2}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (2498852071250 \sqrt {2}-3531015707557\right )}} \left (\left (9832420-6895071 \sqrt {2}\right ) x-2937349 \sqrt {2}+3957722\right )}{\sqrt {2 x^2-x+3}}\right )}{29791000}-\frac {4}{125} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 221
Rule 633
Rule 985
Rule 1043
Rule 1049
Rule 1068
Rule 1080
Rule 1090
Rubi steps
\begin {align*} \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {1}{62} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (-\frac {195}{2}+35 x+40 x^2\right )}{\left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {\left (\frac {66735}{4}-7375 x-25840 x^2\right ) \sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx}{9610}\\ &=\frac {(11359-12920 x) \sqrt {3-x+2 x^2}}{48050}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-6782705+2898425 x-307520 x^2}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{961000}\\ &=\frac {(11359-12920 x) \sqrt {3-x+2 x^2}}{48050}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-33298485+15414685 x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{4805000}+\frac {8}{125} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {(11359-12920 x) \sqrt {3-x+2 x^2}}{48050}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}+\frac {1}{125} \left (4 \sqrt {\frac {2}{23}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )-\frac {\int \frac {605 \left (885694-605427 \sqrt {2}\right )-605 \left (325160-280267 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{105710000 \sqrt {2}}+\frac {\int \frac {605 \left (885694+605427 \sqrt {2}\right )-605 \left (325160+280267 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{105710000 \sqrt {2}}\\ &=\frac {(11359-12920 x) \sqrt {3-x+2 x^2}}{48050}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}-\frac {4}{125} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )-\frac {\left (1331 \left (4997704142500-3531015707557 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-22693550 \left (3531015707557-2498852071250 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {605 \left (3957722-2937349 \sqrt {2}\right )+605 \left (9832420-6895071 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{192200}-\frac {\left (1331 \left (4997704142500+3531015707557 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-22693550 \left (3531015707557+2498852071250 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {605 \left (3957722+2937349 \sqrt {2}\right )+605 \left (9832420+6895071 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{192200}\\ &=\frac {(11359-12920 x) \sqrt {3-x+2 x^2}}{48050}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}-\frac {4}{125} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {\sqrt {\frac {11}{31} \left (3531015707557+2498852071250 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (3531015707557+2498852071250 \sqrt {2}\right )}} \left (3957722+2937349 \sqrt {2}+\left (9832420+6895071 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{961000}-\frac {\sqrt {\frac {11}{31} \left (-3531015707557+2498852071250 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (-3531015707557+2498852071250 \sqrt {2}\right )}} \left (3957722-2937349 \sqrt {2}+\left (9832420-6895071 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{961000}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.86, size = 616, normalized size = 2.19 \begin {gather*} \frac {\frac {15812500 \sqrt {3-x+2 x^2} \left (22552+69621 x+93872 x^2+97155 x^3\right )}{\left (2+3 x+5 x^2\right )^2}-4420600000 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )+972532000 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {3781 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+630 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+150 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+682 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {4978708507 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-165870920 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+1110955025 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-11 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {492740319684 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-128644699540 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+55365920925 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{138143750000} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(119457\) vs.
\(2(220)=440\).
time = 1.09, size = 119458, normalized size = 425.12
method | result | size |
trager | \(\text {Expression too large to display}\) | \(614\) |
risch | \(\frac {11 \left (97155 x^{3}+93872 x^{2}+69621 x +22552\right ) \sqrt {2 x^{2}-x +3}}{96100 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {4 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{125}+\frac {\sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (132861440 \sqrt {2}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+187960123 \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+197090660657 \arctanh \left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right ) \sqrt {2}-271286828868 \arctanh \left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )\right )}{923521000 \sqrt {\frac {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\) | \(740\) |
default | \(\text {Expression too large to display}\) | \(119458\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2240 vs.
\(2 (220) = 440\).
time = 3.51, size = 2240, normalized size = 7.97 \begin {gather*} \text {Too large to display} \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 {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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \frac {{\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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