Optimal. Leaf size=232 \[ \frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {2}{25} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {\sqrt {\frac {11}{31} \left (3169333+2265350 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (3169333+2265350 \sqrt {2}\right )}} \left (3514+2963 \sqrt {2}+\left (9440+6477 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1550}-\frac {\sqrt {\frac {11}{31} \left (-3169333+2265350 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (-3169333+2265350 \sqrt {2}\right )}} \left (3514-2963 \sqrt {2}+\left (9440-6477 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1550} \]
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Rubi [A]
time = 0.37, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {985, 1080,
1090, 633, 221, 1049, 1043, 212, 210} \begin {gather*} \frac {\sqrt {\frac {11}{31} \left (3169333+2265350 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {11}{62 \left (3169333+2265350 \sqrt {2}\right )}} \left (\left (9440+6477 \sqrt {2}\right ) x+2963 \sqrt {2}+3514\right )}{\sqrt {2 x^2-x+3}}\right )}{1550}+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}+\frac {4}{155} (4-5 x) \sqrt {2 x^2-x+3}-\frac {\sqrt {\frac {11}{31} \left (2265350 \sqrt {2}-3169333\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (2265350 \sqrt {2}-3169333\right )}} \left (\left (9440-6477 \sqrt {2}\right ) x-2963 \sqrt {2}+3514\right )}{\sqrt {2 x^2-x+3}}\right )}{1550}-\frac {2}{25} \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 1080
Rule 1090
Rubi steps
\begin {align*} \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \int \frac {\sqrt {3-x+2 x^2} \left (-\frac {69}{2}+13 x+40 x^2\right )}{2+3 x+5 x^2} \, dx\\ &=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {13070-5750 x+2480 x^2}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{3100}\\ &=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {60390-36190 x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{15500}+\frac {4}{25} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{25} \left (2 \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 {-1210 \left (878-549 \sqrt {2}\right )+1210 \left (220-329 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{341000 \sqrt {2}}-\frac {\int \frac {-1210 \left (878+549 \sqrt {2}\right )+1210 \left (220+329 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{341000 \sqrt {2}}\\ &=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {2}{25} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {1}{155} \left (1331 \left (4530700-3169333 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-90774200 \left (3169333-2265350 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {-1210 \left (3514-2963 \sqrt {2}\right )-1210 \left (9440-6477 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )+\frac {1}{155} \left (1331 \left (4530700+3169333 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-90774200 \left (3169333+2265350 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {-1210 \left (3514+2963 \sqrt {2}\right )-1210 \left (9440+6477 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )\\ &=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {2}{25} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {\sqrt {\frac {11}{31} \left (3169333+2265350 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (3169333+2265350 \sqrt {2}\right )}} \left (3514+2963 \sqrt {2}+\left (9440+6477 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1550}-\frac {\sqrt {\frac {11}{31} \left (-3169333+2265350 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (-3169333+2265350 \sqrt {2}\right )}} \left (3514-2963 \sqrt {2}+\left (9440-6477 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1550}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.52, size = 416, normalized size = 1.79 \begin {gather*} \frac {50 \left (\frac {55 (7+13 x) \sqrt {3-x+2 x^2}}{2+3 x+5 x^2}-62 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )\right )+682 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {999 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+310 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+100 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+11 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-72888 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+8230 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2025 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{38750} \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. \(28184\) vs.
\(2(175)=350\).
time = 0.98, size = 28185, normalized size = 121.49
method | result | size |
trager | \(\text {Expression too large to display}\) | \(603\) |
risch | \(\frac {11 \left (7+13 x \right ) \sqrt {2 x^{2}-x +3}}{155 \left (5 x^{2}+3 x +2\right )}+\frac {2 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{25}+\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 (126130 \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}}+178601 \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}}+193755859 \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}-248376216 \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 )}{1489550 \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}}}\) | \(730\) |
default | \(\text {Expression too large to display}\) | \(28185\) |
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. 2150 vs.
\(2 (175) = 350\).
time = 2.79, size = 2150, normalized size = 9.27 \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 {3}{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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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 )}^{3/2}}{{\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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