Optimal. Leaf size=255 \[ -\frac {(1277+2240 x) \sqrt {3-x+2 x^2}}{7750}+\frac {4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {4799 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2500 \sqrt {2}}+\frac {11 \sqrt {\frac {11}{31} \left (224510383+194487500 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (224510383+194487500 \sqrt {2}\right )}} \left (21136+33287 \sqrt {2}+\left (87710+54423 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{38750}-\frac {11 \sqrt {\frac {11}{31} \left (-224510383+194487500 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (-224510383+194487500 \sqrt {2}\right )}} \left (21136-33287 \sqrt {2}+\left (87710-54423 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{38750} \]
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Rubi [A]
time = 0.43, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps
used = 11, 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 {11 \sqrt {\frac {11}{31} \left (224510383+194487500 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {11}{62 \left (224510383+194487500 \sqrt {2}\right )}} \left (\left (87710+54423 \sqrt {2}\right ) x+33287 \sqrt {2}+21136\right )}{\sqrt {2 x^2-x+3}}\right )}{38750}+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}+\frac {4}{155} (4-5 x) \left (2 x^2-x+3\right )^{3/2}-\frac {(2240 x+1277) \sqrt {2 x^2-x+3}}{7750}-\frac {11 \sqrt {\frac {11}{31} \left (194487500 \sqrt {2}-224510383\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (194487500 \sqrt {2}-224510383\right )}} \left (\left (87710-54423 \sqrt {2}\right ) x-33287 \sqrt {2}+21136\right )}{\sqrt {2 x^2-x+3}}\right )}{38750}-\frac {4799 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2500 \sqrt {2}} \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 )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (-\frac {75}{2}+15 x+80 x^2\right )}{2+3 x+5 x^2} \, dx\\ &=\frac {4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {\left (87660-54300 x-53760 x^2\right ) \sqrt {3-x+2 x^2}}{2+3 x+5 x^2} \, dx}{18600}\\ &=-\frac {(1277+2240 x) \sqrt {3-x+2 x^2}}{7750}+\frac {4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-29217120+21064200 x-17852280 x^2}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{1860000}\\ &=-\frac {(1277+2240 x) \sqrt {3-x+2 x^2}}{7750}+\frac {4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-110381040+158877840 x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{9300000}+\frac {4799 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{2500}\\ &=-\frac {(1277+2240 x) \sqrt {3-x+2 x^2}}{7750}+\frac {4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {319440 \left (9272+3801 \sqrt {2}\right )+319440 \left (1670-5471 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{204600000 \sqrt {2}}-\frac {\int \frac {319440 \left (9272-3801 \sqrt {2}\right )+319440 \left (1670+5471 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{204600000 \sqrt {2}}+\frac {4799 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{2500 \sqrt {46}}\\ &=-\frac {(1277+2240 x) \sqrt {3-x+2 x^2}}{7750}+\frac {4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {4799 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2500 \sqrt {2}}-\frac {\left (3865224 \left (388975000-224510383 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-6326598643200 \left (224510383-194487500 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {319440 \left (21136-33287 \sqrt {2}\right )+319440 \left (87710-54423 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{3875}-\frac {\left (3865224 \left (388975000+224510383 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-6326598643200 \left (224510383+194487500 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {319440 \left (21136+33287 \sqrt {2}\right )+319440 \left (87710+54423 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{3875}\\ &=-\frac {(1277+2240 x) \sqrt {3-x+2 x^2}}{7750}+\frac {4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {4799 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2500 \sqrt {2}}+\frac {11 \sqrt {\frac {11}{31} \left (224510383+194487500 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (224510383+194487500 \sqrt {2}\right )}} \left (21136+33287 \sqrt {2}+\left (87710+54423 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{38750}-\frac {11 \sqrt {\frac {11}{31} \left (-224510383+194487500 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (-224510383+194487500 \sqrt {2}\right )}} \left (21136-33287 \sqrt {2}+\left (87710-54423 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{38750}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.72, size = 433, normalized size = 1.70 \begin {gather*} \frac {\frac {500 \sqrt {3-x+2 x^2} \left (8996+9289 x-12555 x^2+3100 x^3\right )}{2+3 x+5 x^2}-3719225 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )+30008 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {5237 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+2880 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2225 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-242 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {639994 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-22980 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+1175 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{3875000} \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. \(40027\) vs.
\(2(194)=388\).
time = 0.96, size = 40028, normalized size = 156.97
method | result | size |
trager | \(\text {Expression too large to display}\) | \(614\) |
risch | \(\frac {\left (3100 x^{3}-12555 x^{2}+9289 x +8996\right ) \sqrt {2 x^{2}-x +3}}{38750 x^{2}+23250 x +15500}+\frac {4799 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{5000}+\frac {11 \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 (1114345 \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}}+1584599 \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}}+1982813041 \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}-1820897034 \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 )}{37238750 \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 | \(\text {Expression too large to display}\) | \(40028\) |
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. 2161 vs.
\(2 (194) = 388\).
time = 3.76, size = 2161, normalized size = 8.47 \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 )^{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 )}^{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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