Optimal. Leaf size=199 \[ \frac {3 (3+61 x) \sqrt {3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}+\frac {1}{49} \sqrt {5} \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )-\frac {\sqrt {\frac {325022311+39132731 \sqrt {11}}{1397}} \tanh ^{-1}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{2156}+\frac {\sqrt {\frac {325022311-39132731 \sqrt {11}}{1397}} \tanh ^{-1}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{2156} \]
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Rubi [A]
time = 0.16, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1068, 1090,
633, 221, 1046, 738, 212} \begin {gather*} \frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}-\frac {\sqrt {\frac {325022311+39132731 \sqrt {11}}{1397}} \tanh ^{-1}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{2156}+\frac {\sqrt {\frac {325022311-39132731 \sqrt {11}}{1397}} \tanh ^{-1}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{2156}+\frac {1}{49} \sqrt {5} \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 633
Rule 738
Rule 1046
Rule 1068
Rule 1090
Rubi steps
\begin {align*} \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^2} \, dx &=\frac {3 (3+61 x) \sqrt {3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}-\frac {1}{308} \int \frac {-948-188 x+220 x^2}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx\\ &=\frac {3 (3+61 x) \sqrt {3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}+\frac {\int \frac {6416+436 x}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx}{2156}+\frac {5}{49} \int \frac {1}{\sqrt {3+2 x+5 x^2}} \, dx\\ &=\frac {3 (3+61 x) \sqrt {3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}+\frac {1}{98} \sqrt {\frac {5}{14}} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{56}}} \, dx,x,2+10 x\right )+\frac {\left (1199-11446 \sqrt {11}\right ) \int \frac {1}{\left (4-2 \sqrt {11}-14 x\right ) \sqrt {3+2 x+5 x^2}} \, dx}{5929}+\frac {\left (1199+11446 \sqrt {11}\right ) \int \frac {1}{\left (4+2 \sqrt {11}-14 x\right ) \sqrt {3+2 x+5 x^2}} \, dx}{5929}\\ &=\frac {3 (3+61 x) \sqrt {3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}+\frac {1}{49} \sqrt {5} \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )-\frac {\left (2 \left (1199-11446 \sqrt {11}\right )\right ) \text {Subst}\left (\int \frac {1}{2352+112 \left (4-2 \sqrt {11}\right )+20 \left (4-2 \sqrt {11}\right )^2-x^2} \, dx,x,\frac {-84-2 \left (4-2 \sqrt {11}\right )-\left (28+10 \left (4-2 \sqrt {11}\right )\right ) x}{\sqrt {3+2 x+5 x^2}}\right )}{5929}-\frac {\left (2 \left (1199+11446 \sqrt {11}\right )\right ) \text {Subst}\left (\int \frac {1}{2352+112 \left (4+2 \sqrt {11}\right )+20 \left (4+2 \sqrt {11}\right )^2-x^2} \, dx,x,\frac {-84-2 \left (4+2 \sqrt {11}\right )-\left (28+10 \left (4+2 \sqrt {11}\right )\right ) x}{\sqrt {3+2 x+5 x^2}}\right )}{5929}\\ &=\frac {3 (3+61 x) \sqrt {3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}+\frac {1}{49} \sqrt {5} \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )-\frac {\sqrt {\frac {325022311+39132731 \sqrt {11}}{1397}} \tanh ^{-1}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{2156}+\frac {\sqrt {\frac {325022311-39132731 \sqrt {11}}{1397}} \tanh ^{-1}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{2156}\\ \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 = 427, normalized size = 2.15 \begin {gather*} \frac {-\frac {5145 (3+61 x) \sqrt {3+2 x+5 x^2}}{-1-4 x+7 x^2}-5390 \sqrt {5} \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )-55 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {-314239 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )+28462 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}-11221 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]-6 \sqrt {5} \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {599633 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )-391895 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+21462 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]}{264110} \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. \(1083\) vs.
\(2(146)=292\).
time = 0.79, size = 1084, normalized size = 5.45
method | result | size |
risch | \(-\frac {3 \left (3+61 x \right ) \sqrt {5 x^{2}+2 x +3}}{154 \left (7 x^{2}-4 x -1\right )}+\frac {\sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{49}+\frac {\left (-11446+109 \sqrt {11}\right ) \sqrt {11}\, \arctanh \left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{11858 \sqrt {250-34 \sqrt {11}}}+\frac {\left (11446+109 \sqrt {11}\right ) \sqrt {11}\, \arctanh \left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{11858 \sqrt {250+34 \sqrt {11}}}\) | \(235\) |
trager | \(\text {Expression too large to display}\) | \(512\) |
default | \(\text {Expression too large to display}\) | \(1084\) |
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. 378 vs.
\(2 (145) = 290\).
time = 0.38, size = 378, normalized size = 1.90 \begin {gather*} -\frac {\sqrt {1397} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {39132731 \, \sqrt {11} + 325022311} \log \left (-\frac {\sqrt {1397} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {39132731 \, \sqrt {11} + 325022311} {\left (16943 \, \sqrt {11} + 235367\right )} + 26119953475 \, \sqrt {11} {\left (x + 3\right )} - 78359860425 \, x + 130599767375}{x}\right ) - \sqrt {1397} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {39132731 \, \sqrt {11} + 325022311} \log \left (\frac {\sqrt {1397} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {39132731 \, \sqrt {11} + 325022311} {\left (16943 \, \sqrt {11} + 235367\right )} - 26119953475 \, \sqrt {11} {\left (x + 3\right )} + 78359860425 \, x - 130599767375}{x}\right ) + \sqrt {1397} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {-39132731 \, \sqrt {11} + 325022311} \log \left (\frac {\sqrt {1397} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (16943 \, \sqrt {11} - 235367\right )} \sqrt {-39132731 \, \sqrt {11} + 325022311} + 26119953475 \, \sqrt {11} {\left (x + 3\right )} + 78359860425 \, x - 130599767375}{x}\right ) - \sqrt {1397} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {-39132731 \, \sqrt {11} + 325022311} \log \left (-\frac {\sqrt {1397} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (16943 \, \sqrt {11} - 235367\right )} \sqrt {-39132731 \, \sqrt {11} + 325022311} - 26119953475 \, \sqrt {11} {\left (x + 3\right )} - 78359860425 \, x + 130599767375}{x}\right ) - 61468 \, \sqrt {5} {\left (7 \, x^{2} - 4 \, x - 1\right )} \log \left (-\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) + 117348 \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (61 \, x + 3\right )}}{6023864 \, {\left (7 \, x^{2} - 4 \, x - 1\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 (x^{2} + 5 x + 2\right ) \sqrt {5 x^{2} + 2 x + 3}}{\left (7 x^{2} - 4 x - 1\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 = -0.01 \begin {gather*} \int \frac {\left (x^2+5\,x+2\right )\,\sqrt {5\,x^2+2\,x+3}}{{\left (-7\,x^2+4\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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