Optimal. Leaf size=266 \[ -\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]
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Rubi [A]
time = 0.26, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {723, 842, 840,
1183, 648, 632, 210, 642} \begin {gather*} \frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \text {ArcTan}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \text {ArcTan}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {16}{49 \sqrt {2 x+1}}-\frac {4}{21 (2 x+1)^{3/2}}-\frac {1}{49} \sqrt {\frac {1}{434} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 723
Rule 840
Rule 842
Rule 1183
Rubi steps
\begin {align*} \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx &=-\frac {4}{21 (1+2 x)^{3/2}}+\frac {1}{7} \int \frac {-1-10 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {1}{49} \int \frac {-39-40 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {2}{49} \text {Subst}\left (\int \frac {-38-40 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {\text {Subst}\left (\int \frac {-38 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-38+8 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{49 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {-38 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-38+8 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{49 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}-\frac {\left (140+19 \sqrt {35}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{1715}-\frac {\left (140+19 \sqrt {35}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{1715}-\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \text {Subst}\left (\int \frac {-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \text {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}-\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {\left (2 \left (140+19 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{1715}+\frac {\left (2 \left (140+19 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{1715}\\ &=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-2 \sqrt {1+2 x}\right )\right )-\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )\right )-\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.65, size = 119, normalized size = 0.45 \begin {gather*} \frac {2 \left (-\frac {434 (19+24 x)}{(1+2 x)^{3/2}}-3 \sqrt {217 \left (7162-199 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-3 \sqrt {217 \left (7162+199 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{31899} \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. \(397\) vs.
\(2(176)=352\).
time = 1.76, size = 398, normalized size = 1.50
method | result | size |
derivativedivides | \(-\frac {4}{21 \left (2 x +1\right )^{\frac {3}{2}}}-\frac {16}{49 \sqrt {2 x +1}}+\frac {\left (-945 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+10 x +5+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}\right )}{106330}+\frac {2 \left (-1178 \sqrt {5}\, \sqrt {7}-\frac {\left (-945 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{10}\right ) \arctan \left (\frac {10 \sqrt {2 x +1}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (945 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+\sqrt {5}\, \sqrt {7}+10 x +5\right )}{106330}+\frac {2 \left (-1178 \sqrt {5}\, \sqrt {7}+\frac {\left (945 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{10}\right ) \arctan \left (\frac {-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(398\) |
default | \(-\frac {4}{21 \left (2 x +1\right )^{\frac {3}{2}}}-\frac {16}{49 \sqrt {2 x +1}}+\frac {\left (-945 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+10 x +5+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}\right )}{106330}+\frac {2 \left (-1178 \sqrt {5}\, \sqrt {7}-\frac {\left (-945 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{10}\right ) \arctan \left (\frac {10 \sqrt {2 x +1}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (945 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+\sqrt {5}\, \sqrt {7}+10 x +5\right )}{106330}+\frac {2 \left (-1178 \sqrt {5}\, \sqrt {7}+\frac {\left (945 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}}{10}\right ) \arctan \left (\frac {-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(398\) |
trager | \(-\frac {4 \left (24 x +19\right )}{147 \left (2 x +1\right )^{\frac {3}{2}}}+\frac {\RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right ) \ln \left (\frac {5132701 x \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{5}+374431547 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{3} x +37655576 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{3}+31739505 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} \sqrt {2 x +1}+6784080780 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right ) x +1262449632 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )-301062125 \sqrt {2 x +1}}{217 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} x +6565 x -796}\right )}{49}+\frac {\RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) \ln \left (\frac {733243 \RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{4} x +43311371 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} \RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) x -983924655 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} \sqrt {2 x +1}-5379368 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} \RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right )+633205440 \RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) x -74281021535 \sqrt {2 x +1}-174737920 \RootOf \left (\textit {\_Z}^{2}+47089 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right )}{217 \RootOf \left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} x +7759 x +796}\right )}{10633}\) | \(435\) |
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. 567 vs.
\(2 (179) = 358\).
time = 2.79, size = 567, normalized size = 2.13 \begin {gather*} -\frac {74028 \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {7162 \, \sqrt {35} + 42875} \arctan \left (\frac {1}{326335010575} \, \sqrt {1085} \sqrt {217} \sqrt {199} 35^{\frac {3}{4}} \sqrt {2} \sqrt {\sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (4 \, \sqrt {35} \sqrt {31} - 19 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 431830 \, x + 43183 \, \sqrt {35} + 215915} \sqrt {7162 \, \sqrt {35} + 42875} {\left (19 \, \sqrt {35} - 140\right )} - \frac {1}{1511405} \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} {\left (19 \, \sqrt {35} - 140\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 74028 \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {7162 \, \sqrt {35} + 42875} \arctan \left (\frac {1}{799520775908750} \, \sqrt {217} \sqrt {199} 35^{\frac {3}{4}} \sqrt {2} \sqrt {-6512712500 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (4 \, \sqrt {35} \sqrt {31} - 19 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 2812384638875000 \, x + 281238463887500 \, \sqrt {35} + 1406192319437500} \sqrt {7162 \, \sqrt {35} + 42875} {\left (19 \, \sqrt {35} - 140\right )} - \frac {1}{1511405} \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} {\left (19 \, \sqrt {35} - 140\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 3 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (7162 \, \sqrt {35} \sqrt {31} {\left (4 \, x^{2} + 4 \, x + 1\right )} - 42875 \, \sqrt {31} {\left (4 \, x^{2} + 4 \, x + 1\right )}\right )} \sqrt {7162 \, \sqrt {35} + 42875} \log \left (\frac {6512712500}{199} \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (4 \, \sqrt {35} \sqrt {31} - 19 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 14132586125000 \, x + 1413258612500 \, \sqrt {35} + 7066293062500\right ) - 3 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (7162 \, \sqrt {35} \sqrt {31} {\left (4 \, x^{2} + 4 \, x + 1\right )} - 42875 \, \sqrt {31} {\left (4 \, x^{2} + 4 \, x + 1\right )}\right )} \sqrt {7162 \, \sqrt {35} + 42875} \log \left (-\frac {6512712500}{199} \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {2} {\left (4 \, \sqrt {35} \sqrt {31} - 19 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {7162 \, \sqrt {35} + 42875} + 14132586125000 \, x + 1413258612500 \, \sqrt {35} + 7066293062500\right ) + 374828440 \, {\left (24 \, x + 19\right )} \sqrt {2 \, x + 1}}{13774945170 \, {\left (4 \, 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 {1}{\left (2 x + 1\right )^{\frac {5}{2}} \cdot \left (5 x^{2} + 3 x + 2\right )}\, 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. 599 vs.
\(2 (179) = 358\).
time = 1.71, size = 599, normalized size = 2.25 \begin {gather*} -\frac {1}{91177975} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 4655 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 9310 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) - \frac {1}{91177975} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 4655 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 9310 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (-\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} - \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) - \frac {1}{182355950} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 4655 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 9310 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) + \frac {1}{182355950} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 4655 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 9310 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (-2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) - \frac {4 \, {\left (24 \, x + 19\right )}}{147 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 187, normalized size = 0.70 \begin {gather*} -\frac {\frac {32\,x}{49}+\frac {76}{147}}{{\left (2\,x+1\right )}^{3/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{720600125\,\left (-\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}-\frac {50944\,\sqrt {31}\,\sqrt {217}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{22338603875\,\left (-\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}\right )\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{10633}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{720600125\,\left (\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}+\frac {50944\,\sqrt {31}\,\sqrt {217}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{22338603875\,\left (\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}\right )\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{10633} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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