Optimal. Leaf size=314 \[ \frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \sqrt {35}\right ) \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \sqrt {35}\right ) \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{47089}-\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178} \]
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Rubi [A]
time = 0.27, antiderivative size = 314, 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 = {754, 836, 840,
1183, 648, 632, 210, 642} \begin {gather*} -\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \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 )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \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 )}{47089}+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (7920 x+9227)}{94178 \left (5 x^2+3 x+2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (64681225 \sqrt {35}-250141922\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (64681225 \sqrt {35}-250141922\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{94178} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 754
Rule 836
Rule 840
Rule 1183
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {1}{434} \int \frac {271+100 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {26097+7920 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{94178}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {44274+7920 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{47089}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {44274 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (44274-1584 \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 )}{94178 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {44274 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (44274-1584 \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 )}{94178 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\left (3 \left (9240+7379 \sqrt {35}\right )\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 )}{3296230}+\frac {\left (3 \left (9240+7379 \sqrt {35}\right )\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 )}{3296230}-\frac {\left (3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )}\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 )}{94178}+\frac {\left (3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )}\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 )}{94178}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}-\frac {\left (3 \left (9240+7379 \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 )}{1648115}-\frac {\left (3 \left (9240+7379 \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 )}{1648115}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (250141922+64681225 \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 )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (250141922+64681225 \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 )}{47089}-\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.71, size = 143, normalized size = 0.46 \begin {gather*} \frac {\frac {217 \sqrt {1+2 x} \left (26483+47861 x+69895 x^2+39600 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+3 \sqrt {217 \left (250141922+52010281 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 (250141922-52010281 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{10218313} \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. \(690\) vs.
\(2(224)=448\).
time = 3.20, size = 691, normalized size = 2.20 Too large to display
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. 653 vs.
\(2 (227) = 454\).
time = 2.57, size = 653, normalized size = 2.08 \begin {gather*} -\frac {19347824532 \cdot 97578096035^{\frac {1}{4}} \sqrt {105602} \sqrt {217} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {250141922 \, \sqrt {35} + 2263842875} \arctan \left (\frac {1}{15121769925583791519919258475683975} \cdot 97578096035^{\frac {3}{4}} \sqrt {1677751} \sqrt {105602} \sqrt {37715} \sqrt {217} \sqrt {97578096035^{\frac {1}{4}} \sqrt {105602} \sqrt {217} {\left (264 \, \sqrt {35} \sqrt {31} - 7379 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {250141922 \, \sqrt {35} + 2263842875} + 5959242818165770 \, x + 595924281816577 \, \sqrt {35} + 2979621409082885} \sqrt {250141922 \, \sqrt {35} + 2263842875} {\left (7379 \, \sqrt {35} - 9240\right )} - \frac {1}{1101288930146897876195} \cdot 97578096035^{\frac {3}{4}} \sqrt {105602} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {250141922 \, \sqrt {35} + 2263842875} {\left (7379 \, \sqrt {35} - 9240\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 19347824532 \cdot 97578096035^{\frac {1}{4}} \sqrt {105602} \sqrt {217} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {250141922 \, \sqrt {35} + 2263842875} \arctan \left (\frac {1}{389007531335643036849922924286970256875} \cdot 97578096035^{\frac {3}{4}} \sqrt {1677751} \sqrt {105602} \sqrt {217} \sqrt {-24958867696875 \cdot 97578096035^{\frac {1}{4}} \sqrt {105602} \sqrt {217} {\left (264 \, \sqrt {35} \sqrt {31} - 7379 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {250141922 \, \sqrt {35} + 2263842875} + 148735953072151976291860968750 \, x + 14873595307215197629186096875 \, \sqrt {35} + 74367976536075988145930484375} \sqrt {250141922 \, \sqrt {35} + 2263842875} {\left (7379 \, \sqrt {35} - 9240\right )} - \frac {1}{1101288930146897876195} \cdot 97578096035^{\frac {3}{4}} \sqrt {105602} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {250141922 \, \sqrt {35} + 2263842875} {\left (7379 \, \sqrt {35} - 9240\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) - 3 \cdot 97578096035^{\frac {1}{4}} \sqrt {105602} \sqrt {217} {\left (250141922 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 2263842875 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {250141922 \, \sqrt {35} + 2263842875} \log \left (\frac {24958867696875}{1677751} \cdot 97578096035^{\frac {1}{4}} \sqrt {105602} \sqrt {217} {\left (264 \, \sqrt {35} \sqrt {31} - 7379 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {250141922 \, \sqrt {35} + 2263842875} + 88651982965381618781250 \, x + 8865198296538161878125 \, \sqrt {35} + 44325991482690809390625\right ) + 3 \cdot 97578096035^{\frac {1}{4}} \sqrt {105602} \sqrt {217} {\left (250141922 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 2263842875 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {250141922 \, \sqrt {35} + 2263842875} \log \left (-\frac {24958867696875}{1677751} \cdot 97578096035^{\frac {1}{4}} \sqrt {105602} \sqrt {217} {\left (264 \, \sqrt {35} \sqrt {31} - 7379 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {250141922 \, \sqrt {35} + 2263842875} + 88651982965381618781250 \, x + 8865198296538161878125 \, \sqrt {35} + 44325991482690809390625\right ) - 1293155691541972090 \, {\left (39600 \, x^{3} + 69895 \, x^{2} + 47861 \, x + 26483\right )} \sqrt {2 \, x + 1}}{121786816718039847492020 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\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}{\sqrt {2 x + 1} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 642 vs.
\(2 (227) = 454\).
time = 2.34, size = 642, normalized size = 2.04 \begin {gather*} \frac {3}{175244067950} \, \sqrt {31} {\left (13860 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 66 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 132 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 27720 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 1807855 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 3615710 \, \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 {3}{175244067950} \, \sqrt {31} {\left (13860 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 66 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 132 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 27720 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 1807855 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 3615710 \, \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 {3}{350488135900} \, \sqrt {31} {\left (66 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 13860 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 27720 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 132 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 1807855 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 3615710 \, \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 {3}{350488135900} \, \sqrt {31} {\left (66 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 13860 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 27720 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 132 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 1807855 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 3615710 \, \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 {2 \, {\left (19800 \, {\left (2 \, x + 1\right )}^{\frac {7}{2}} + 10495 \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} + 15332 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} + 60305 \, \sqrt {2 \, x + 1}\right )}}{47089 \, {\left (5 \, {\left (2 \, x + 1\right )}^{2} - 8 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 246, normalized size = 0.78 \begin {gather*} -\frac {\frac {3446\,\sqrt {2\,x+1}}{33635}+\frac {30664\,{\left (2\,x+1\right )}^{3/2}}{1177225}+\frac {4198\,{\left (2\,x+1\right )}^{5/2}}{235445}+\frac {1584\,{\left (2\,x+1\right )}^{7/2}}{47089}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{665489348040125\,\left (\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}+\frac {46760544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{20630169789243875\,\left (\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}\right )\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{10218313}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{665489348040125\,\left (-\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}-\frac {46760544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{20630169789243875\,\left (-\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}\right )\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{10218313} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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