Optimal. Leaf size=300 \[ \frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{6727}+\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{6727}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454} \]
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Rubi [A]
time = 0.26, antiderivative size = 300, 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 = {750, 836, 840,
1183, 648, 632, 210, 642} \begin {gather*} -\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \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 )}{6727}+\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \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 )}{6727}+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (1790 x+599)}{13454 \left (5 x^2+3 x+2\right )}+\frac {\sqrt {\frac {1}{434} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{13454} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 750
Rule 836
Rule 840
Rule 1183
Rubi steps
\begin {align*} \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}-\frac {1}{62} \int \frac {-27-50 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-1439-1790 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{13454}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\text {Subst}\left (\int \frac {-1088-1790 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{6727}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\text {Subst}\left (\int \frac {-1088 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-1088+358 \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 )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\text {Subst}\left (\int \frac {-1088 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-1088+358 \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 )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}+\frac {\left (6265+544 \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 )}{470890}+\frac {\left (6265+544 \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 )}{470890}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \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 )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \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 )}{13454}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\left (6265+544 \sqrt {35}\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 )}{235445}-\frac {\left (6265+544 \sqrt {35}\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 )}{235445}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \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 )}{6727}+\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \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 )}{6727}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.66, size = 141, normalized size = 0.47 \begin {gather*} \frac {\frac {217 \sqrt {1+2 x} \left (1849+7547 x+8365 x^2+8950 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+\sqrt {217 \left (9651062-825499 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {217 \left (9651062+825499 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{1459759} \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. \(698\) vs.
\(2(210)=420\).
time = 2.73, size = 699, normalized size = 2.33 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. 652 vs.
\(2 (213) = 426\).
time = 2.58, size = 652, normalized size = 2.17 \begin {gather*} \frac {102361876 \cdot 121835^{\frac {1}{4}} \sqrt {217} \sqrt {118} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {9651062 \, \sqrt {35} + 63240625} \arctan \left (\frac {1}{37203599497016727197675} \cdot 121835^{\frac {3}{4}} \sqrt {26629} \sqrt {2065} \sqrt {217} \sqrt {118} \sqrt {121835^{\frac {1}{4}} \sqrt {217} \sqrt {118} {\left (179 \, \sqrt {35} \sqrt {31} - 544 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} + 105688636970 \, x + 10568863697 \, \sqrt {35} + 52844318485} \sqrt {9651062 \, \sqrt {35} + 63240625} {\left (544 \, \sqrt {35} - 6265\right )} - \frac {1}{21824703534305} \cdot 121835^{\frac {3}{4}} \sqrt {217} \sqrt {118} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} {\left (544 \, \sqrt {35} - 6265\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 102361876 \cdot 121835^{\frac {1}{4}} \sqrt {217} \sqrt {118} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {9651062 \, \sqrt {35} + 63240625} \arctan \left (\frac {1}{1139360234596137270428796875} \cdot 121835^{\frac {3}{4}} \sqrt {26629} \sqrt {217} \sqrt {118} \sqrt {-1936744140625 \cdot 121835^{\frac {1}{4}} \sqrt {217} \sqrt {118} {\left (179 \, \sqrt {35} \sqrt {31} - 544 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} + 204691848382290253906250 \, x + 20469184838229025390625 \, \sqrt {35} + 102345924191145126953125} \sqrt {9651062 \, \sqrt {35} + 63240625} {\left (544 \, \sqrt {35} - 6265\right )} - \frac {1}{21824703534305} \cdot 121835^{\frac {3}{4}} \sqrt {217} \sqrt {118} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} {\left (544 \, \sqrt {35} - 6265\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 121835^{\frac {1}{4}} \sqrt {217} \sqrt {118} {\left (9651062 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 63240625 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {9651062 \, \sqrt {35} + 63240625} \log \left (\frac {1936744140625}{26629} \cdot 121835^{\frac {1}{4}} \sqrt {217} \sqrt {118} {\left (179 \, \sqrt {35} \sqrt {31} - 544 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} + 7686801922050781250 \, x + 768680192205078125 \, \sqrt {35} + 3843400961025390625\right ) - 121835^{\frac {1}{4}} \sqrt {217} \sqrt {118} {\left (9651062 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 63240625 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {9651062 \, \sqrt {35} + 63240625} \log \left (-\frac {1936744140625}{26629} \cdot 121835^{\frac {1}{4}} \sqrt {217} \sqrt {118} {\left (179 \, \sqrt {35} \sqrt {31} - 544 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} + 7686801922050781250 \, x + 768680192205078125 \, \sqrt {35} + 3843400961025390625\right ) + 22934434222490 \, {\left (8950 \, x^{3} + 8365 \, x^{2} + 7547 \, x + 1849\right )} \sqrt {2 \, x + 1}}{308559878029380460 \, {\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 [A]
time = 4.86, size = 199, normalized size = 0.66 \begin {gather*} \frac {286400 \left (2 x + 1\right )^{\frac {7}{2}}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} - \frac {323840 \left (2 x + 1\right )^{\frac {5}{2}}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} + \frac {754496 \left (2 x + 1\right )^{\frac {3}{2}}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} - \frac {243712 \sqrt {2 x + 1}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} + 64 \operatorname {RootSum} {\left (75465931487403231630327808 t^{4} + 9053854476152406016 t^{2} + 333142578125, \left ( t \mapsto t \log {\left (\frac {21632117045402271744 t^{3}}{158378125} + \frac {10865340674816 t}{1108646875} + \sqrt {2 x + 1} \right )} \right )\right )} \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.
\(2 (213) = 426\).
time = 1.69, size = 642, normalized size = 2.14 \begin {gather*} \frac {1}{100139467400} \, \sqrt {31} {\left (37590 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 179 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 358 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 75180 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 533120 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 1066240 \, \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}{100139467400} \, \sqrt {31} {\left (37590 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 179 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 358 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 75180 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 533120 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 1066240 \, \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}{200278934800} \, \sqrt {31} {\left (179 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 37590 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 75180 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 358 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 533120 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 1066240 \, \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}{200278934800} \, \sqrt {31} {\left (179 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 37590 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 75180 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 358 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 533120 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 1066240 \, \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 (4475 \, {\left (2 \, x + 1\right )}^{\frac {7}{2}} - 5060 \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} + 11789 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} - 3808 \, \sqrt {2 \, x + 1}\right )}}{6727 \, {\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 = 1.06, size = 245, normalized size = 0.82 \begin {gather*} \frac {\frac {1088\,\sqrt {2\,x+1}}{24025}-\frac {23578\,{\left (2\,x+1\right )}^{3/2}}{168175}+\frac {2024\,{\left (2\,x+1\right )}^{5/2}}{33635}-\frac {358\,{\left (2\,x+1\right )}^{7/2}}{6727}}{\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 {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{1940202180875\,\left (-\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}-\frac {27488\,\sqrt {31}\,\sqrt {217}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{60146267607125\,\left (-\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}\right )\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{1459759}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{1940202180875\,\left (\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}+\frac {27488\,\sqrt {31}\,\sqrt {217}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{60146267607125\,\left (\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}\right )\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{1459759} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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