Optimal. Leaf size=105 \[ 2 \sqrt {\frac {10}{\sqrt {35}-2}} \tan ^{-1}\left (\frac {\sqrt {20 x+10}+\sqrt {2+\sqrt {35}}}{\sqrt {\sqrt {35}-2}}\right )-2 \sqrt {\frac {10}{\sqrt {35}-2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {35}}-\sqrt {20 x+10}}{\sqrt {\sqrt {35}-2}}\right ) \]
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Rubi [A] time = 0.24, antiderivative size = 115, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {826, 1161, 618, 204} \begin {gather*} 2 \sqrt {\frac {10}{\sqrt {35}-2}} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-2 \sqrt {\frac {10}{\sqrt {35}-2}} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 826
Rule 1161
Rubi steps
\begin {align*} \int \frac {5+\sqrt {35}+10 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {-10+2 \left (5+\sqrt {35}\right )+10 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=2 \operatorname {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 )+2 \operatorname {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 )\\ &=-\left (4 \operatorname {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 )\right )-4 \operatorname {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 )\\ &=-2 \sqrt {\frac {10}{-2+\sqrt {35}}} \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 )+2 \sqrt {\frac {10}{-2+\sqrt {35}}} \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 )\\ \end {align*}
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Mathematica [C] time = 0.27, size = 141, normalized size = 1.34 \begin {gather*} \frac {2}{217} \left (\sqrt {2-i \sqrt {31}} \left (31 \sqrt {7}-7 i \sqrt {155}-2 i \sqrt {217}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+\sqrt {2+i \sqrt {31}} \left (31 \sqrt {7}+7 i \sqrt {155}+2 i \sqrt {217}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.38, size = 73, normalized size = 0.70 \begin {gather*} 2 \sqrt {\frac {1}{31} \left (20+10 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{434} \left (70+35 \sqrt {35}\right )} (2 x+1)-\sqrt {\frac {1}{62} \left (14+7 \sqrt {35}\right )}}{\sqrt {2 x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 52, normalized size = 0.50 \begin {gather*} -\frac {2}{31} \, \sqrt {31} \sqrt {10 \, \sqrt {35} + 20} \arctan \left (-\frac {{\left (5 \, \sqrt {31} {\left (2 \, x + 1\right )} - \sqrt {35} \sqrt {31}\right )} \sqrt {10 \, \sqrt {35} + 20}}{310 \, \sqrt {2 \, x + 1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.01, size = 313, normalized size = 2.98 \begin {gather*} \frac {1}{7443100} \, \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 )} + 980 \, \sqrt {35} \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 1960 \, \sqrt {35} \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}{7443100} \, \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 )} + 980 \, \sqrt {35} \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 1960 \, \sqrt {35} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 111, normalized size = 1.06 \begin {gather*} \frac {20 \arctan \left (\frac {10 \sqrt {2 x +1}-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {20 \arctan \left (\frac {10 \sqrt {2 x +1}+\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}} \end {gather*}
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*} \int \frac {10 \, x + \sqrt {35} + 5}{{\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {2 \, x + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.27, size = 143, normalized size = 1.36 \begin {gather*} 2\,\sqrt {\frac {10\,\sqrt {35}}{31}+\frac {20}{31}}\,\left (\mathrm {atan}\left (\frac {\sqrt {434}\,\left (39\,\sqrt {35}+140\right )\,\sqrt {2\,x+1}\,{\left (\sqrt {35}-2\right )}^2\,\sqrt {\sqrt {35}+2}}{417074}\right )+\mathrm {atan}\left (\frac {31\,\sqrt {2\,x+1}\,\left (\frac {\sqrt {\frac {10\,\sqrt {35}}{31}+\frac {20}{31}}\,\left (10000\,\sqrt {35}+20000\right )}{39\,\sqrt {35}+140}-\frac {\sqrt {434}\,\left (\frac {390000\,\sqrt {35}}{31}+\frac {1400000}{31}\right )\,{\left (\sqrt {35}-2\right )}^2\,\sqrt {\sqrt {35}+2}}{417074}\right )}{10000}+\frac {\sqrt {434}\,\left (\frac {200000\,\sqrt {35}}{31}+\frac {1950000}{31}\right )\,{\left (2\,x+1\right )}^{3/2}\,{\left (\sqrt {35}-2\right )}^2\,\sqrt {\sqrt {35}+2}}{134540000}\right )\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 {10 x + 5 + \sqrt {35}}{\sqrt {2 x + 1} \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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