Optimal. Leaf size=270 \[ \frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}+\frac {1}{31} \sqrt {\frac {1}{434} \left (47 \sqrt {35}-218\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{31} \sqrt {\frac {1}{434} \left (47 \sqrt {35}-218\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]
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Rubi [A] time = 0.34, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {736, 826, 1169, 634, 618, 204, 628} \[ \frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}+\frac {1}{31} \sqrt {\frac {1}{434} \left (47 \sqrt {35}-218\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{31} \sqrt {\frac {1}{434} \left (47 \sqrt {35}-218\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 736
Rule 826
Rule 1169
Rubi steps
\begin {align*} \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \int \frac {-7-10 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {2}{31} \operatorname {Subst}\left (\int \frac {-4-10 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-4 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-4+2 \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 )}{31 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {-4 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-4+2 \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 )}{31 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}+\frac {\left (35+2 \sqrt {35}\right ) \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 )}{1085}+\frac {\left (35+2 \sqrt {35}\right ) \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 )}{1085}+\frac {1}{31} \sqrt {\frac {1}{434} \left (-218+47 \sqrt {35}\right )} \operatorname {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}{31} \sqrt {\frac {1}{434} \left (-218+47 \sqrt {35}\right )} \operatorname {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 {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \sqrt {\frac {1}{434} \left (-218+47 \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}{31} \sqrt {\frac {1}{434} \left (-218+47 \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 (35+2 \sqrt {35}\right )\right ) \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 )}{1085}-\frac {\left (2 \left (35+2 \sqrt {35}\right )\right ) \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 )}{1085}\\ &=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \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}{31} \sqrt {\frac {2}{217} \left (218+47 \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}{31} \sqrt {\frac {1}{434} \left (-218+47 \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}{31} \sqrt {\frac {1}{434} \left (-218+47 \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] time = 0.36, size = 145, normalized size = 0.54 \[ \frac {1}{31} \left (\frac {\sqrt {2 x+1} (10 x+3)}{5 x^2+3 x+2}+\frac {2 \sqrt {10-5 i \sqrt {31}} \left (62-39 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+2 \sqrt {10+5 i \sqrt {31}} \left (62+39 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )}{1085}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 526, normalized size = 1.95 \[ \frac {3844 \cdot 77315^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {20492 \, \sqrt {35} + 154630} \arctan \left (\frac {1}{26522397764975} \cdot 77315^{\frac {3}{4}} \sqrt {1645} \sqrt {217} \sqrt {77315^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} - 2 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 3161690 \, x + 316169 \, \sqrt {35} + 1580845} \sqrt {20492 \, \sqrt {35} + 154630} {\left (2 \, \sqrt {35} - 35\right )} - \frac {1}{520098005} \cdot 77315^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} {\left (2 \, \sqrt {35} - 35\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 3844 \cdot 77315^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {20492 \, \sqrt {35} + 154630} \arctan \left (\frac {1}{53044795529950} \cdot 77315^{\frac {3}{4}} \sqrt {217} \sqrt {-6580 \cdot 77315^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} - 2 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 20803920200 \, x + 2080392020 \, \sqrt {35} + 10401960100} \sqrt {20492 \, \sqrt {35} + 154630} {\left (2 \, \sqrt {35} - 35\right )} - \frac {1}{520098005} \cdot 77315^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} {\left (2 \, \sqrt {35} - 35\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 77315^{\frac {1}{4}} \sqrt {217} {\left (218 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} - 1645 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {20492 \, \sqrt {35} + 154630} \log \left (6580 \cdot 77315^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} - 2 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 20803920200 \, x + 2080392020 \, \sqrt {35} + 10401960100\right ) - 77315^{\frac {1}{4}} \sqrt {217} {\left (218 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} - 1645 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {20492 \, \sqrt {35} + 154630} \log \left (-6580 \cdot 77315^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} - 2 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {20492 \, \sqrt {35} + 154630} + 20803920200 \, x + 2080392020 \, \sqrt {35} + 10401960100\right ) + 686086730 \, {\left (10 \, x + 3\right )} \sqrt {2 \, x + 1}}{21268688630 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.29, size = 622, normalized size = 2.30 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.94, size = 972, normalized size = 3.60 \[ \frac {218 \sqrt {20+10 \sqrt {35}}\, \sqrt {2 \sqrt {35}+4}\, \sqrt {35}\, \arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {2 x +1}}{\sqrt {-20+10 \sqrt {35}}}\right )}{6727 \left (2 \sqrt {5}-5 \sqrt {7}\right ) \sqrt {-20+10 \sqrt {35}}}-\frac {235 \sqrt {20+10 \sqrt {35}}\, \sqrt {2 \sqrt {35}+4}\, \arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {2 x +1}}{\sqrt {-20+10 \sqrt {35}}}\right )}{961 \left (2 \sqrt {5}-5 \sqrt {7}\right ) \sqrt {-20+10 \sqrt {35}}}-\frac {40 \sqrt {5}\, \arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {2 x +1}}{\sqrt {-20+10 \sqrt {35}}}\right )}{31 \left (2 \sqrt {5}-5 \sqrt {7}\right ) \sqrt {-20+10 \sqrt {35}}}+\frac {80 \sqrt {7}\, \arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {2 x +1}}{\sqrt {-20+10 \sqrt {35}}}\right )}{217 \left (2 \sqrt {5}-5 \sqrt {7}\right ) \sqrt {-20+10 \sqrt {35}}}+\frac {218 \sqrt {20+10 \sqrt {35}}\, \sqrt {2 \sqrt {35}+4}\, \sqrt {35}\, \arctan \left (\frac {10 \sqrt {2 x +1}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{6727 \left (2 \sqrt {5}-5 \sqrt {7}\right ) \sqrt {-20+10 \sqrt {35}}}-\frac {235 \sqrt {20+10 \sqrt {35}}\, \sqrt {2 \sqrt {35}+4}\, \arctan \left (\frac {10 \sqrt {2 x +1}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{961 \left (2 \sqrt {5}-5 \sqrt {7}\right ) \sqrt {-20+10 \sqrt {35}}}-\frac {40 \sqrt {5}\, \arctan \left (\frac {10 \sqrt {2 x +1}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{31 \left (2 \sqrt {5}-5 \sqrt {7}\right ) \sqrt {-20+10 \sqrt {35}}}+\frac {80 \sqrt {7}\, \arctan \left (\frac {10 \sqrt {2 x +1}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{217 \left (2 \sqrt {5}-5 \sqrt {7}\right ) \sqrt {-20+10 \sqrt {35}}}+\frac {109 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}\, \ln \left (10 x +5+\sqrt {35}-\sqrt {2 x +1}\, \sqrt {20+10 \sqrt {35}}\right )}{6727 \left (2 \sqrt {5}-5 \sqrt {7}\right )}-\frac {235 \sqrt {2 \sqrt {35}+4}\, \ln \left (10 x +5+\sqrt {35}-\sqrt {2 x +1}\, \sqrt {20+10 \sqrt {35}}\right )}{1922 \left (2 \sqrt {5}-5 \sqrt {7}\right )}-\frac {109 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}\, \ln \left (10 x +5+\sqrt {35}+\sqrt {2 x +1}\, \sqrt {20+10 \sqrt {35}}\right )}{6727 \left (2 \sqrt {5}-5 \sqrt {7}\right )}+\frac {235 \sqrt {2 \sqrt {35}+4}\, \ln \left (10 x +5+\sqrt {35}+\sqrt {2 x +1}\, \sqrt {20+10 \sqrt {35}}\right )}{1922 \left (2 \sqrt {5}-5 \sqrt {7}\right )}+\frac {\frac {2 \left (-5425 \sqrt {7}+2170 \sqrt {5}\right ) \sqrt {2 x +1}}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}+\frac {5 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1085 \sqrt {5}+310 \sqrt {7}\right )}{6727 \left (50 \sqrt {5}-125 \sqrt {7}\right )}}{2 x +\frac {\sqrt {5}\, \sqrt {7}}{5}+\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}}{5}+1}-\frac {5 \left (-\frac {2 \left (-5425 \sqrt {7}+2170 \sqrt {5}\right ) \sqrt {2 x +1}}{25 \left (2 \sqrt {5}-5 \sqrt {7}\right )}+\frac {\sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1085 \sqrt {5}+310 \sqrt {7}\right )}{50 \sqrt {5}-125 \sqrt {7}}\right )}{6727 \left (2 x +\frac {\sqrt {5}\, \sqrt {7}}{5}-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}}{5}+1\right )} \]
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 \[ \int \frac {\sqrt {2 \, x + 1}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 208, normalized size = 0.77 \[ -\frac {\frac {8\,\sqrt {2\,x+1}}{155}-\frac {4\,{\left (2\,x+1\right )}^{3/2}}{31}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{5886125\,\left (-\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}-\frac {256\,\sqrt {31}\,\sqrt {217}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{182469875\,\left (-\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}\right )\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{6727}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{5886125\,\left (\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}+\frac {256\,\sqrt {31}\,\sqrt {217}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{182469875\,\left (\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}\right )\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{6727} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.85, size = 83, normalized size = 0.31 \[ \frac {80 \left (2 x + 1\right )^{\frac {3}{2}}}{- 992 x + 620 \left (2 x + 1\right )^{2} + 372} - \frac {32 \sqrt {2 x + 1}}{- 992 x + 620 \left (2 x + 1\right )^{2} + 372} + 16 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left (t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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