Optimal. Leaf size=35 \[ \log \left (2 x^2+2 \sqrt {x^4+2 x^3-2 x^2-3 x-4}+2 x-3\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 39, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1680, 12, 1107, 621, 206} \begin {gather*} -\tanh ^{-1}\left (\frac {7-4 \left (x+\frac {1}{2}\right )^2}{\sqrt {16 \left (x+\frac {1}{2}\right )^4-56 \left (x+\frac {1}{2}\right )^2-51}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 621
Rule 1107
Rule 1680
Rubi steps
\begin {align*} \int \frac {1+2 x}{\sqrt {-4-3 x-2 x^2+2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {-51-56 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {-51-56 x^2+16 x^4}} \, dx,x,\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-51-56 x+16 x^2}} \, dx,x,\left (\frac {1}{2}+x\right )^2\right )\\ &=8 \operatorname {Subst}\left (\int \frac {1}{64-x^2} \, dx,x,\frac {8 \left (-7+4 \left (\frac {1}{2}+x\right )^2\right )}{\sqrt {-51-56 \left (\frac {1}{2}+x\right )^2+(1+2 x)^4}}\right )\\ &=-\tanh ^{-1}\left (\frac {7-(1+2 x)^2}{\sqrt {-51-14 (1+2 x)^2+(1+2 x)^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.98, size = 505, normalized size = 14.43 \begin {gather*} \frac {6 \sqrt [6]{-1} \left (-2 x+\sqrt {17}-1\right ) \left ((-1)^{2/3}-x\right ) \sqrt {\frac {i \left (2 x+\sqrt {17}+1\right )}{\left (\sqrt {17}-i \sqrt {3}\right ) \left ((-1)^{2/3}-x\right )}} \sqrt {\frac {-\sqrt {17} x+2 (-1)^{2/3} x+x-\sqrt [3]{-1} \sqrt {17}+\sqrt [3]{-1}-2}{\left (\sqrt {17}+i \sqrt {3}\right ) \left ((-1)^{2/3}-x\right )}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {2 i \sqrt {17} x+2 \sqrt {3} x-\sqrt {51}+i \sqrt {17}+\sqrt {3}+3 i}{2 i \sqrt {17} x-2 \sqrt {3} x+\sqrt {51}+i \sqrt {17}-\sqrt {3}+3 i}}\right )|\frac {7 i-\sqrt {51}}{7 i+\sqrt {51}}\right )-2 \Pi \left (-\frac {\sqrt {3}-i \sqrt {17}}{\sqrt {3}+i \sqrt {17}};\sin ^{-1}\left (\sqrt {\frac {2 i \sqrt {17} x+2 \sqrt {3} x-\sqrt {51}+i \sqrt {17}+\sqrt {3}+3 i}{2 i \sqrt {17} x-2 \sqrt {3} x+\sqrt {51}+i \sqrt {17}-\sqrt {3}+3 i}}\right )|\frac {7 i-\sqrt {51}}{7 i+\sqrt {51}}\right )\right )}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt {3}+i \sqrt {17}\right ) \sqrt {\frac {-2 x+\sqrt {17}-1}{\left (\sqrt {3}-i \sqrt {17}\right ) \left ((-1)^{2/3}-x\right )}} \sqrt {x^4+2 x^3-2 x^2-3 x-4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.19, size = 35, normalized size = 1.00 \begin {gather*} \log \left (2 x^2+2 \sqrt {x^4+2 x^3-2 x^2-3 x-4}+2 x-3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 33, normalized size = 0.94 \begin {gather*} \log \left (2 \, x^{2} + 2 \, x + 2 \, \sqrt {x^{4} + 2 \, x^{3} - 2 \, x^{2} - 3 \, x - 4} - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 35, normalized size = 1.00 \begin {gather*} -\log \left ({\left | -2 \, x^{2} - 2 \, x + 2 \, \sqrt {{\left (x^{2} + x\right )}^{2} - 3 \, x^{2} - 3 \, x - 4} + 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.39, size = 782, normalized size = 22.34
result 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*} \int \frac {2 \, x + 1}{\sqrt {x^{4} + 2 \, x^{3} - 2 \, x^{2} - 3 \, x - 4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {2\,x+1}{\sqrt {x^4+2\,x^3-2\,x^2-3\,x-4}} \,d x \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 {2 x + 1}{\sqrt {\left (x^{2} + x - 4\right ) \left (x^{2} + x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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