Optimal. Leaf size=31 \[ \frac {1}{2} \log \left (x^2+\sqrt {x^4+8 x^3+16 x^2+13}+4 x\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 0.65, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1680, 1107, 619, 215} \begin {gather*} -\frac {1}{2} \sinh ^{-1}\left (\frac {4-(x+2)^2}{\sqrt {13}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 619
Rule 1107
Rule 1680
Rubi steps
\begin {align*} \int \frac {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {29-8 x^2+x^4}} \, dx,x,2+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {29-8 x+x^2}} \, dx,x,(2+x)^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{52}}} \, dx,x,2 x (4+x)\right )}{4 \sqrt {13}}\\ &=\frac {1}{2} \sinh ^{-1}\left (\frac {x (4+x)}{\sqrt {13}}\right )\\ \end {align*}
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Mathematica [C] time = 2.28, size = 898, normalized size = 28.97 \begin {gather*} \frac {2 \left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \sqrt {\frac {\sqrt {4-i \sqrt {13}} \left (-x+\sqrt {4+i \sqrt {13}}-2\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (-x+\sqrt {4-i \sqrt {13}}-2\right )}} \left (x-\sqrt {4-i \sqrt {13}}+2\right )^2 \sqrt {\frac {\left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x+\sqrt {4-i \sqrt {13}}+2\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x-\sqrt {4-i \sqrt {13}}+2\right )}} \sqrt {-\frac {\sqrt {4-i \sqrt {13}} \left (x+\sqrt {4+i \sqrt {13}}+2\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-x+\sqrt {4-i \sqrt {13}}-2\right )}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) x-2 i \sqrt {4+i \sqrt {13}}+2 i \sqrt {4-i \sqrt {13}}-i \sqrt {29}+\sqrt {13}+4 i}{-i \left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) x-2 i \sqrt {4+i \sqrt {13}}-2 i \sqrt {4-i \sqrt {13}}+i \sqrt {29}+\sqrt {13}+4 i}}\right )|\frac {1}{13} \left (-45-8 \sqrt {29}\right )\right )-2 \Pi \left (-\frac {\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}}{\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}};\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) x-2 i \sqrt {4+i \sqrt {13}}+2 i \sqrt {4-i \sqrt {13}}-i \sqrt {29}+\sqrt {13}+4 i}{-i \left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) x-2 i \sqrt {4+i \sqrt {13}}-2 i \sqrt {4-i \sqrt {13}}+i \sqrt {29}+\sqrt {13}+4 i}}\right )|\frac {1}{13} \left (-45-8 \sqrt {29}\right )\right )\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \sqrt {x^4+8 x^3+16 x^2+13}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.13, size = 31, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (4 x+x^2+\sqrt {13+16 x^2+8 x^3+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} + 4 \, x + \sqrt {x^{4} + 8 \, x^{3} + 16 \, x^{2} + 13}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 25, normalized size = 0.81 \begin {gather*} -\frac {1}{2} \, \log \left (-x^{2} - 4 \, x + \sqrt {{\left (x^{2} + 4 \, x\right )}^{2} + 13}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 30, normalized size = 0.97
method | result | size |
trager | \(-\frac {\ln \left (-x^{2}+\sqrt {x^{4}+8 x^{3}+16 x^{2}+13}-4 x \right )}{2}\) | \(30\) |
default | \(\frac {4 \left (-\sqrt {4+i \sqrt {13}}-\sqrt {4-i \sqrt {13}}\right ) \sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (x +2-\sqrt {4+i \sqrt {13}}\right )^{2} \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2+\sqrt {4-i \sqrt {13}}\right )}{\left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2-\sqrt {4-i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \sqrt {4+i \sqrt {13}}\, \sqrt {\left (x +2+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4-i \sqrt {13}}\right ) \left (x +2-\sqrt {4-i \sqrt {13}}\right )}}+\frac {2 \left (-\sqrt {4+i \sqrt {13}}-\sqrt {4-i \sqrt {13}}\right ) \sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (x +2-\sqrt {4+i \sqrt {13}}\right )^{2} \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2+\sqrt {4-i \sqrt {13}}\right )}{\left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2-\sqrt {4-i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (\left (-2+\sqrt {4+i \sqrt {13}}\right ) \EllipticF \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )-2 \sqrt {4+i \sqrt {13}}\, \EllipticPi \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \frac {\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}}{\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \sqrt {4+i \sqrt {13}}\, \sqrt {\left (x +2+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4-i \sqrt {13}}\right ) \left (x +2-\sqrt {4-i \sqrt {13}}\right )}}\) | \(1252\) |
elliptic | \(\frac {4 \left (-\sqrt {4+i \sqrt {13}}-\sqrt {4-i \sqrt {13}}\right ) \sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (x +2-\sqrt {4+i \sqrt {13}}\right )^{2} \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2+\sqrt {4-i \sqrt {13}}\right )}{\left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2-\sqrt {4-i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \sqrt {4+i \sqrt {13}}\, \sqrt {\left (x +2+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4-i \sqrt {13}}\right ) \left (x +2-\sqrt {4-i \sqrt {13}}\right )}}+\frac {2 \left (-\sqrt {4+i \sqrt {13}}-\sqrt {4-i \sqrt {13}}\right ) \sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (x +2-\sqrt {4+i \sqrt {13}}\right )^{2} \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2+\sqrt {4-i \sqrt {13}}\right )}{\left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \sqrt {\frac {\sqrt {4+i \sqrt {13}}\, \left (x +2-\sqrt {4-i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}\, \left (\left (-2+\sqrt {4+i \sqrt {13}}\right ) \EllipticF \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )-2 \sqrt {4+i \sqrt {13}}\, \EllipticPi \left (\sqrt {\frac {\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4+i \sqrt {13}}\right )}{\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right )}}, \frac {\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}}{\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}}, \sqrt {-\frac {\left (\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )^{2}}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \left (-\sqrt {4-i \sqrt {13}}+\sqrt {4+i \sqrt {13}}\right )}}\right )\right )}{\left (\sqrt {4-i \sqrt {13}}-\sqrt {4+i \sqrt {13}}\right ) \sqrt {4+i \sqrt {13}}\, \sqrt {\left (x +2+\sqrt {4+i \sqrt {13}}\right ) \left (x +2-\sqrt {4+i \sqrt {13}}\right ) \left (x +2+\sqrt {4-i \sqrt {13}}\right ) \left (x +2-\sqrt {4-i \sqrt {13}}\right )}}\) | \(1252\) |
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 {x + 2}{\sqrt {x^{4} + 8 \, x^{3} + 16 \, x^{2} + 13}}\,{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 {x+2}{\sqrt {x^4+8\,x^3+16\,x^2+13}} \,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 {x + 2}{\sqrt {x^{4} + 8 x^{3} + 16 x^{2} + 13}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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