Optimal. Leaf size=269 \[ \frac{1}{28} \left (7+5 i \sqrt{7}\right ) x^2+\frac{1}{28} \left (7-5 i \sqrt{7}\right ) x^2-\frac{1}{56} \left (35+9 i \sqrt{7}\right ) \log \left (4 x^2+\left (1-i \sqrt{7}\right ) x+4\right )-\frac{1}{56} \left (35-9 i \sqrt{7}\right ) \log \left (4 x^2+\left (1+i \sqrt{7}\right ) x+4\right )+\frac{1}{14} \left (7+5 i \sqrt{7}\right ) x+\frac{1}{14} \left (7-5 i \sqrt{7}\right ) x-\frac{\left (\sqrt{7}+53 i\right ) \tan ^{-1}\left (\frac{8 x-i \sqrt{7}+1}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35+i \sqrt{7}\right )}}+\frac{\left (-\sqrt{7}+53 i\right ) \tan ^{-1}\left (\frac{8 x+i \sqrt{7}+1}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35-i \sqrt{7}\right )}} \]
[Out]
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Rubi [A] time = 1.04639, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ \frac{1}{28} \left (7+5 i \sqrt{7}\right ) x^2+\frac{1}{28} \left (7-5 i \sqrt{7}\right ) x^2-\frac{1}{56} \left (35+9 i \sqrt{7}\right ) \log \left (4 x^2+\left (1-i \sqrt{7}\right ) x+4\right )-\frac{1}{56} \left (35-9 i \sqrt{7}\right ) \log \left (4 x^2+\left (1+i \sqrt{7}\right ) x+4\right )+\frac{1}{14} \left (7+5 i \sqrt{7}\right ) x+\frac{1}{14} \left (7-5 i \sqrt{7}\right ) x-\frac{\left (\sqrt{7}+53 i\right ) \tan ^{-1}\left (\frac{8 x-i \sqrt{7}+1}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35+i \sqrt{7}\right )}}+\frac{\left (-\sqrt{7}+53 i\right ) \tan ^{-1}\left (\frac{8 x+i \sqrt{7}+1}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35-i \sqrt{7}\right )}} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(5 + x + 3*x^2 + 2*x^3))/(2 + x + 5*x^2 + x^3 + 2*x^4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \left (\frac{5}{8} + \frac{9 \sqrt{7} i}{56}\right ) \log{\left (4 x^{2} + x \left (1 - \sqrt{7} i\right ) + 4 \right )} - \left (\frac{5}{8} - \frac{9 \sqrt{7} i}{56}\right ) \log{\left (4 x^{2} + x \left (1 + \sqrt{7} i\right ) + 4 \right )} + \frac{\left (7 - 53 \sqrt{7} i\right ) \operatorname{atan}{\left (\frac{8 x + 1 + \sqrt{7} i}{\sqrt{35 + 4 \sqrt{77}} - i \sqrt{-35 + 4 \sqrt{77}}} \right )}}{14 \left (- \sqrt{35 + 4 \sqrt{77}} + i \sqrt{-35 + 4 \sqrt{77}}\right )} - \frac{\left (7 + 53 \sqrt{7} i\right ) \operatorname{atan}{\left (\frac{8 x + 1 - \sqrt{7} i}{\sqrt{35 + 4 \sqrt{77}} + i \sqrt{-35 + 4 \sqrt{77}}} \right )}}{14 \left (\sqrt{35 + 4 \sqrt{77}} + i \sqrt{-35 + 4 \sqrt{77}}\right )} - \int \left (- \frac{1}{2}\right )\, dx + \int \frac{1}{2}\, dx + \left (\frac{1}{2} - \frac{5 \sqrt{7} i}{14}\right ) \int x\, dx + \left (\frac{1}{2} + \frac{5 \sqrt{7} i}{14}\right ) \int x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(2*x**3+3*x**2+x+5)/(2*x**4+x**3+5*x**2+x+2),x)
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Mathematica [C] time = 0.0269464, size = 101, normalized size = 0.38 \[ -\text{RootSum}\left [2 \text{$\#$1}^4+\text{$\#$1}^3+5 \text{$\#$1}^2+\text{$\#$1}+2\&,\frac{5 \text{$\#$1}^3 \log (x-\text{$\#$1})+\text{$\#$1}^2 \log (x-\text{$\#$1})+3 \text{$\#$1} \log (x-\text{$\#$1})+2 \log (x-\text{$\#$1})}{8 \text{$\#$1}^3+3 \text{$\#$1}^2+10 \text{$\#$1}+1}\&\right ]+\frac{x^2}{2}+x \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(5 + x + 3*x^2 + 2*x^3))/(2 + x + 5*x^2 + x^3 + 2*x^4),x]
[Out]
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Maple [C] time = 0.009, size = 67, normalized size = 0.3 \[ x+{\frac{{x}^{2}}{2}}+\sum _{{\it \_R}={\it RootOf} \left ( 2\,{{\it \_Z}}^{4}+{{\it \_Z}}^{3}+5\,{{\it \_Z}}^{2}+{\it \_Z}+2 \right ) }{\frac{ \left ( -5\,{{\it \_R}}^{3}-{{\it \_R}}^{2}-3\,{\it \_R}-2 \right ) \ln \left ( x-{\it \_R} \right ) }{8\,{{\it \_R}}^{3}+3\,{{\it \_R}}^{2}+10\,{\it \_R}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(2*x^3+3*x^2+x+5)/(2*x^4+x^3+5*x^2+x+2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{1}{2} \, x^{2} + x - \int \frac{5 \, x^{3} + x^{2} + 3 \, x + 2}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^3 + 3*x^2 + x + 5)*x^2/(2*x^4 + x^3 + 5*x^2 + x + 2),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^3 + 3*x^2 + x + 5)*x^2/(2*x^4 + x^3 + 5*x^2 + x + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.97885, size = 53, normalized size = 0.2 \[ \frac{x^{2}}{2} + x + \operatorname{RootSum}{\left (686 t^{4} + 1715 t^{3} + 1372 t^{2} + 448 t + 256, \left ( t \mapsto t \log{\left (\frac{5145 t^{3}}{4192} + \frac{1421 t^{2}}{8384} - \frac{2541 t}{2096} + x + \frac{17}{262} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(2*x**3+3*x**2+x+5)/(2*x**4+x**3+5*x**2+x+2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x^{3} + 3 \, x^{2} + x + 5\right )} x^{2}}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^3 + 3*x^2 + x + 5)*x^2/(2*x^4 + x^3 + 5*x^2 + x + 2),x, algorithm="giac")
[Out]