3.251 \(\int \frac{x^2 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx\)

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]

((7 - (5*I)*Sqrt[7])*x)/14 + ((7 + (5*I)*Sqrt[7])*x)/14 + ((7 - (5*I)*Sqrt[7])*x
^2)/28 + ((7 + (5*I)*Sqrt[7])*x^2)/28 - ((53*I + Sqrt[7])*ArcTan[(1 - I*Sqrt[7]
+ 8*x)/Sqrt[2*(35 + I*Sqrt[7])]])/(2*Sqrt[14*(35 + I*Sqrt[7])]) + ((53*I - Sqrt[
7])*ArcTan[(1 + I*Sqrt[7] + 8*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(2*Sqrt[14*(35 - I*S
qrt[7])]) - ((35 + (9*I)*Sqrt[7])*Log[4 + (1 - I*Sqrt[7])*x + 4*x^2])/56 - ((35
- (9*I)*Sqrt[7])*Log[4 + (1 + I*Sqrt[7])*x + 4*x^2])/56

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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]

((7 - (5*I)*Sqrt[7])*x)/14 + ((7 + (5*I)*Sqrt[7])*x)/14 + ((7 - (5*I)*Sqrt[7])*x
^2)/28 + ((7 + (5*I)*Sqrt[7])*x^2)/28 - ((53*I + Sqrt[7])*ArcTan[(1 - I*Sqrt[7]
+ 8*x)/Sqrt[2*(35 + I*Sqrt[7])]])/(2*Sqrt[14*(35 + I*Sqrt[7])]) + ((53*I - Sqrt[
7])*ArcTan[(1 + I*Sqrt[7] + 8*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(2*Sqrt[14*(35 - I*S
qrt[7])]) - ((35 + (9*I)*Sqrt[7])*Log[4 + (1 - I*Sqrt[7])*x + 4*x^2])/56 - ((35
- (9*I)*Sqrt[7])*Log[4 + (1 + I*Sqrt[7])*x + 4*x^2])/56

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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)

[Out]

-(5/8 + 9*sqrt(7)*I/56)*log(4*x**2 + x*(1 - sqrt(7)*I) + 4) - (5/8 - 9*sqrt(7)*I
/56)*log(4*x**2 + x*(1 + sqrt(7)*I) + 4) + (7 - 53*sqrt(7)*I)*atan((8*x + 1 + sq
rt(7)*I)/(sqrt(35 + 4*sqrt(77)) - I*sqrt(-35 + 4*sqrt(77))))/(14*(-sqrt(35 + 4*s
qrt(77)) + I*sqrt(-35 + 4*sqrt(77)))) - (7 + 53*sqrt(7)*I)*atan((8*x + 1 - sqrt(
7)*I)/(sqrt(35 + 4*sqrt(77)) + I*sqrt(-35 + 4*sqrt(77))))/(14*(sqrt(35 + 4*sqrt(
77)) + I*sqrt(-35 + 4*sqrt(77)))) - Integral(-1/2, x) + Integral(1/2, x) + (1/2
- 5*sqrt(7)*I/14)*Integral(x, x) + (1/2 + 5*sqrt(7)*I/14)*Integral(x, 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]

x + x^2/2 - RootSum[2 + #1 + 5*#1^2 + #1^3 + 2*#1^4 & , (2*Log[x - #1] + 3*Log[x
- #1]*#1 + Log[x - #1]*#1^2 + 5*Log[x - #1]*#1^3)/(1 + 10*#1 + 3*#1^2 + 8*#1^3)
& ]

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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]

x+1/2*x^2+sum((-5*_R^3-_R^2-3*_R-2)/(8*_R^3+3*_R^2+10*_R+1)*ln(x-_R),_R=RootOf(2
*_Z^4+_Z^3+5*_Z^2+_Z+2))

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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]

1/2*x^2 + x - integrate((5*x^3 + x^2 + 3*x + 2)/(2*x^4 + x^3 + 5*x^2 + x + 2), x
)

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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]

Exception raised: NotImplementedError

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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]

x**2/2 + x + RootSum(686*_t**4 + 1715*_t**3 + 1372*_t**2 + 448*_t + 256, Lambda(
_t, _t*log(5145*_t**3/4192 + 1421*_t**2/8384 - 2541*_t/2096 + x + 17/262)))

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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]

integrate((2*x^3 + 3*x^2 + x + 5)*x^2/(2*x^4 + x^3 + 5*x^2 + x + 2), x)