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

Optimal. Leaf size=245 \[ -\frac{1}{56} \left (35-9 i \sqrt{7}\right ) \log \left (4 i x^2+\left (-\sqrt{7}+i\right ) x+4 i\right )-\frac{1}{56} \left (35+9 i \sqrt{7}\right ) \log \left (4 i x^2+\left (\sqrt{7}+i\right ) x+4 i\right )+\frac{1}{28} \left (35+9 i \sqrt{7}\right ) \log (x)+\frac{1}{28} \left (35-9 i \sqrt{7}\right ) \log (x)-\frac{\left (53+i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x-\sqrt{7}+i}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35-i \sqrt{7}\right )}}+\frac{\left (53-i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x+\sqrt{7}+i}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35+i \sqrt{7}\right )}} \]

[Out]

-((53 + I*Sqrt[7])*ArcTanh[(I - Sqrt[7] + (8*I)*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(2
*Sqrt[14*(35 - I*Sqrt[7])]) + ((53 - I*Sqrt[7])*ArcTanh[(I + Sqrt[7] + (8*I)*x)/
Sqrt[2*(35 + I*Sqrt[7])]])/(2*Sqrt[14*(35 + I*Sqrt[7])]) + ((35 - (9*I)*Sqrt[7])
*Log[x])/28 + ((35 + (9*I)*Sqrt[7])*Log[x])/28 - ((35 - (9*I)*Sqrt[7])*Log[4*I +
(I - Sqrt[7])*x + (4*I)*x^2])/56 - ((35 + (9*I)*Sqrt[7])*Log[4*I + (I + Sqrt[7]
)*x + (4*I)*x^2])/56

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Rubi [A]  time = 1.19434, antiderivative size = 245, 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 \[ -\frac{1}{56} \left (35-9 i \sqrt{7}\right ) \log \left (4 i x^2+\left (-\sqrt{7}+i\right ) x+4 i\right )-\frac{1}{56} \left (35+9 i \sqrt{7}\right ) \log \left (4 i x^2+\left (\sqrt{7}+i\right ) x+4 i\right )+\frac{1}{28} \left (35+9 i \sqrt{7}\right ) \log (x)+\frac{1}{28} \left (35-9 i \sqrt{7}\right ) \log (x)-\frac{\left (53+i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x-\sqrt{7}+i}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35-i \sqrt{7}\right )}}+\frac{\left (53-i \sqrt{7}\right ) \tanh ^{-1}\left (\frac{8 i x+\sqrt{7}+i}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{2 \sqrt{14 \left (35+i \sqrt{7}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[(5 + x + 3*x^2 + 2*x^3)/(x*(2 + x + 5*x^2 + x^3 + 2*x^4)),x]

[Out]

-((53 + I*Sqrt[7])*ArcTanh[(I - Sqrt[7] + (8*I)*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(2
*Sqrt[14*(35 - I*Sqrt[7])]) + ((53 - I*Sqrt[7])*ArcTanh[(I + Sqrt[7] + (8*I)*x)/
Sqrt[2*(35 + I*Sqrt[7])]])/(2*Sqrt[14*(35 + I*Sqrt[7])]) + ((35 - (9*I)*Sqrt[7])
*Log[x])/28 + ((35 + (9*I)*Sqrt[7])*Log[x])/28 - ((35 - (9*I)*Sqrt[7])*Log[4*I +
(I - Sqrt[7])*x + (4*I)*x^2])/56 - ((35 + (9*I)*Sqrt[7])*Log[4*I + (I + Sqrt[7]
)*x + (4*I)*x^2])/56

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Rubi in Sympy [A]  time = 115.025, size = 253, normalized size = 1.03 \[ \left (\frac{5}{4} - \frac{9 \sqrt{7} i}{28}\right ) \log{\left (x \right )} + \left (\frac{5}{4} + \frac{9 \sqrt{7} i}{28}\right ) \log{\left (x \right )} - \left (\frac{5}{8} + \frac{9 \sqrt{7} i}{56}\right ) \log{\left (4 x^{2} + x \left (1 - \sqrt{7} i\right ) + 4 \right )} - \left (\frac{5}{8} - \frac{9 \sqrt{7} i}{56}\right ) \log{\left (4 x^{2} + x \left (1 + \sqrt{7} i\right ) + 4 \right )} - \frac{\left (\frac{1}{2} - \frac{53 \sqrt{7} i}{14}\right ) \operatorname{atan}{\left (\frac{8 x + 1 + \sqrt{7} i}{\sqrt{35 + 4 \sqrt{77}} - i \sqrt{-35 + 4 \sqrt{77}}} \right )}}{- \sqrt{35 + 4 \sqrt{77}} + i \sqrt{-35 + 4 \sqrt{77}}} + \frac{\left (7 + 53 \sqrt{7} i\right ) \operatorname{atan}{\left (\frac{8 x + 1 - \sqrt{7} i}{\sqrt{35 + 4 \sqrt{77}} + i \sqrt{-35 + 4 \sqrt{77}}} \right )}}{14 \left (\sqrt{35 + 4 \sqrt{77}} + i \sqrt{-35 + 4 \sqrt{77}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**3+3*x**2+x+5)/x/(2*x**4+x**3+5*x**2+x+2),x)

[Out]

(5/4 - 9*sqrt(7)*I/28)*log(x) + (5/4 + 9*sqrt(7)*I/28)*log(x) - (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) - (1/2 - 53*sqrt(7)*I/14)*atan((8*x + 1 + sqrt(7)*I)/(sq
rt(35 + 4*sqrt(77)) - I*sqrt(-35 + 4*sqrt(77))))/(-sqrt(35 + 4*sqrt(77)) + I*sqr
t(-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))))

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Mathematica [C]  time = 0.0282212, size = 101, normalized size = 0.41 \[ \frac{5 \log (x)}{2}-\frac{1}{2} \text{RootSum}\left [2 \text{$\#$1}^4+\text{$\#$1}^3+5 \text{$\#$1}^2+\text{$\#$1}+2\&,\frac{10 \text{$\#$1}^3 \log (x-\text{$\#$1})+\text{$\#$1}^2 \log (x-\text{$\#$1})+19 \text{$\#$1} \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{8 \text{$\#$1}^3+3 \text{$\#$1}^2+10 \text{$\#$1}+1}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(5 + x + 3*x^2 + 2*x^3)/(x*(2 + x + 5*x^2 + x^3 + 2*x^4)),x]

[Out]

(5*Log[x])/2 - RootSum[2 + #1 + 5*#1^2 + #1^3 + 2*#1^4 & , (3*Log[x - #1] + 19*L
og[x - #1]*#1 + Log[x - #1]*#1^2 + 10*Log[x - #1]*#1^3)/(1 + 10*#1 + 3*#1^2 + 8*
#1^3) & ]/2

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Maple [C]  time = 0.012, size = 67, normalized size = 0.3 \[{\frac{5\,\ln \left ( x \right ) }{2}}+{\frac{1}{2}\sum _{{\it \_R}={\it RootOf} \left ( 2\,{{\it \_Z}}^{4}+{{\it \_Z}}^{3}+5\,{{\it \_Z}}^{2}+{\it \_Z}+2 \right ) }{\frac{ \left ( -10\,{{\it \_R}}^{3}-{{\it \_R}}^{2}-19\,{\it \_R}-3 \right ) \ln \left ( x-{\it \_R} \right ) }{8\,{{\it \_R}}^{3}+3\,{{\it \_R}}^{2}+10\,{\it \_R}+1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^3+3*x^2+x+5)/x/(2*x^4+x^3+5*x^2+x+2),x)

[Out]

5/2*ln(x)+1/2*sum((-10*_R^3-_R^2-19*_R-3)/(8*_R^3+3*_R^2+10*_R+1)*ln(x-_R),_R=Ro
otOf(2*_Z^4+_Z^3+5*_Z^2+_Z+2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{2} \, \int \frac{10 \, x^{3} + x^{2} + 19 \, x + 3}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} + \frac{5}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 + 3*x^2 + x + 5)/((2*x^4 + x^3 + 5*x^2 + x + 2)*x),x, algorithm="maxima")

[Out]

-1/2*integrate((10*x^3 + x^2 + 19*x + 3)/(2*x^4 + x^3 + 5*x^2 + x + 2), x) + 5/2
*log(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)/((2*x^4 + x^3 + 5*x^2 + x + 2)*x),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 30.5423, size = 60, normalized size = 0.24 \[ \frac{5 \log{\left (x \right )}}{2} + \operatorname{RootSum}{\left (686 t^{4} + 1715 t^{3} + 1372 t^{2} + 448 t + 256, \left ( t \mapsto t \log{\left (- \frac{160344611 t^{4}}{532759184} - \frac{16880402 t^{3}}{33297449} + \frac{4010520787 t^{2}}{2131036736} + \frac{1537535671 t}{532759184} + x + \frac{46660495}{66594898} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**3+3*x**2+x+5)/x/(2*x**4+x**3+5*x**2+x+2),x)

[Out]

5*log(x)/2 + RootSum(686*_t**4 + 1715*_t**3 + 1372*_t**2 + 448*_t + 256, Lambda(
_t, _t*log(-160344611*_t**4/532759184 - 16880402*_t**3/33297449 + 4010520787*_t*
*2/2131036736 + 1537535671*_t/532759184 + x + 46660495/66594898)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x^{3} + 3 \, x^{2} + x + 5}{{\left (2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 + 3*x^2 + x + 5)/((2*x^4 + x^3 + 5*x^2 + x + 2)*x),x, algorithm="giac")

[Out]

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