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

Optimal. Leaf size=307 \[ \frac{1}{42} \left (7+5 i \sqrt{7}\right ) x^3+\frac{1}{42} \left (7-5 i \sqrt{7}\right ) x^3+\frac{1}{28} \left (7+5 i \sqrt{7}\right ) x^2+\frac{1}{28} \left (7-5 i \sqrt{7}\right ) x^2+\frac{3}{112} \left (7-11 i \sqrt{7}\right ) \log \left (4 x^2+\left (1-i \sqrt{7}\right ) x+4\right )+\frac{3}{112} \left (7+11 i \sqrt{7}\right ) \log \left (4 x^2+\left (1+i \sqrt{7}\right ) x+4\right )-\frac{1}{28} \left (35+9 i \sqrt{7}\right ) x-\frac{1}{28} \left (35-9 i \sqrt{7}\right ) x+\frac{11 \left (5 \sqrt{7}+9 i\right ) \tan ^{-1}\left (\frac{8 x-i \sqrt{7}+1}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{4 \sqrt{14 \left (35+i \sqrt{7}\right )}}-\frac{11 \left (-5 \sqrt{7}+9 i\right ) \tan ^{-1}\left (\frac{8 x+i \sqrt{7}+1}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{4 \sqrt{14 \left (35-i \sqrt{7}\right )}} \]

[Out]

-((35 - (9*I)*Sqrt[7])*x)/28 - ((35 + (9*I)*Sqrt[7])*x)/28 + ((7 - (5*I)*Sqrt[7]
)*x^2)/28 + ((7 + (5*I)*Sqrt[7])*x^2)/28 + ((7 - (5*I)*Sqrt[7])*x^3)/42 + ((7 +
(5*I)*Sqrt[7])*x^3)/42 + (11*(9*I + 5*Sqrt[7])*ArcTan[(1 - I*Sqrt[7] + 8*x)/Sqrt
[2*(35 + I*Sqrt[7])]])/(4*Sqrt[14*(35 + I*Sqrt[7])]) - (11*(9*I - 5*Sqrt[7])*Arc
Tan[(1 + I*Sqrt[7] + 8*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(4*Sqrt[14*(35 - I*Sqrt[7])
]) + (3*(7 - (11*I)*Sqrt[7])*Log[4 + (1 - I*Sqrt[7])*x + 4*x^2])/112 + (3*(7 + (
11*I)*Sqrt[7])*Log[4 + (1 + I*Sqrt[7])*x + 4*x^2])/112

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Rubi [A]  time = 1.30302, antiderivative size = 307, 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 \[ \frac{1}{42} \left (7+5 i \sqrt{7}\right ) x^3+\frac{1}{42} \left (7-5 i \sqrt{7}\right ) x^3+\frac{1}{28} \left (7+5 i \sqrt{7}\right ) x^2+\frac{1}{28} \left (7-5 i \sqrt{7}\right ) x^2+\frac{3}{112} \left (7-11 i \sqrt{7}\right ) \log \left (4 x^2+\left (1-i \sqrt{7}\right ) x+4\right )+\frac{3}{112} \left (7+11 i \sqrt{7}\right ) \log \left (4 x^2+\left (1+i \sqrt{7}\right ) x+4\right )-\frac{1}{28} \left (35+9 i \sqrt{7}\right ) x-\frac{1}{28} \left (35-9 i \sqrt{7}\right ) x+\frac{11 \left (5 \sqrt{7}+9 i\right ) \tan ^{-1}\left (\frac{8 x-i \sqrt{7}+1}{\sqrt{2 \left (35+i \sqrt{7}\right )}}\right )}{4 \sqrt{14 \left (35+i \sqrt{7}\right )}}-\frac{11 \left (-5 \sqrt{7}+9 i\right ) \tan ^{-1}\left (\frac{8 x+i \sqrt{7}+1}{\sqrt{2 \left (35-i \sqrt{7}\right )}}\right )}{4 \sqrt{14 \left (35-i \sqrt{7}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

-((35 - (9*I)*Sqrt[7])*x)/28 - ((35 + (9*I)*Sqrt[7])*x)/28 + ((7 - (5*I)*Sqrt[7]
)*x^2)/28 + ((7 + (5*I)*Sqrt[7])*x^2)/28 + ((7 - (5*I)*Sqrt[7])*x^3)/42 + ((7 +
(5*I)*Sqrt[7])*x^3)/42 + (11*(9*I + 5*Sqrt[7])*ArcTan[(1 - I*Sqrt[7] + 8*x)/Sqrt
[2*(35 + I*Sqrt[7])]])/(4*Sqrt[14*(35 + I*Sqrt[7])]) - (11*(9*I - 5*Sqrt[7])*Arc
Tan[(1 + I*Sqrt[7] + 8*x)/Sqrt[2*(35 - I*Sqrt[7])]])/(4*Sqrt[14*(35 - I*Sqrt[7])
]) + (3*(7 - (11*I)*Sqrt[7])*Log[4 + (1 - I*Sqrt[7])*x + 4*x^2])/112 + (3*(7 + (
11*I)*Sqrt[7])*Log[4 + (1 + I*Sqrt[7])*x + 4*x^2])/112

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ x^{3} \left (\frac{1}{6} - \frac{5 \sqrt{7} i}{42}\right ) + x^{3} \left (\frac{1}{6} + \frac{5 \sqrt{7} i}{42}\right ) + \left (\frac{3}{16} - \frac{33 \sqrt{7} i}{112}\right ) \log{\left (4 x^{2} + x \left (1 - \sqrt{7} i\right ) + 4 \right )} + \left (\frac{3}{16} + \frac{33 \sqrt{7} i}{112}\right ) \log{\left (4 x^{2} + x \left (1 + \sqrt{7} i\right ) + 4 \right )} - \frac{\left (\frac{55}{4} - \frac{99 \sqrt{7} i}{28}\right ) \operatorname{atan}{\left (\frac{8 x + 1 + \sqrt{7} i}{\sqrt{35 + 4 \sqrt{77}} - i \sqrt{-35 + 4 \sqrt{77}}} \right )}}{- \sqrt{35 + 4 \sqrt{77}} + i \sqrt{-35 + 4 \sqrt{77}}} + \frac{\left (\frac{55}{4} + \frac{99 \sqrt{7} i}{28}\right ) \operatorname{atan}{\left (\frac{8 x + 1 - \sqrt{7} i}{\sqrt{35 + 4 \sqrt{77}} + i \sqrt{-35 + 4 \sqrt{77}}} \right )}}{\sqrt{35 + 4 \sqrt{77}} + i \sqrt{-35 + 4 \sqrt{77}}} + \int \left (- \frac{5}{4}\right )\, dx - \int \frac{5}{4}\, dx + \left (\frac{1}{2} - \frac{5 \sqrt{7} i}{14}\right ) \int x\, dx + \left (\frac{1}{2} + \frac{5 \sqrt{7} i}{14}\right ) \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*(1/6 - 5*sqrt(7)*I/42) + x**3*(1/6 + 5*sqrt(7)*I/42) + (3/16 - 33*sqrt(7)*I
/112)*log(4*x**2 + x*(1 - sqrt(7)*I) + 4) + (3/16 + 33*sqrt(7)*I/112)*log(4*x**2
+ x*(1 + sqrt(7)*I) + 4) - (55/4 - 99*sqrt(7)*I/28)*atan((8*x + 1 + sqrt(7)*I)/
(sqrt(35 + 4*sqrt(77)) - I*sqrt(-35 + 4*sqrt(77))))/(-sqrt(35 + 4*sqrt(77)) + I*
sqrt(-35 + 4*sqrt(77))) + (55/4 + 99*sqrt(7)*I/28)*atan((8*x + 1 - sqrt(7)*I)/(s
qrt(35 + 4*sqrt(77)) + I*sqrt(-35 + 4*sqrt(77))))/(sqrt(35 + 4*sqrt(77)) + I*sqr
t(-35 + 4*sqrt(77))) + Integral(-5/4, x) - Integral(5/4, 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 = 0.032629, size = 109, normalized size = 0.36 \[ \frac{1}{6} \left (3 \text{RootSum}\left [2 \text{$\#$1}^4+\text{$\#$1}^3+5 \text{$\#$1}^2+\text{$\#$1}+2\&,\frac{3 \text{$\#$1}^3 \log (x-\text{$\#$1})+19 \text{$\#$1}^2 \log (x-\text{$\#$1})+\text{$\#$1} \log (x-\text{$\#$1})+10 \log (x-\text{$\#$1})}{8 \text{$\#$1}^3+3 \text{$\#$1}^2+10 \text{$\#$1}+1}\&\right ]+x \left (2 x^2+3 x-15\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(-15 + 3*x + 2*x^2) + 3*RootSum[2 + #1 + 5*#1^2 + #1^3 + 2*#1^4 & , (10*Log[x
- #1] + Log[x - #1]*#1 + 19*Log[x - #1]*#1^2 + 3*Log[x - #1]*#1^3)/(1 + 10*#1 +
3*#1^2 + 8*#1^3) & ])/6

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Maple [C]  time = 0.01, size = 74, normalized size = 0.2 \[{\frac{{x}^{3}}{3}}+{\frac{{x}^{2}}{2}}-{\frac{5\,x}{2}}+{\frac{1}{2}\sum _{{\it \_R}={\it RootOf} \left ( 2\,{{\it \_Z}}^{4}+{{\it \_Z}}^{3}+5\,{{\it \_Z}}^{2}+{\it \_Z}+2 \right ) }{\frac{ \left ( 3\,{{\it \_R}}^{3}+19\,{{\it \_R}}^{2}+{\it \_R}+10 \right ) \ln \left ( x-{\it \_R} \right ) }{8\,{{\it \_R}}^{3}+3\,{{\it \_R}}^{2}+10\,{\it \_R}+1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3+1/2*x^2-5/2*x+1/2*sum((3*_R^3+19*_R^2+_R+10)/(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. \[ \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{5}{2} \, x + \frac{1}{2} \, \int \frac{3 \, x^{3} + 19 \, x^{2} + x + 10}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 3.4075, size = 61, normalized size = 0.2 \[ \frac{x^{3}}{3} + \frac{x^{2}}{2} - \frac{5 x}{2} + \operatorname{RootSum}{\left (1372 t^{4} - 1029 t^{3} + 3136 t^{2} + 2688 t + 512, \left ( t \mapsto t \log{\left (\frac{5831 t^{3}}{1936} - \frac{23765 t^{2}}{7744} + \frac{2065 t}{242} + x + \frac{415}{121} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/3 + x**2/2 - 5*x/2 + RootSum(1372*_t**4 - 1029*_t**3 + 3136*_t**2 + 2688*_t
+ 512, Lambda(_t, _t*log(5831*_t**3/1936 - 23765*_t**2/7744 + 2065*_t/242 + x +
415/121)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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