3.393 \(\int \frac{\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx\)

Optimal. Leaf size=168 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}{\sqrt [6]{b}}\right )}{3 b^{5/6} \sqrt{\sqrt [3]{a}+\sqrt [3]{b}}}+\frac{\tan ^{-1}\left (\frac{x \sqrt{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a}}}{\sqrt [6]{b}}\right )}{3 b^{5/6} \sqrt{\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a}}}+\frac{\tan ^{-1}\left (\frac{x \sqrt{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b}}}{\sqrt [6]{b}}\right )}{3 b^{5/6} \sqrt{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b}}} \]

[Out]

ArcTan[(Sqrt[a^(1/3) + b^(1/3)]*x)/b^(1/6)]/(3*Sqrt[a^(1/3) + b^(1/3)]*b^(5/6))
+ ArcTan[(Sqrt[-((-1)^(1/3)*a^(1/3)) + b^(1/3)]*x)/b^(1/6)]/(3*Sqrt[-((-1)^(1/3)
*a^(1/3)) + b^(1/3)]*b^(5/6)) + ArcTan[(Sqrt[(-1)^(2/3)*a^(1/3) + b^(1/3)]*x)/b^
(1/6)]/(3*Sqrt[(-1)^(2/3)*a^(1/3) + b^(1/3)]*b^(5/6))

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Rubi [F]  time = 0.675554, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[(1 + x^2)^2/(a*x^6 + b*(1 + x^2)^3),x]

[Out]

Defer[Int][(a*x^6 + b*(1 + x^2)^3)^(-1), x] + 2*Defer[Int][x^2/(a*x^6 + b*(1 + x
^2)^3), x] + Defer[Int][x^4/(a*x^6 + b*(1 + x^2)^3), x]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+1)**2/(a*x**6+b*(x**2+1)**3),x)

[Out]

Timed out

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Mathematica [C]  time = 0.0963126, size = 95, normalized size = 0.57 \[ \frac{1}{6} \text{RootSum}\left [\text{$\#$1}^6 a+\text{$\#$1}^6 b+3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b+b\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+2 \text{$\#$1}^2 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{\text{$\#$1}^5 a+\text{$\#$1}^5 b+2 \text{$\#$1}^3 b+\text{$\#$1} b}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^2)^2/(a*x^6 + b*(1 + x^2)^3),x]

[Out]

RootSum[b + 3*b*#1^2 + 3*b*#1^4 + a*#1^6 + b*#1^6 & , (Log[x - #1] + 2*Log[x - #
1]*#1^2 + Log[x - #1]*#1^4)/(b*#1 + 2*b*#1^3 + a*#1^5 + b*#1^5) & ]/6

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Maple [C]  time = 0.235, size = 67, normalized size = 0.4 \[{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a+b \right ){{\it \_Z}}^{6}+3\,b{{\it \_Z}}^{4}+3\,b{{\it \_Z}}^{2}+b \right ) }{\frac{ \left ({{\it \_R}}^{4}+2\,{{\it \_R}}^{2}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}a+{{\it \_R}}^{5}b+2\,{{\it \_R}}^{3}b+{\it \_R}\,b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)^2/(a*x^6+b*(x^2+1)^3),x)

[Out]

1/6*sum((_R^4+2*_R^2+1)/(_R^5*a+_R^5*b+2*_R^3*b+_R*b)*ln(x-_R),_R=RootOf((a+b)*_
Z^6+3*b*_Z^4+3*b*_Z^2+b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} + 1\right )}^{2}}{a x^{6} +{\left (x^{2} + 1\right )}^{3} b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b),x, algorithm="maxima")

[Out]

integrate((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b), 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((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 6.13289, size = 42, normalized size = 0.25 \[ \operatorname{RootSum}{\left (t^{6} \left (46656 a b^{5} + 46656 b^{6}\right ) + 3888 t^{4} b^{4} + 108 t^{2} b^{2} + 1, \left ( t \mapsto t \log{\left (6 t b + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+1)**2/(a*x**6+b*(x**2+1)**3),x)

[Out]

RootSum(_t**6*(46656*a*b**5 + 46656*b**6) + 3888*_t**4*b**4 + 108*_t**2*b**2 + 1
, Lambda(_t, _t*log(6*_t*b + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} + 1\right )}^{2}}{a x^{6} +{\left (x^{2} + 1\right )}^{3} b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b),x, algorithm="giac")

[Out]

integrate((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b), x)