3.903 \(\int \frac{x^2}{a-b+2 a x^2+a x^4} \, dx\)

Optimal. Leaf size=109 \[ \frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}} \]

[Out]

-(Sqrt[Sqrt[a] - Sqrt[b]]*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^(3/4
)*Sqrt[b]) + (Sqrt[Sqrt[a] + Sqrt[b]]*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[b]]
])/(2*a^(3/4)*Sqrt[b])

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Rubi [A]  time = 0.128659, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a - b + 2*a*x^2 + a*x^4),x]

[Out]

-(Sqrt[Sqrt[a] - Sqrt[b]]*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^(3/4
)*Sqrt[b]) + (Sqrt[Sqrt[a] + Sqrt[b]]*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[b]]
])/(2*a^(3/4)*Sqrt[b])

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Rubi in Sympy [A]  time = 19.6964, size = 94, normalized size = 0.86 \[ - \frac{\sqrt{\sqrt{a} - \sqrt{b}} \operatorname{atan}{\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a} - \sqrt{b}}} \right )}}{2 a^{\frac{3}{4}} \sqrt{b}} + \frac{\sqrt{\sqrt{a} + \sqrt{b}} \operatorname{atan}{\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a} + \sqrt{b}}} \right )}}{2 a^{\frac{3}{4}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(a*x**4+2*a*x**2+a-b),x)

[Out]

-sqrt(sqrt(a) - sqrt(b))*atan(a**(1/4)*x/sqrt(sqrt(a) - sqrt(b)))/(2*a**(3/4)*sq
rt(b)) + sqrt(sqrt(a) + sqrt(b))*atan(a**(1/4)*x/sqrt(sqrt(a) + sqrt(b)))/(2*a**
(3/4)*sqrt(b))

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Mathematica [A]  time = 0.189585, size = 128, normalized size = 1.17 \[ \frac{\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a-\sqrt{a} \sqrt{b}}}\right )}{\sqrt{a-\sqrt{a} \sqrt{b}}}}{2 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a - b + 2*a*x^2 + a*x^4),x]

[Out]

(-(((Sqrt[a] - Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - Sqrt[a]*Sqrt[b]]])/Sqrt[a -
Sqrt[a]*Sqrt[b]]) + ((Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a + Sqrt[a]*Sqr
t[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*Sqrt[b])

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Maple [A]  time = 0.017, size = 134, normalized size = 1.2 \[{\frac{1}{2}\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}}+{\frac{a}{2}\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}}-{\frac{1}{2}{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}}+{\frac{a}{2}{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a*x^4+2*a*x^2+a-b),x)

[Out]

1/2/(((a*b)^(1/2)+a)*a)^(1/2)*arctan(x*a/(((a*b)^(1/2)+a)*a)^(1/2))+1/2/(a*b)^(1
/2)/(((a*b)^(1/2)+a)*a)^(1/2)*arctan(x*a/(((a*b)^(1/2)+a)*a)^(1/2))*a-1/2/(((a*b
)^(1/2)-a)*a)^(1/2)*arctanh(x*a/(((a*b)^(1/2)-a)*a)^(1/2))+1/2/(a*b)^(1/2)/(((a*
b)^(1/2)-a)*a)^(1/2)*arctanh(x*a/(((a*b)^(1/2)-a)*a)^(1/2))*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{a x^{4} + 2 \, a x^{2} + a - b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="maxima")

[Out]

integrate(x^2/(a*x^4 + 2*a*x^2 + a - b), x)

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Fricas [A]  time = 0.277647, size = 360, normalized size = 3.3 \[ \frac{1}{4} \, \sqrt{-\frac{a b \sqrt{\frac{1}{a^{3} b}} + 1}{a b}} \log \left (a^{2} b \sqrt{-\frac{a b \sqrt{\frac{1}{a^{3} b}} + 1}{a b}} \sqrt{\frac{1}{a^{3} b}} + x\right ) - \frac{1}{4} \, \sqrt{-\frac{a b \sqrt{\frac{1}{a^{3} b}} + 1}{a b}} \log \left (-a^{2} b \sqrt{-\frac{a b \sqrt{\frac{1}{a^{3} b}} + 1}{a b}} \sqrt{\frac{1}{a^{3} b}} + x\right ) - \frac{1}{4} \, \sqrt{\frac{a b \sqrt{\frac{1}{a^{3} b}} - 1}{a b}} \log \left (a^{2} b \sqrt{\frac{a b \sqrt{\frac{1}{a^{3} b}} - 1}{a b}} \sqrt{\frac{1}{a^{3} b}} + x\right ) + \frac{1}{4} \, \sqrt{\frac{a b \sqrt{\frac{1}{a^{3} b}} - 1}{a b}} \log \left (-a^{2} b \sqrt{\frac{a b \sqrt{\frac{1}{a^{3} b}} - 1}{a b}} \sqrt{\frac{1}{a^{3} b}} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="fricas")

[Out]

1/4*sqrt(-(a*b*sqrt(1/(a^3*b)) + 1)/(a*b))*log(a^2*b*sqrt(-(a*b*sqrt(1/(a^3*b))
+ 1)/(a*b))*sqrt(1/(a^3*b)) + x) - 1/4*sqrt(-(a*b*sqrt(1/(a^3*b)) + 1)/(a*b))*lo
g(-a^2*b*sqrt(-(a*b*sqrt(1/(a^3*b)) + 1)/(a*b))*sqrt(1/(a^3*b)) + x) - 1/4*sqrt(
(a*b*sqrt(1/(a^3*b)) - 1)/(a*b))*log(a^2*b*sqrt((a*b*sqrt(1/(a^3*b)) - 1)/(a*b))
*sqrt(1/(a^3*b)) + x) + 1/4*sqrt((a*b*sqrt(1/(a^3*b)) - 1)/(a*b))*log(-a^2*b*sqr
t((a*b*sqrt(1/(a^3*b)) - 1)/(a*b))*sqrt(1/(a^3*b)) + x)

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Sympy [A]  time = 1.07186, size = 44, normalized size = 0.4 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{2} + 32 t^{2} a^{2} b + a - b, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} b - 4 t a + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a*x**4+2*a*x**2+a-b),x)

[Out]

RootSum(256*_t**4*a**3*b**2 + 32*_t**2*a**2*b + a - b, Lambda(_t, _t*log(-64*_t*
*3*a**2*b - 4*_t*a + x)))

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GIAC/XCAS [A]  time = 2.25456, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a*x^4 + 2*a*x^2 + a - b),x, algorithm="giac")

[Out]

Done