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

Optimal. Leaf size=331 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}+\sqrt{a}}} \]

[Out]

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

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Rubi [A]  time = 0.619689, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}+\sqrt{a}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 65.3461, size = 294, normalized size = 0.89 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt [4]{a} x - \frac{\sqrt{- 2 \sqrt{a} + 2 \sqrt{a + b}}}{2}\right )}{\sqrt{\sqrt{a} + \sqrt{a + b}}} \right )}}{4 a^{\frac{3}{4}} \sqrt{\sqrt{a} + \sqrt{a + b}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt [4]{a} x + \frac{\sqrt{- 2 \sqrt{a} + 2 \sqrt{a + b}}}{2}\right )}{\sqrt{\sqrt{a} + \sqrt{a + b}}} \right )}}{4 a^{\frac{3}{4}} \sqrt{\sqrt{a} + \sqrt{a + b}}} + \frac{\sqrt{2} \log{\left (x^{2} + \frac{\sqrt{a + b}}{\sqrt{a}} - \frac{\sqrt{2} x \sqrt{- \sqrt{a} + \sqrt{a + b}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}} \sqrt{- \sqrt{a} + \sqrt{a + b}}} - \frac{\sqrt{2} \log{\left (x^{2} + \frac{\sqrt{a + b}}{\sqrt{a}} + \frac{\sqrt{2} x \sqrt{- \sqrt{a} + \sqrt{a + b}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}} \sqrt{- \sqrt{a} + \sqrt{a + b}}} \]

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(2)*atan(sqrt(2)*(a**(1/4)*x - sqrt(-2*sqrt(a) + 2*sqrt(a + b))/2)/sqrt(sqrt
(a) + sqrt(a + b)))/(4*a**(3/4)*sqrt(sqrt(a) + sqrt(a + b))) + sqrt(2)*atan(sqrt
(2)*(a**(1/4)*x + sqrt(-2*sqrt(a) + 2*sqrt(a + b))/2)/sqrt(sqrt(a) + sqrt(a + b)
))/(4*a**(3/4)*sqrt(sqrt(a) + sqrt(a + b))) + sqrt(2)*log(x**2 + sqrt(a + b)/sqr
t(a) - sqrt(2)*x*sqrt(-sqrt(a) + sqrt(a + b))/a**(1/4))/(8*a**(3/4)*sqrt(-sqrt(a
) + sqrt(a + b))) - sqrt(2)*log(x**2 + sqrt(a + b)/sqrt(a) + sqrt(2)*x*sqrt(-sqr
t(a) + sqrt(a + b))/a**(1/4))/(8*a**(3/4)*sqrt(-sqrt(a) + sqrt(a + b)))

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Mathematica [C]  time = 0.198852, size = 143, normalized size = 0.43 \[ \frac{\frac{\left (\sqrt{b}+i \sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a-i \sqrt{a} \sqrt{b}}}\right )}{\sqrt{a-i \sqrt{a} \sqrt{b}}}+\frac{\left (\sqrt{b}-i \sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right )}{\sqrt{a+i \sqrt{a} \sqrt{b}}}}{2 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(((I*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[b]]])/Sqrt[a
- I*Sqrt[a]*Sqrt[b]] + (((-I)*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*S
qrt[a]*Sqrt[b]]])/Sqrt[a + I*Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*Sqrt[b])

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Maple [B]  time = 0.049, size = 724, normalized size = 2.2 \[ \text{result too large to display} \]

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]

1/8/a^(3/2)/b*ln(-a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-2*a)^(1/2)*x-(a+b)^(1/2))*(2*(a
^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/4/a^(3/2)/b/(4*a^(1/2)*(a+b)^(1/2)-2*
(a*(a+b))^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(
4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2
)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)+1/8/a^(1/2)/b*ln(-a^(1/2)*x^2+(2
*(a*(a+b))^(1/2)-2*a)^(1/2)*x-(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/4/a^(
1/2)/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2)*x+
(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/
2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/8/a^(3/2)/b*ln
(a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-2*a)^(1/2)*x+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a
)^(1/2)*(a^2+a*b)^(1/2)+1/4/a^(3/2)/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2
*a)^(1/2)*arctan((2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1
/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/
2)-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/8/a^(1/2)/b*ln(a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-2*
a)^(1/2)*x+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)+1/4/a^(1/2)/b/(4*a^(1/2)*(
a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2
*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/
2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)

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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 = 0.280333, size = 377, normalized size = 1.14 \[ \frac{1}{4} \, \sqrt{\frac{a b \sqrt{-\frac{1}{a^{3} b}} + 1}{a b}} \log \left (a^{2} b \sqrt{\frac{a b \sqrt{-\frac{1}{a^{3} b}} + 1}{a b}} \sqrt{-\frac{1}{a^{3} b}} + x\right ) - \frac{1}{4} \, \sqrt{\frac{a b \sqrt{-\frac{1}{a^{3} b}} + 1}{a b}} \log \left (-a^{2} b \sqrt{\frac{a b \sqrt{-\frac{1}{a^{3} b}} + 1}{a b}} \sqrt{-\frac{1}{a^{3} b}} + x\right ) - \frac{1}{4} \, \sqrt{-\frac{a b \sqrt{-\frac{1}{a^{3} b}} - 1}{a b}} \log \left (a^{2} b \sqrt{-\frac{a b \sqrt{-\frac{1}{a^{3} b}} - 1}{a b}} \sqrt{-\frac{1}{a^{3} b}} + x\right ) + \frac{1}{4} \, \sqrt{-\frac{a b \sqrt{-\frac{1}{a^{3} b}} - 1}{a b}} \log \left (-a^{2} b \sqrt{-\frac{a b \sqrt{-\frac{1}{a^{3} b}} - 1}{a b}} \sqrt{-\frac{1}{a^{3} b}} + x\right ) \]

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))*l
og(-a^2*b*sqrt((a*b*sqrt(-1/(a^3*b)) + 1)/(a*b))*sqrt(-1/(a^3*b)) + x) - 1/4*sqr
t(-(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*sqrt(-(a*b*sqrt(-1/(a^3*b)) - 1)/(a*b))*sqrt(-1/(a^3*b)) + x)

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Sympy [A]  time = 1.07097, size = 44, normalized size = 0.13 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{2} - 32 t^{2} a^{2} b + a + b, \left ( t \mapsto t \log{\left (64 t^{3} a^{2} b - 4 t a + x \right )} \right )\right )} \]

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 = 2.24044, size = 1, normalized size = 0. \[ \mathit{Done} \]

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