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

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Rubi [A]  time = 0.0537356, 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, Rules used = {1130, 205} \[ \frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{b}} \]

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

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^2}{a-b+2 a x^2+a x^4} \, dx &=-\left (\frac{1}{2} \left (-1+\frac{\sqrt{a}}{\sqrt{b}}\right ) \int \frac{1}{a-\sqrt{a} \sqrt{b}+a x^2} \, dx\right )+\frac{1}{2} \left (1+\frac{\sqrt{a}}{\sqrt{b}}\right ) \int \frac{1}{a+\sqrt{a} \sqrt{b}+a x^2} \, dx\\ &=-\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}}\\ \end{align*}

Mathematica [A]  time = 0.104889, 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]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*Sqrt[b])

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Maple [A]  time = 0.133, size = 134, normalized size = 1.2 \begin{align*} -{\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}}}}+{\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}}}} \end{align*}

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

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

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 [B]  time = 1.47349, size = 617, normalized size = 5.66 \begin{align*} \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 ) \end{align*}

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))*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) + 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 = 0.373015, size = 44, normalized size = 0.4 \begin{align*} \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 )} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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]

Exception raised: NotImplementedError