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)*Sqr
t[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 = 0.255547, 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, Rules used = {1129, 634, 618, 204, 628} \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{\sqrt{a+b}+\sqrt{a}}} \]

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)*Sqr
t[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]])

Rule 1129

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/
c, 2]}, Dist[1/(2*c*r), Int[x^(m - 1)/(q - r*x + x^2), x], x] - Dist[1/(2*c*r), Int[x^(m - 1)/(q + r*x + x^2),
x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^2}{a+b+2 a x^2+a x^4} \, dx &=\frac{\int \frac{x}{\frac{\sqrt{a+b}}{\sqrt{a}}-\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{2 \sqrt{2} a^{3/4} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\int \frac{x}{\frac{\sqrt{a+b}}{\sqrt{a}}+\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{2 \sqrt{2} a^{3/4} \sqrt{-\sqrt{a}+\sqrt{a+b}}}\\ &=\frac{\int \frac{1}{\frac{\sqrt{a+b}}{\sqrt{a}}-\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 a}+\frac{\int \frac{1}{\frac{\sqrt{a+b}}{\sqrt{a}}+\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 a}+\frac{\int \frac{-\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}{\sqrt [4]{a}}+2 x}{\frac{\sqrt{a+b}}{\sqrt{a}}-\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 \sqrt{2} a^{3/4} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\int \frac{\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}{\sqrt [4]{a}}+2 x}{\frac{\sqrt{a+b}}{\sqrt{a}}+\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 \sqrt{2} a^{3/4} \sqrt{-\sqrt{a}+\sqrt{a+b}}}\\ &=\frac{\log \left (\sqrt{a+b}-\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a}+\sqrt{a+b}} x+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\log \left (\sqrt{a+b}+\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a}+\sqrt{a+b}} x+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\frac{\sqrt{a+b}}{\sqrt{a}}\right )-x^2} \, dx,x,-\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}{\sqrt [4]{a}}+2 x\right )}{2 a}-\frac{\operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\frac{\sqrt{a+b}}{\sqrt{a}}\right )-x^2} \, dx,x,\frac{\sqrt{2} \sqrt{-\sqrt{a}+\sqrt{a+b}}}{\sqrt [4]{a}}+2 x\right )}{2 a}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-\sqrt{a}+\sqrt{a+b}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{a+b}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{\sqrt{a}+\sqrt{a+b}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{-\sqrt{a}+\sqrt{a+b}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{a+b}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{\sqrt{a}+\sqrt{a+b}}}+\frac{\log \left (\sqrt{a+b}-\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a}+\sqrt{a+b}} x+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt{-\sqrt{a}+\sqrt{a+b}}}-\frac{\log \left (\sqrt{a+b}+\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a}+\sqrt{a+b}} x+\sqrt{a} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt{-\sqrt{a}+\sqrt{a+b}}}\\ \end{align*}

Mathematica [C]  time = 0.116752, 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*Sqrt[a]*Sqrt[b]]])/Sqrt[a + I*Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*Sq
rt[b])

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Maple [B]  time = 0.159, size = 724, normalized size = 2.2 \begin{align*} \text{result too large to display} \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]

1/8/a^(3/2)/b*ln(-x^2*a^(1/2)+x*(2*(a*(a+b))^(1/2)-2*a)^(1/2)-(a+b)^(1/2))*(a^2+a*b)^(1/2)*(2*(a^2+a*b)^(1/2)-
2*a)^(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*x*a^(1/2)+(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)*(a^2+a*b
)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)+1/8/a^(1/2)/b*ln(-x^2*a^(1/2)+x*(2*(a*(a+b))^(1/2)-2*a)^(1/2)-(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*
x*a^(1/2)+(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(x^2*a^(1/2)+x*(2*(a*(a+b))^(1/2)-2*a)^(1/2)+(a+b)
^(1/2))*(a^2+a*b)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(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*x*a^(1/2)+(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)*(a^2+a*b)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/8/a^(1/2)/b*ln(x^2*a^(1/2)+x
*(2*(a*(a+b))^(1/2)-2*a)^(1/2)+(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*x*a^(1/2)+(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. \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 [A]  time = 1.52766, size = 633, normalized size = 1.91 \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))*sqr
t(-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 = 0.355501, size = 44, normalized size = 0.13 \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