3.663 \(\int \frac{1}{(a^2+2 a b x^2+b^2 x^4)^{2/3}} \, dx\)

Optimal. Leaf size=609 \[ \frac{3^{3/4} a \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} \left (\frac{b x^2}{a}+1\right )^{4/3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{\sqrt{2} b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}}+\frac{3 x \left (\frac{b x^2}{a}+1\right )^{4/3}}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )}+\frac{3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} \left (\frac{b x^2}{a}+1\right )^{4/3} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}} \]

[Out]

(3*x*(a + b*x^2))/(2*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)) + (3*x*(1 + (b*x^2)/a)^(4/3))/(2*(a^2 + 2*a*b*x^2 +
b^2*x^4)^(2/3)*(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))) - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*a*(1 + (b*x^2)/a)^(4/3)*(
1 - (1 + (b*x^2)/a)^(1/3))*Sqrt[(1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - Sqrt[3] - (1 + (b*x^2
)/a)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))],
-7 + 4*Sqrt[3]])/(4*b*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*Sqrt[-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1
+ (b*x^2)/a)^(1/3))^2)]) + (3^(3/4)*a*(1 + (b*x^2)/a)^(4/3)*(1 - (1 + (b*x^2)/a)^(1/3))*Sqrt[(1 + (1 + (b*x^2
)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (
1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[2]*b*x*(a^2 + 2*a*b*x^2 +
b^2*x^4)^(2/3)*Sqrt[-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))^2)])

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Rubi [A]  time = 0.366157, antiderivative size = 609, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1089, 199, 235, 304, 219, 1879} \[ \frac{3 x \left (\frac{b x^2}{a}+1\right )^{4/3}}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )}+\frac{3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac{3^{3/4} a \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} \left (\frac{b x^2}{a}+1\right )^{4/3} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{2} b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} \left (\frac{b x^2}{a}+1\right )^{4/3} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(a + b*x^2))/(2*a*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)) + (3*x*(1 + (b*x^2)/a)^(4/3))/(2*(a^2 + 2*a*b*x^2 +
b^2*x^4)^(2/3)*(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))) - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*a*(1 + (b*x^2)/a)^(4/3)*(
1 - (1 + (b*x^2)/a)^(1/3))*Sqrt[(1 + (1 + (b*x^2)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - Sqrt[3] - (1 + (b*x^2
)/a)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))],
-7 + 4*Sqrt[3]])/(4*b*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*Sqrt[-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1
+ (b*x^2)/a)^(1/3))^2)]) + (3^(3/4)*a*(1 + (b*x^2)/a)^(4/3)*(1 - (1 + (b*x^2)/a)^(1/3))*Sqrt[(1 + (1 + (b*x^2
)/a)^(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (
1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[2]*b*x*(a^2 + 2*a*b*x^2 +
b^2*x^4)^(2/3)*Sqrt[-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))^2)])

Rule 1089

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^2 + c*x^4)^FracPart[p]
)/(1 + (2*c*x^2)/b)^(2*FracPart[p]), Int[(1 + (2*c*x^2)/b)^(2*p), x], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2
- 4*a*c, 0] &&  !IntegerQ[2*p]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 235

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Dist[(3*Sqrt[b*x^2])/(2*b*x), Subst[Int[x/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a
+ b*x^3], x], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 1879

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 + Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 + Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 - Sqrt[3])*s + r*x)), x
] + Simp[(3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx &=\frac{\left (1+\frac{b x^2}{a}\right )^{4/3} \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{4/3}} \, dx}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=\frac{3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac{\left (1+\frac{b x^2}{a}\right )^{4/3} \int \frac{1}{\sqrt [3]{1+\frac{b x^2}{a}}} \, dx}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=\frac{3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac{\left (3 a \sqrt{\frac{b x^2}{a}} \left (1+\frac{b x^2}{a}\right )^{4/3}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1+\frac{b x^2}{a}}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=\frac{3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac{\left (3 a \sqrt{\frac{b x^2}{a}} \left (1+\frac{b x^2}{a}\right )^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1+\frac{b x^2}{a}}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac{\left (3 \sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )} a \sqrt{\frac{b x^2}{a}} \left (1+\frac{b x^2}{a}\right )^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1+\frac{b x^2}{a}}\right )}{2 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=\frac{3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac{3 x \left (1+\frac{b x^2}{a}\right )^{4/3}}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}\right )}-\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a \left (1+\frac{b x^2}{a}\right )^{4/3} \left (1-\sqrt [3]{1+\frac{b x^2}{a}}\right ) \sqrt{\frac{1+\sqrt [3]{1+\frac{b x^2}{a}}+\left (1+\frac{b x^2}{a}\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}\right )^2}} E\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}}{1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}}\right )|-7+4 \sqrt{3}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{1+\frac{b x^2}{a}}}{\left (1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}\right )^2}}}+\frac{3^{3/4} a \left (1+\frac{b x^2}{a}\right )^{4/3} \left (1-\sqrt [3]{1+\frac{b x^2}{a}}\right ) \sqrt{\frac{1+\sqrt [3]{1+\frac{b x^2}{a}}+\left (1+\frac{b x^2}{a}\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}}{1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}}\right )|-7+4 \sqrt{3}\right )}{\sqrt{2} b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{1+\frac{b x^2}{a}}}{\left (1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}\right )^2}}}\\ \end{align*}

Mathematica [C]  time = 0.0215587, size = 64, normalized size = 0.11 \[ -\frac{x \left (a+b x^2\right ) \left (\sqrt [3]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )-3\right )}{2 a \left (\left (a+b x^2\right )^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-2/3),x]

[Out]

-(x*(a + b*x^2)*(-3 + (1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)]))/(2*a*((a + b*x^2)
^2)^(2/3))

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Maple [F]  time = 0.176, size = 0, normalized size = 0. \begin{align*} \int \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{-{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x)

[Out]

int(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x, algorithm="maxima")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-2/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{2}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)^(-2/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(2/3),x)

[Out]

Integral((a**2 + 2*a*b*x**2 + b**2*x**4)**(-2/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x, algorithm="giac")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-2/3), x)