3.7.62 \(\int \frac {x^2}{(a^2+2 a b x^2+b^2 x^4)^{2/3}} \, dx\) [662]

Optimal. Leaf size=618 \[ -\frac {3 x \left (a+b x^2\right )}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac {9 a x \left (1+\frac {b x^2}{a}\right )^{4/3}}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )}+\frac {9 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^2 \left (1+\frac {b x^2}{a}\right )^{4/3} \left (1-\sqrt [3]{1+\frac {b x^2}{a}}\right ) \sqrt {\frac {1+\sqrt [3]{1+\frac {b x^2}{a}}+\left (1+\frac {b x^2}{a}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}}{1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}}\right )|-7+4 \sqrt {3}\right )}{4 b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt {-\frac {1-\sqrt [3]{1+\frac {b x^2}{a}}}{\left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )^2}}}-\frac {3\ 3^{3/4} a^2 \left (1+\frac {b x^2}{a}\right )^{4/3} \left (1-\sqrt [3]{1+\frac {b x^2}{a}}\right ) \sqrt {\frac {1+\sqrt [3]{1+\frac {b x^2}{a}}+\left (1+\frac {b x^2}{a}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}}{1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {2} b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt {-\frac {1-\sqrt [3]{1+\frac {b x^2}{a}}}{\left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )^2}}} \]

[Out]

-3/2*x*(b*x^2+a)/b/(b^2*x^4+2*a*b*x^2+a^2)^(2/3)-9/2*a*x*(1+b*x^2/a)^(4/3)/b/(b^2*x^4+2*a*b*x^2+a^2)^(2/3)/(1-
(1+b*x^2/a)^(1/3)-3^(1/2))-3/2*3^(3/4)*a^2*(1+b*x^2/a)^(4/3)*(1-(1+b*x^2/a)^(1/3))*EllipticF((1-(1+b*x^2/a)^(1
/3)+3^(1/2))/(1-(1+b*x^2/a)^(1/3)-3^(1/2)),2*I-I*3^(1/2))*((1+(1+b*x^2/a)^(1/3)+(1+b*x^2/a)^(2/3))/(1-(1+b*x^2
/a)^(1/3)-3^(1/2))^2)^(1/2)/b^2/x/(b^2*x^4+2*a*b*x^2+a^2)^(2/3)*2^(1/2)/((-1+(1+b*x^2/a)^(1/3))/(1-(1+b*x^2/a)
^(1/3)-3^(1/2))^2)^(1/2)+9/4*3^(1/4)*a^2*(1+b*x^2/a)^(4/3)*(1-(1+b*x^2/a)^(1/3))*EllipticE((1-(1+b*x^2/a)^(1/3
)+3^(1/2))/(1-(1+b*x^2/a)^(1/3)-3^(1/2)),2*I-I*3^(1/2))*((1+(1+b*x^2/a)^(1/3)+(1+b*x^2/a)^(2/3))/(1-(1+b*x^2/a
)^(1/3)-3^(1/2))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))/b^2/x/(b^2*x^4+2*a*b*x^2+a^2)^(2/3)/((-1+(1+b*x^2/a)^(1/3)
)/(1-(1+b*x^2/a)^(1/3)-3^(1/2))^2)^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 618, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1127, 294, 241, 310, 225, 1893} \begin {gather*} -\frac {3\ 3^{3/4} a^2 \left (\frac {b x^2}{a}+1\right )^{4/3} \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}} F\left (\text {ArcSin}\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^2 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 {9 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^2 \left (\frac {b x^2}{a}+1\right )^{4/3} \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}} E\left (\text {ArcSin}\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^2 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 {9 a x \left (\frac {b x^2}{a}+1\right )^{4/3}}{2 b \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 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x*(a + b*x^2))/(2*b*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)) - (9*a*x*(1 + (b*x^2)/a)^(4/3))/(2*b*(a^2 + 2*a*b*x
^2 + b^2*x^4)^(2/3)*(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))) + (9*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^2*(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^2*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)*Sqrt[-((1 - (1 + (b*x^2)/a)^(1/3))/(1 - Sqr
t[3] - (1 + (b*x^2)/a)^(1/3))^2)]) - (3*3^(3/4)*a^2*(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^2*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 225

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

Rule 241

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 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/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 1127

Int[((d_.)*(x_))^(m_.)*((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[(d*x)^m*(1 + 2*c*(x^2/b))^(2*p), x], x] /; FreeQ[{
a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[2*p]

Rule 1893

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/(
(1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], 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 {x^2}{\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 {x^2}{\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 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac {\left (3 a \left (1+\frac {b x^2}{a}\right )^{4/3}\right ) \int \frac {1}{\sqrt [3]{1+\frac {b x^2}{a}}} \, dx}{2 b \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 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac {\left (9 a^2 \sqrt {\frac {b x^2}{a}} \left (1+\frac {b x^2}{a}\right )^{4/3}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+\frac {b x^2}{a}}\right )}{4 b^2 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 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac {\left (9 a^2 \sqrt {\frac {b x^2}{a}} \left (1+\frac {b x^2}{a}\right )^{4/3}\right ) \text {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^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac {\left (9 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )} a^2 \sqrt {\frac {b x^2}{a}} \left (1+\frac {b x^2}{a}\right )^{4/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+\frac {b x^2}{a}}\right )}{2 b^2 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 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac {9 a x \left (1+\frac {b x^2}{a}\right )^{4/3}}{2 b \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 {9 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^2 \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^2 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^{3/4} a^2 \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^2 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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 5.75, size = 64, normalized size = 0.10 \begin {gather*} \frac {3 x \left (a+b x^2\right ) \left (-1+\sqrt [3]{1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )\right )}{2 b \left (\left (a+b x^2\right )^2\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {2}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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