3.2.14 \(\int \frac {(a-b x^2)^{2/3}}{(3 a+b x^2)^3} \, dx\) [114]

Optimal. Leaf size=818 \[ \frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}+\frac {x \left (a-b x^2\right )^{2/3}}{36 a^2 \left (3 a+b x^2\right )}-\frac {x}{36 a^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{216\ 2^{2/3} a^{11/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{72\ 2^{2/3} a^{11/6} \sqrt {b}}-\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{24\ 3^{3/4} a^{5/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{18 \sqrt {2} \sqrt [4]{3} a^{5/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]

[Out]

1/12*x*(-b*x^2+a)^(2/3)/a/(b*x^2+3*a)^2+1/36*x*(-b*x^2+a)^(2/3)/a^2/(b*x^2+3*a)-1/36*x/a^2/(-(-b*x^2+a)^(1/3)+
a^(1/3)*(1-3^(1/2)))+1/144*arctanh(x*b^(1/2)/a^(1/6)/(a^(1/3)+2^(1/3)*(-b*x^2+a)^(1/3)))*2^(1/3)/a^(11/6)/b^(1
/2)-1/432*arctanh(x*b^(1/2)/a^(1/2))*2^(1/3)/a^(11/6)/b^(1/2)+1/432*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(-b*x^2+a)
^(1/3))*3^(1/2)/x/b^(1/2))*2^(1/3)/a^(11/6)*3^(1/2)/b^(1/2)+1/432*arctan(3^(1/2)*a^(1/2)/x/b^(1/2))*2^(1/3)/a^
(11/6)*3^(1/2)/b^(1/2)+1/108*(a^(1/3)-(-b*x^2+a)^(1/3))*EllipticF((-(-b*x^2+a)^(1/3)+a^(1/3)*(1+3^(1/2)))/(-(-
b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b*x^2+a)^(2/3))/(-(-b*
x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)*3^(3/4)/a^(5/3)/b/x*2^(1/2)/(-a^(1/3)*(a^(1/3)-(-b*x^2+a)^(1/3))/(-
(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)-1/72*(a^(1/3)-(-b*x^2+a)^(1/3))*EllipticE((-(-b*x^2+a)^(1/3)+a^
(1/3)*(1+3^(1/2)))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(
-b*x^2+a)^(2/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*3^(1/4)/a^(5/3)/b/
x/(-a^(1/3)*(a^(1/3)-(-b*x^2+a)^(1/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 0.46, antiderivative size = 818, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {423, 541, 544, 241, 310, 225, 1893, 402} \begin {gather*} \frac {\left (a-b x^2\right )^{2/3} x}{36 a^2 \left (b x^2+3 a\right )}-\frac {x}{36 a^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {\left (a-b x^2\right )^{2/3} x}{12 a \left (b x^2+3 a\right )^2}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{216\ 2^{2/3} a^{11/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{72\ 2^{2/3} a^{11/6} \sqrt {b}}-\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\text {ArcSin}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{24\ 3^{3/4} a^{5/3} b \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}+\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\text {ArcSin}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{18 \sqrt {2} \sqrt [4]{3} a^{5/3} b \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a - b*x^2)^(2/3))/(12*a*(3*a + b*x^2)^2) + (x*(a - b*x^2)^(2/3))/(36*a^2*(3*a + b*x^2)) - x/(36*a^2*((1 -
Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(72*2^(2/3)*Sqrt[3]*a^(11/6)*Sq
rt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(72*2^(2/3)*Sqrt[3]*a^(11
/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(216*2^(2/3)*a^(11/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(
1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(72*2^(2/3)*a^(11/6)*Sqrt[b]) - (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^
(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/
3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3)
)], -7 + 4*Sqrt[3]])/(24*3^(3/4)*a^(5/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(
1/3) - (a - b*x^2)^(1/3))^2)]) + ((a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a
- b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a -
b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(18*Sqrt[2]*3^(1/4)*a^(5/3)*b*x*
Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(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 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 402

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[q*(ArcTan
[Sqrt[3]/(q*x)]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d)), x] + (Simp[q*(ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*
x^2)^(1/3))]/(2*2^(2/3)*a^(1/3)*d)), x] - Simp[q*(ArcTanh[q*x]/(6*2^(2/3)*a^(1/3)*d)), x] + Simp[q*(ArcTan[Sqr
t[3]*((a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3))/(a^(1/3)*q*x))]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d)), x])] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]

Rule 423

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-x)*(a + b*x^n)^(p + 1)*((
c + d*x^n)^q/(a*n*(p + 1))), x] + Dist[1/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(n*
(p + 1) + 1) + d*(n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[
p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 544

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

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 {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^3} \, dx &=\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}-\frac {\int \frac {-3 a+\frac {5 b x^2}{3}}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2} \, dx}{12 a}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}+\frac {x \left (a-b x^2\right )^{2/3}}{36 a^2 \left (3 a+b x^2\right )}+\frac {\int \frac {16 a^2 b+\frac {8}{3} a b^2 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{288 a^3 b}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}+\frac {x \left (a-b x^2\right )^{2/3}}{36 a^2 \left (3 a+b x^2\right )}+\frac {\int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{108 a^2}+\frac {\int \frac {1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{36 a}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}+\frac {x \left (a-b x^2\right )^{2/3}}{36 a^2 \left (3 a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{216\ 2^{2/3} a^{11/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{72\ 2^{2/3} a^{11/6} \sqrt {b}}-\frac {\sqrt {-b x^2} \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{72 a^2 b x}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}+\frac {x \left (a-b x^2\right )^{2/3}}{36 a^2 \left (3 a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{216\ 2^{2/3} a^{11/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{72\ 2^{2/3} a^{11/6} \sqrt {b}}+\frac {\sqrt {-b x^2} \text {Subst}\left (\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{72 a^2 b x}-\frac {\left (\sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{36 a^{5/3} b x}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}+\frac {x \left (a-b x^2\right )^{2/3}}{36 a^2 \left (3 a+b x^2\right )}-\frac {x}{36 a^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{216\ 2^{2/3} a^{11/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{72\ 2^{2/3} a^{11/6} \sqrt {b}}-\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{24\ 3^{3/4} a^{5/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{18 \sqrt {2} \sqrt [4]{3} a^{5/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 10.29, size = 252, normalized size = 0.31 \begin {gather*} \frac {\frac {b x^3 \sqrt [3]{1-\frac {b x^2}{a}} F_1\left (\frac {3}{2};\frac {1}{3},1;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{a^3}+\frac {27 x \left (6-\frac {5 b x^2}{a}-\frac {b^2 x^4}{a^2}+\frac {18 \left (3 a+b x^2\right ) F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{9 a F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+2 b x^2 \left (-F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )}\right )}{\left (3 a+b x^2\right )^2}}{972 \sqrt [3]{a-b x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((b*x^3*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])/a^3 + (27*x*(6 - (5*b*x^2
)/a - (b^2*x^4)/a^2 + (18*(3*a + b*x^2)*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/(9*a*AppellF1[1
/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]
+ AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))/(3*a + b*x^2)^2)/(972*(a - b*x^2)^(1/3))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}}}{\left (b \,x^{2}+3 a \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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