\(\int \frac {(a-b x^2)^{2/3}}{3 a+b x^2} \, dx\) [323]

Optimal result

Integrand size = 24, antiderivative size = 740 \[ \int \frac {\left (a-b x^2\right )^{2/3}}{3 a+b x^2} \, dx=\frac {3 x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}+\frac {\sqrt [3]{2} \sqrt [6]{a} \arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}+\frac {\sqrt [3]{2} \sqrt [6]{a} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}+\frac {\sqrt [3]{2} \sqrt [6]{a} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{\sqrt {b}}+\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{2 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {\sqrt {2} 3^{3/4} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 5.64 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a-b x^2\right )^{2/3}}{3 a+b x^2} \, dx=\frac {9 a x \left (a-b x^2\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {2}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{\left (3 a+b x^2\right ) \left (9 a \operatorname {AppellF1}\left (\frac {1}{2},-\frac {2}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )-2 b x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},-\frac {2}{3},2,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )\right )} \] Input:

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

Output:

(9*a*x*(a - b*x^2)^(2/3)*AppellF1[1/2, -2/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^ 
2)/a])/((3*a + b*x^2)*(9*a*AppellF1[1/2, -2/3, 1, 3/2, (b*x^2)/a, -1/3*(b* 
x^2)/a] - 2*b*x^2*(AppellF1[3/2, -2/3, 2, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] 
+ 2*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])))
 

Rubi [A] (warning: unable to verify)

Time = 0.63 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {301, 233, 305, 833, 760, 2418}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a-b x^2\right )^{2/3}}{3 a+b x^2} \, dx\)

\(\Big \downarrow \) 301

\(\displaystyle 4 a \int \frac {1}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}dx-\int \frac {1}{\sqrt [3]{a-b x^2}}dx\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3 \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{2 b x}+4 a \int \frac {1}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}dx\)

\(\Big \downarrow \) 305

\(\displaystyle \frac {3 \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{2 b x}+4 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}\right )\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3 \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{2 b x}+4 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}\right )\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3 \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{2 b x}+4 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}\right )\)

\(\Big \downarrow \) 2418

\(\displaystyle 4 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}\right )+\frac {3 \sqrt {-b x^2} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{2 b x}\)

Input:

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

Output:

4*a*(ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt 
[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqr 
t[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a] 
]/(6*2^(2/3)*a^(5/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^ 
(1/3)*(a - b*x^2)^(1/3)))]/(2*2^(2/3)*a^(5/6)*Sqrt[b])) + (3*Sqrt[-(b*x^2) 
]*((-2*Sqrt[-(b*x^2)])/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3)) + (3^(1 
/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(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]])/(S 
qrt[-(b*x^2)]*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)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*a 
^(1/3)*(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]*E 
llipticF[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]])/(3^(1/4)*Sqrt[-(b*x^2)]*S 
qrt[-((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)])))/(2*b*x)
 

Defintions of rubi rules used

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[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 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 305

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit 
h[{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[Sqrt[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 760

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]*Sqrt[(- 
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 833

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

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[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] + S 
imp[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 - S 
qrt[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*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}}}{b \,x^{2}+3 a}d x\]

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{2/3}}{3 a+b x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a-b x^2\right )^{2/3}}{3 a+b x^2} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {2}{3}}}{3 a + b x^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\left (a-b x^2\right )^{2/3}}{3 a+b x^2} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {2}{3}}}{b x^{2} + 3 \, a} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\left (a-b x^2\right )^{2/3}}{3 a+b x^2} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {2}{3}}}{b x^{2} + 3 \, a} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{2/3}}{3 a+b x^2} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{2/3}}{b\,x^2+3\,a} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a-b x^2\right )^{2/3}}{3 a+b x^2} \, dx=\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}}}{b \,x^{2}+3 a}d x \] Input:

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

Output:

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