\(\int \frac {(3 a+b x^2)^2}{\sqrt [3]{a-b x^2}} \, dx\) [334]

Optimal result

Integrand size = 24, antiderivative size = 591 \[ \int \frac {\left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}} \, dx=-\frac {261}{91} a x \left (a-b x^2\right )^{2/3}-\frac {3}{13} b x^3 \left (a-b x^2\right )^{2/3}-\frac {3240 a^2 x}{91 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {1620 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {1080 \sqrt {2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \] Output:

-261/91*a*x*(-b*x^2+a)^(2/3)-3/13*b*x^3*(-b*x^2+a)^(2/3)-3240*a^2*x/(91*(1 
-3^(1/2))*a^(1/3)-91*(-b*x^2+a)^(1/3))-1620/91*3^(1/4)*(1/2*6^(1/2)+1/2*2^ 
(1/2))*a^(7/3)*(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)*Ellip 
ticE(((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)+1080/91*2^(1/2)*3^(3/4)*a^(7/3)*(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 5 vs. order 4 in optimal.

Time = 12.95 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.27 \[ \int \frac {\left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}} \, dx=\frac {x \sqrt [3]{1-\frac {b x^2}{a}} \left (63 a \left (45 a^2+10 a b x^2+b^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {7}{2},\frac {b x^2}{a}\right )+8 b x^2 \left (18 a^2+9 a b x^2+b^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {3}{2},\frac {9}{2},\frac {b x^2}{a}\right )+4 b \left (3 a x+b x^3\right )^2 \, _3F_2\left (\frac {4}{3},\frac {3}{2},2;1,\frac {9}{2};\frac {b x^2}{a}\right )\right )}{315 a \sqrt [3]{a-b x^2}} \] Input:

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

Output:

(x*(1 - (b*x^2)/a)^(1/3)*(63*a*(45*a^2 + 10*a*b*x^2 + b^2*x^4)*Hypergeomet 
ric2F1[1/3, 1/2, 7/2, (b*x^2)/a] + 8*b*x^2*(18*a^2 + 9*a*b*x^2 + b^2*x^4)* 
Hypergeometric2F1[4/3, 3/2, 9/2, (b*x^2)/a] + 4*b*(3*a*x + b*x^3)^2*Hyperg 
eometricPFQ[{4/3, 3/2, 2}, {1, 9/2}, (b*x^2)/a]))/(315*a*(a - b*x^2)^(1/3) 
)
 

Rubi [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {318, 27, 299, 233, 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.

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

\(\Big \downarrow \) 318

\(\displaystyle -\frac {3 \int -\frac {2 a b \left (11 b x^2+21 a\right )}{\sqrt [3]{a-b x^2}}dx}{13 b}-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{13} a \int \frac {11 b x^2+21 a}{\sqrt [3]{a-b x^2}}dx-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {6}{13} a \left (\frac {180}{7} a \int \frac {1}{\sqrt [3]{a-b x^2}}dx-\frac {33}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {6}{13} a \left (-\frac {270 a \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{7 b x}-\frac {33}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {6}{13} a \left (-\frac {270 a \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{7 b x}-\frac {33}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {6}{13} a \left (-\frac {270 a \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{7 b x}-\frac {33}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {6}{13} a \left (-\frac {270 a \sqrt {-b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{7 b x}-\frac {33}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\)

Input:

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

Output:

(-3*x*(a - b*x^2)^(2/3)*(3*a + b*x^2))/13 + (6*a*((-33*x*(a - b*x^2)^(2/3) 
)/7 - (270*a*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 + Sqr 
t[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]])/(Sqrt[-(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]*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]]) 
/(3^(1/4)*Sqrt[-(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)])))/(7*b*x)))/13
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 299

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

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

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]

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

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {1}{3}}} \,d x } \] Input:

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

Output:

integral(-(b^2*x^4 + 6*a*b*x^2 + 9*a^2)*(-b*x^2 + a)^(2/3)/(b*x^2 - a), x)
 

Sympy [A] (verification not implemented)

Time = 1.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.16 \[ \int \frac {\left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}} \, dx=9 a^{\frac {5}{3}} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + 2 a^{\frac {2}{3}} b x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {b^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5 \sqrt [3]{a}} \] Input:

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

Output:

9*a**(5/3)*x*hyper((1/3, 1/2), (3/2,), b*x**2*exp_polar(2*I*pi)/a) + 2*a** 
(2/3)*b*x**3*hyper((1/3, 3/2), (5/2,), b*x**2*exp_polar(2*I*pi)/a) + b**2* 
x**5*hyper((1/3, 5/2), (7/2,), b*x**2*exp_polar(2*I*pi)/a)/(5*a**(1/3))
 

Maxima [F]

\[ \int \frac {\left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {1}{3}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {1}{3}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}} \, dx=\int \frac {{\left (b\,x^2+3\,a\right )}^2}{{\left (a-b\,x^2\right )}^{1/3}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}} \, dx=\left (\int \frac {x^{4}}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}}}d x \right ) b^{2}+6 \left (\int \frac {x^{2}}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}}}d x \right ) a b +9 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}}}d x \right ) a^{2} \] Input:

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

Output:

int(x**4/(a - b*x**2)**(1/3),x)*b**2 + 6*int(x**2/(a - b*x**2)**(1/3),x)*a 
*b + 9*int(1/(a - b*x**2)**(1/3),x)*a**2