\(\int x^2 (a+b x^2)^{2/3} \, dx\) [734]

Optimal result

Integrand size = 15, antiderivative size = 577 \[ \int x^2 \left (a+b x^2\right )^{2/3} \, dx=\frac {12 a x \left (a+b x^2\right )^{2/3}}{91 b}+\frac {3}{13} x^3 \left (a+b x^2\right )^{2/3}+\frac {36 a^2 x}{91 b \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac {18 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{91 b^2 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {12 \sqrt {2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{91 b^2 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \] Output:

12/91*a*x*(b*x^2+a)^(2/3)/b+3/13*x^3*(b*x^2+a)^(2/3)+36/91*a^2*x/b/((1-3^( 
1/2))*a^(1/3)-(b*x^2+a)^(1/3))-18/91*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)*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^2/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)+12/91*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^2/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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 5.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.11 \[ \int x^2 \left (a+b x^2\right )^{2/3} \, dx=\frac {3 x \left (a+b x^2\right )^{2/3} \left (a+b x^2-\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{\left (1+\frac {b x^2}{a}\right )^{2/3}}\right )}{13 b} \] Input:

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

Output:

(3*x*(a + b*x^2)^(2/3)*(a + b*x^2 - (a*Hypergeometric2F1[-2/3, 1/2, 3/2, - 
((b*x^2)/a)])/(1 + (b*x^2)/a)^(2/3)))/(13*b)
 

Rubi [A] (warning: unable to verify)

Time = 0.48 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {248, 262, 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.

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

\(\Big \downarrow \) 248

\(\displaystyle \frac {4}{13} a \int \frac {x^2}{\sqrt [3]{b x^2+a}}dx+\frac {3}{13} x^3 \left (a+b x^2\right )^{2/3}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {4}{13} a \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {3 a \int \frac {1}{\sqrt [3]{b x^2+a}}dx}{7 b}\right )+\frac {3}{13} x^3 \left (a+b x^2\right )^{2/3}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {4}{13} a \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{14 b^2 x}\right )+\frac {3}{13} x^3 \left (a+b x^2\right )^{2/3}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {4}{13} a \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{14 b^2 x}\right )+\frac {3}{13} x^3 \left (a+b x^2\right )^{2/3}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {4}{13} a \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{14 b^2 x}\right )+\frac {3}{13} x^3 \left (a+b x^2\right )^{2/3}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {4}{13} a \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{14 b^2 x}\right )+\frac {3}{13} x^3 \left (a+b x^2\right )^{2/3}\)

Input:

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

Output:

(3*x^3*(a + b*x^2)^(2/3))/13 + (4*a*((3*x*(a + b*x^2)^(2/3))/(7*b) - (9*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 + 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]])/(Sqrt[b*x^2]*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqr 
t[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)])))/(14*b^2*x)))/13
 

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 248

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

rule 262

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.05 \[ \int x^2 \left (a+b x^2\right )^{2/3} \, dx=\frac {a^{\frac {2}{3}} x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} \] Input:

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

Output:

a**(2/3)*x**3*hyper((-2/3, 3/2), (5/2,), b*x**2*exp_polar(I*pi)/a)/3
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (a+b x^2\right )^{2/3} \, dx=\frac {\frac {12 \left (b \,x^{2}+a \right )^{\frac {2}{3}} a x}{91}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {2}{3}} b \,x^{3}}{13}-\frac {12 \left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}}}d x \right ) a^{2}}{91}}{b} \] Input:

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

Output:

(3*(4*(a + b*x**2)**(2/3)*a*x + 7*(a + b*x**2)**(2/3)*b*x**3 - 4*int((a + 
b*x**2)**(2/3)/(a + b*x**2),x)*a**2))/(91*b)