\(\int \frac {x^3}{\sqrt {a-b x^3} \sqrt {a+b x^3}} \, dx\) [882]

Optimal result

Integrand size = 27, antiderivative size = 551 \[ \int \frac {x^3}{\sqrt {a-b x^3} \sqrt {a+b x^3}} \, dx=\frac {a^2-b^2 x^6}{b^{4/3} \left (\left (1+\sqrt {3}\right ) a^{2/3}-b^{2/3} x^2\right ) \sqrt {a-b x^3} \sqrt {a+b x^3}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{2/3} \left (a^{2/3}-b^{2/3} x^2\right ) \sqrt {\frac {a^{4/3}+a^{2/3} b^{2/3} x^2+b^{4/3} x^4}{\left (\left (1+\sqrt {3}\right ) a^{2/3}-b^{2/3} x^2\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) a^{2/3}-b^{2/3} x^2}{\left (1+\sqrt {3}\right ) a^{2/3}-b^{2/3} x^2}\right )|-7-4 \sqrt {3}\right )}{2 b^{4/3} \sqrt {\frac {a^{2/3} \left (a^{2/3}-b^{2/3} x^2\right )}{\left (\left (1+\sqrt {3}\right ) a^{2/3}-b^{2/3} x^2\right )^2}} \sqrt {a-b x^3} \sqrt {a+b x^3}}+\frac {\sqrt {2} a^{2/3} \left (a^{2/3}-b^{2/3} x^2\right ) \sqrt {\frac {a^{4/3}+a^{2/3} b^{2/3} x^2+b^{4/3} x^4}{\left (\left (1+\sqrt {3}\right ) a^{2/3}-b^{2/3} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) a^{2/3}-b^{2/3} x^2}{\left (1+\sqrt {3}\right ) a^{2/3}-b^{2/3} x^2}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{4/3} \sqrt {\frac {a^{2/3} \left (a^{2/3}-b^{2/3} x^2\right )}{\left (\left (1+\sqrt {3}\right ) a^{2/3}-b^{2/3} x^2\right )^2}} \sqrt {a-b x^3} \sqrt {a+b x^3}} \] Output:

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

Mathematica [C] (verified)

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

Time = 1.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.13 \[ \int \frac {x^3}{\sqrt {a-b x^3} \sqrt {a+b x^3}} \, dx=\frac {x^4 \sqrt {1-\frac {b^2 x^6}{a^2}} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\frac {b^2 x^6}{a^2}\right )}{4 \sqrt {a-b x^3} \sqrt {a+b x^3}} \] Input:

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

Output:

(x^4*Sqrt[1 - (b^2*x^6)/a^2]*HypergeometricPFQ[{1/2, 2/3}, {5/3}, (b^2*x^6 
)/a^2])/(4*Sqrt[a - b*x^3]*Sqrt[a + b*x^3])
 

Rubi [A] (warning: unable to verify)

Time = 1.01 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {808, 890, 832, 759, 2416}

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.

\(\displaystyle \int \frac {x^3}{\sqrt {a-b x^3} \sqrt {a+b x^3}} \, dx\)

\(\Big \downarrow \) 808

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

\(\Big \downarrow \) 890

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

\(\Big \downarrow \) 832

\(\displaystyle \frac {\sqrt {a^2-b^2 x^6} \left (\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {a^2-b^2 x^6}}dx^2}{b^{2/3}}-\frac {\int \frac {\left (1-\sqrt {3}\right ) a^{2/3}-b^{2/3} x^2}{\sqrt {a^2-b^2 x^6}}dx^2}{b^{2/3}}\right )}{2 \sqrt {a-b \left (x^2\right )^{3/2}} \sqrt {a+b \left (x^2\right )^{3/2}}}\)

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 2416

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

Input:

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

Output:

(Sqrt[a^2 - b^2*x^6]*(-(((-2*Sqrt[a^2 - b^2*x^6])/(b^(2/3)*((1 + Sqrt[3])* 
a^(2/3) - b^(2/3)*x^2)) + (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(2/3)*(a^(2/3) - b^ 
(2/3)*x^2)*Sqrt[(a^(4/3) + a^(2/3)*b^(2/3)*x^2 + b^(4/3)*x^4)/((1 + Sqrt[3 
])*a^(2/3) - b^(2/3)*x^2)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(2/3) - b^( 
2/3)*x^2)/((1 + Sqrt[3])*a^(2/3) - b^(2/3)*x^2)], -7 - 4*Sqrt[3]])/(b^(2/3 
)*Sqrt[(a^(2/3)*(a^(2/3) - b^(2/3)*x^2))/((1 + Sqrt[3])*a^(2/3) - b^(2/3)* 
x^2)^2]*Sqrt[a^2 - b^2*x^6]))/b^(2/3)) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3] 
]*a^(2/3)*(a^(2/3) - b^(2/3)*x^2)*Sqrt[(a^(4/3) + a^(2/3)*b^(2/3)*x^2 + b^ 
(4/3)*x^4)/((1 + Sqrt[3])*a^(2/3) - b^(2/3)*x^2)^2]*EllipticF[ArcSin[((1 - 
 Sqrt[3])*a^(2/3) - b^(2/3)*x^2)/((1 + Sqrt[3])*a^(2/3) - b^(2/3)*x^2)], - 
7 - 4*Sqrt[3]])/(3^(1/4)*b^(4/3)*Sqrt[(a^(2/3)*(a^(2/3) - b^(2/3)*x^2))/(( 
1 + Sqrt[3])*a^(2/3) - b^(2/3)*x^2)^2]*Sqrt[a^2 - b^2*x^6])))/(2*Sqrt[a - 
b*(x^2)^(3/2)]*Sqrt[a + b*(x^2)^(3/2)])
 

Defintions of rubi rules used

rule 759

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] & 
& PosQ[a]
 

rule 808

Int[(x_)^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^( 
p_), x_Symbol] :> With[{k = GCD[m + 1, 2*n]}, Simp[1/k   Subst[Int[x^((m + 
1)/k - 1)*(a1 + b1*x^(n/k))^p*(a2 + b2*x^(n/k))^p, x], x, x^k], x] /; k != 
1] /; FreeQ[{a1, b1, a2, b2, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0 
] && IntegerQ[m]
 

rule 832

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] && PosQ[a]
 

rule 890

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_) 
^(n_))^(p_), x_Symbol] :> Simp[(a1 + b1*x^n)^FracPart[p]*((a2 + b2*x^n)^Fra 
cPart[p]/(a1*a2 + b1*b2*x^(2*n))^FracPart[p])   Int[(c*x)^m*(a1*a2 + b1*b2* 
x^(2*n))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + 
 a1*b2, 0] &&  !IntegerQ[p]
 

rule 2416

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 + 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] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \frac {x^{3}}{\sqrt {-b \,x^{3}+a}\, \sqrt {b \,x^{3}+a}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.06 \[ \int \frac {x^3}{\sqrt {a-b x^3} \sqrt {a+b x^3}} \, dx=\frac {\sqrt {-b^{2}} {\rm weierstrassZeta}\left (0, \frac {4 \, a^{2}}{b^{2}}, {\rm weierstrassPInverse}\left (0, \frac {4 \, a^{2}}{b^{2}}, x^{2}\right )\right )}{b^{2}} \] Input:

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

Output:

sqrt(-b^2)*weierstrassZeta(0, 4*a^2/b^2, weierstrassPInverse(0, 4*a^2/b^2, 
 x^2))/b^2
 

Sympy [A] (verification not implemented)

Time = 2.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.22 \[ \int \frac {x^3}{\sqrt {a-b x^3} \sqrt {a+b x^3}} \, dx=\frac {i \sqrt [3]{a} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{12}, \frac {7}{12}, 1 & \frac {1}{3}, \frac {1}{3}, \frac {5}{6} \\- \frac {1}{6}, \frac {1}{12}, \frac {1}{3}, \frac {7}{12}, \frac {5}{6} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{6}}} \right )}}{12 \pi ^{\frac {3}{2}} b^{\frac {4}{3}}} + \frac {\sqrt [3]{a} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {2}{3}, - \frac {5}{12}, - \frac {1}{6}, \frac {1}{12}, \frac {1}{3}, 1 & \\- \frac {5}{12}, \frac {1}{12} & - \frac {2}{3}, - \frac {1}{6}, - \frac {1}{6}, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{6}}} \right )} e^{- \frac {i \pi }{3}}}{12 \pi ^{\frac {3}{2}} b^{\frac {4}{3}}} \] Input:

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

Output:

I*a**(1/3)*meijerg(((1/12, 7/12, 1), (1/3, 1/3, 5/6)), ((-1/6, 1/12, 1/3, 
7/12, 5/6), (0,)), a**2/(b**2*x**6))/(12*pi**(3/2)*b**(4/3)) + a**(1/3)*me 
ijerg(((-2/3, -5/12, -1/6, 1/12, 1/3, 1), ()), ((-5/12, 1/12), (-2/3, -1/6 
, -1/6, 0)), a**2*exp_polar(-2*I*pi)/(b**2*x**6))*exp(-I*pi/3)/(12*pi**(3/ 
2)*b**(4/3))
 

Maxima [F]

\[ \int \frac {x^3}{\sqrt {a-b x^3} \sqrt {a+b x^3}} \, dx=\int { \frac {x^{3}}{\sqrt {b x^{3} + a} \sqrt {-b x^{3} + a}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {x^3}{\sqrt {a-b x^3} \sqrt {a+b x^3}} \, dx=\int { \frac {x^{3}}{\sqrt {b x^{3} + a} \sqrt {-b x^{3} + a}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\sqrt {a-b x^3} \sqrt {a+b x^3}} \, dx=\int \frac {x^3}{\sqrt {b\,x^3+a}\,\sqrt {a-b\,x^3}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^3}{\sqrt {a-b x^3} \sqrt {a+b x^3}} \, dx=\int \frac {\sqrt {b \,x^{3}+a}\, \sqrt {-b \,x^{3}+a}\, x^{3}}{-b^{2} x^{6}+a^{2}}d x \] Input:

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

Output:

int((sqrt(a + b*x**3)*sqrt(a - b*x**3)*x**3)/(a**2 - b**2*x**6),x)