\(\int x^2 \sqrt {a-b x^4} \, dx\) [180]

Optimal result

Integrand size = 16, antiderivative size = 132 \[ \int x^2 \sqrt {a-b x^4} \, dx=\frac {1}{5} x^3 \sqrt {a-b x^4}+\frac {2 a^{7/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{3/4} \sqrt {a-b x^4}}-\frac {2 a^{7/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{5 b^{3/4} \sqrt {a-b x^4}} \] Output:

1/5*x^3*(-b*x^4+a)^(1/2)+2/5*a^(7/4)*(1-b*x^4/a)^(1/2)*EllipticE(b^(1/4)*x 
/a^(1/4),I)/b^(3/4)/(-b*x^4+a)^(1/2)-2/5*a^(7/4)*(1-b*x^4/a)^(1/2)*Ellipti 
cF(b^(1/4)*x/a^(1/4),I)/b^(3/4)/(-b*x^4+a)^(1/2)
 

Mathematica [C] (verified)

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

Time = 4.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.39 \[ \int x^2 \sqrt {a-b x^4} \, dx=\frac {x^3 \sqrt {a-b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^4}{a}\right )}{3 \sqrt {1-\frac {b x^4}{a}}} \] Input:

Integrate[x^2*Sqrt[a - b*x^4],x]
 

Output:

(x^3*Sqrt[a - b*x^4]*Hypergeometric2F1[-1/2, 3/4, 7/4, (b*x^4)/a])/(3*Sqrt 
[1 - (b*x^4)/a])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {811, 836, 27, 765, 762, 1390, 1389, 327}

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.

\(\displaystyle \int x^2 \sqrt {a-b x^4} \, dx\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {2}{5} a \int \frac {x^2}{\sqrt {a-b x^4}}dx+\frac {1}{5} x^3 \sqrt {a-b x^4}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {2}{5} a \left (\frac {\sqrt {a} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )+\frac {1}{5} x^3 \sqrt {a-b x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{5} a \left (\frac {\int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )+\frac {1}{5} x^3 \sqrt {a-b x^4}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {2}{5} a \left (\frac {\int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}\right )+\frac {1}{5} x^3 \sqrt {a-b x^4}\)

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

Int[x^2*Sqrt[a - b*x^4],x]
 

Output:

(x^3*Sqrt[a - b*x^4])/5 + (2*a*((a^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticE[Arc 
Sin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b*x^4]) - (a^(3/4)*Sqrt[1 
 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a 
- b*x^4])))/5
 

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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 811

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

rule 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {x^{3} \sqrt {-b \,x^{4}+a}}{5}-\frac {2 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\)	\(104\)
risch	\(\frac {x^{3} \sqrt {-b \,x^{4}+a}}{5}-\frac {2 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\)	\(104\)
elliptic	\(\frac {x^{3} \sqrt {-b \,x^{4}+a}}{5}-\frac {2 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\)	\(104\)

Input:

int(x^2*(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/5*x^3*(-b*x^4+a)^(1/2)-2/5*a^(3/2)/(1/a^(1/2)*b^(1/2))^(1/2)*(1-b^(1/2)* 
x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4+a)^(1/2)/b^(1/2)* 
(EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(1/a^(1/2)*b^(1/2))^ 
(1/2),I))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.68 \[ \int x^2 \sqrt {a-b x^4} \, dx=-\frac {2 \, a \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 2 \, a \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (b x^{4} - 2 \, a\right )} \sqrt {-b x^{4} + a}}{5 \, b x} \] Input:

integrate(x^2*(-b*x^4+a)^(1/2),x, algorithm="fricas")
 

Output:

-1/5*(2*a*sqrt(-b)*x*(a/b)^(3/4)*elliptic_e(arcsin((a/b)^(1/4)/x), -1) - 2 
*a*sqrt(-b)*x*(a/b)^(3/4)*elliptic_f(arcsin((a/b)^(1/4)/x), -1) - (b*x^4 - 
 2*a)*sqrt(-b*x^4 + a))/(b*x)
 

Sympy [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.31 \[ \int x^2 \sqrt {a-b x^4} \, dx=\frac {\sqrt {a} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] Input:

integrate(x**2*(-b*x**4+a)**(1/2),x)
 

Output:

sqrt(a)*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b*x**4*exp_polar(2*I*pi 
)/a)/(4*gamma(7/4))
 

Maxima [F]

\[ \int x^2 \sqrt {a-b x^4} \, dx=\int { \sqrt {-b x^{4} + a} x^{2} \,d x } \] Input:

integrate(x^2*(-b*x^4+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(-b*x^4 + a)*x^2, x)
 

Giac [F]

\[ \int x^2 \sqrt {a-b x^4} \, dx=\int { \sqrt {-b x^{4} + a} x^{2} \,d x } \] Input:

integrate(x^2*(-b*x^4+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-b*x^4 + a)*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \sqrt {a-b x^4} \, dx=\int x^2\,\sqrt {a-b\,x^4} \,d x \] Input:

int(x^2*(a - b*x^4)^(1/2),x)
 

Output:

int(x^2*(a - b*x^4)^(1/2), x)
 

Reduce [F]

\[ \int x^2 \sqrt {a-b x^4} \, dx=\frac {\sqrt {-b \,x^{4}+a}\, x^{3}}{5}+\frac {2 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{-b \,x^{4}+a}d x \right ) a}{5} \] Input:

int(x^2*(-b*x^4+a)^(1/2),x)
 

Output:

(sqrt(a - b*x**4)*x**3 + 2*int((sqrt(a - b*x**4)*x**2)/(a - b*x**4),x)*a)/ 
5