\(\int \frac {x^6}{\sqrt {a-b x^4}} \, dx\) [220]

Optimal result

Integrand size = 16, antiderivative size = 135 \[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=-\frac {x^3 \sqrt {a-b x^4}}{5 b}+\frac {3 a^{7/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^4}}-\frac {3 a^{7/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{5 b^{7/4} \sqrt {a-b x^4}} \] Output:

-1/5*x^3*(-b*x^4+a)^(1/2)/b+3/5*a^(7/4)*(1-b*x^4/a)^(1/2)*EllipticE(b^(1/4 
)*x/a^(1/4),I)/b^(7/4)/(-b*x^4+a)^(1/2)-3/5*a^(7/4)*(1-b*x^4/a)^(1/2)*Elli 
pticF(b^(1/4)*x/a^(1/4),I)/b^(7/4)/(-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 = 10.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.49 \[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=\frac {x^3 \left (-a+b x^4+a \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^4}{a}\right )\right )}{5 b \sqrt {a-b x^4}} \] Input:

Integrate[x^6/Sqrt[a - b*x^4],x]
 

Output:

(x^3*(-a + b*x^4 + a*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, 
(b*x^4)/a]))/(5*b*Sqrt[a - b*x^4])
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {843, 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.

\(\displaystyle \int \frac {x^6}{\sqrt {a-b x^4}} \, dx\)

\(\Big \downarrow \) 843

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

Int[x^6/Sqrt[a - b*x^4],x]
 

Output:

-1/5*(x^3*Sqrt[a - b*x^4])/b + (3*a*((a^(3/4)*Sqrt[1 - (b*x^4)/a]*Elliptic 
E[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b*x^4]) - (a^(3/4)*S 
qrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sq 
rt[a - b*x^4])))/(5*b)
 

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 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 843

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

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 = 0.77 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.79

method	result	size
default	\(-\frac {x^{3} \sqrt {-b \,x^{4}+a}}{5 b}-\frac {3 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(107\)
risch	\(-\frac {x^{3} \sqrt {-b \,x^{4}+a}}{5 b}-\frac {3 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(107\)
elliptic	\(-\frac {x^{3} \sqrt {-b \,x^{4}+a}}{5 b}-\frac {3 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(107\)

Input:

int(x^6/(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/5*x^3*(-b*x^4+a)^(1/2)/b-3/5*a^(3/2)/b^(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)*(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 = 89, normalized size of antiderivative = 0.66 \[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=-\frac {3 \, a \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 3 \, a \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (b x^{4} + 3 \, a\right )} \sqrt {-b x^{4} + a}}{5 \, b^{2} x} \] Input:

integrate(x^6/(-b*x^4+a)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.29 \[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=\frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} \] Input:

integrate(x**6/(-b*x**4+a)**(1/2),x)
 

Output:

x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), b*x**4*exp_polar(2*I*pi)/a)/(4* 
sqrt(a)*gamma(11/4))
 

Maxima [F]

\[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=\int { \frac {x^{6}}{\sqrt {-b x^{4} + a}} \,d x } \] Input:

integrate(x^6/(-b*x^4+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^6/sqrt(-b*x^4 + a), x)
 

Giac [F]

\[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=\int { \frac {x^{6}}{\sqrt {-b x^{4} + a}} \,d x } \] Input:

integrate(x^6/(-b*x^4+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^6/sqrt(-b*x^4 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=\int \frac {x^6}{\sqrt {a-b\,x^4}} \,d x \] Input:

int(x^6/(a - b*x^4)^(1/2),x)
 

Output:

int(x^6/(a - b*x^4)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=\frac {-\sqrt {-b \,x^{4}+a}\, x^{3}+3 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{-b \,x^{4}+a}d x \right ) a}{5 b} \] Input:

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

Output:

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