\(\int \frac {x^6}{\sqrt {a-b x^4}} \, dx\) [847]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {327, 313, 230, 227, 1214, 1213, 435} \[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=-\frac {3 a^{7/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{5 b^{7/4} \sqrt {a-b x^4}}+\frac {3 a^{7/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^4}}-\frac {x^3 \sqrt {a-b x^4}}{5 b} \]

[In]

Int[x^6/Sqrt[a - b*x^4],x]

[Out]

-1/5*(x^3*Sqrt[a - b*x^4])/b + (3*a^(7/4)*Sqrt[1 - (b*x^4)/a]*EllipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(5*b
^(7/4)*Sqrt[a - b*x^4]) - (3*a^(7/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(5*b^(7/4
)*Sqrt[a - b*x^4])

Rule 227

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

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[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 313

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

Rule 327

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] - Dist[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 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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 1213

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

Rule 1214

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

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {a-b x^4}}{5 b}+\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}-\frac {\left (3 a^{3/2}\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{5 b^{3/2}}+\frac {\left (3 a^{3/2}\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a-b x^4}} \, dx}{5 b^{3/2}} \\ & = -\frac {x^3 \sqrt {a-b x^4}}{5 b}-\frac {\left (3 a^{3/2} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{5 b^{3/2} \sqrt {a-b x^4}}+\frac {\left (3 a^{3/2} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{5 b^{3/2} \sqrt {a-b x^4}} \\ & = -\frac {x^3 \sqrt {a-b x^4}}{5 b}-\frac {3 a^{7/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^4}}+\frac {\left (3 a^{3/2} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}} \, dx}{5 b^{3/2} \sqrt {a-b x^4}} \\ & = -\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 .\sin ^{-1}\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}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^4}} \\ \end{align*}

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 \[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=\frac {x^3 \left (-a+b x^4+a \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^4}{a}\right )\right )}{5 b \sqrt {a-b x^4}} \]

[In]

Integrate[x^6/Sqrt[a - b*x^4],x]

[Out]

(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])

Maple [A] (verified)

Time = 4.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.79

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

[In]

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

[Out]

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

none

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.66 \[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=-\frac {3 \, a \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 3 \, a \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (b x^{4} + 3 \, a\right )} \sqrt {-b x^{4} + a}}{5 \, b^{2} x} \]

[In]

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

[Out]

-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 = 0.50 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.29 \[ \int \frac {x^6}{\sqrt {a-b x^4}} \, dx=\frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

integrate(x^6/sqrt(-b*x^4 + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^6/sqrt(-b*x^4 + a), x)

Mupad [F(-1)]

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

[In]

int(x^6/(a - b*x^4)^(1/2),x)

[Out]

int(x^6/(a - b*x^4)^(1/2), x)