\(\int \frac {x^8}{\sqrt {a+b x^4}} \, dx\) [821]

Optimal result

Integrand size = 15, antiderivative size = 130 \[ \int \frac {x^8}{\sqrt {a+b x^4}} \, dx=-\frac {5 a x \sqrt {a+b x^4}}{21 b^2}+\frac {x^5 \sqrt {a+b x^4}}{7 b}+\frac {5 a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{42 b^{9/4} \sqrt {a+b x^4}} \]

[Out]

-5/21*a*x*(b*x^4+a)^(1/2)/b^2+1/7*x^5*(b*x^4+a)^(1/2)/b+5/42*a^(7/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2
)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2
))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/b^(9/4)/(b*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {327, 226} \[ \int \frac {x^8}{\sqrt {a+b x^4}} \, dx=\frac {5 a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{42 b^{9/4} \sqrt {a+b x^4}}-\frac {5 a x \sqrt {a+b x^4}}{21 b^2}+\frac {x^5 \sqrt {a+b x^4}}{7 b} \]

[In]

Int[x^8/Sqrt[a + b*x^4],x]

[Out]

(-5*a*x*Sqrt[a + b*x^4])/(21*b^2) + (x^5*Sqrt[a + b*x^4])/(7*b) + (5*a^(7/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a +
b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(42*b^(9/4)*Sqrt[a + b*x^4])

Rule 226

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

Rubi steps \begin{align*} \text {integral}& = \frac {x^5 \sqrt {a+b x^4}}{7 b}-\frac {(5 a) \int \frac {x^4}{\sqrt {a+b x^4}} \, dx}{7 b} \\ & = -\frac {5 a x \sqrt {a+b x^4}}{21 b^2}+\frac {x^5 \sqrt {a+b x^4}}{7 b}+\frac {\left (5 a^2\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{21 b^2} \\ & = -\frac {5 a x \sqrt {a+b x^4}}{21 b^2}+\frac {x^5 \sqrt {a+b x^4}}{7 b}+\frac {5 a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{42 b^{9/4} \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 = 79, normalized size of antiderivative = 0.61 \[ \int \frac {x^8}{\sqrt {a+b x^4}} \, dx=\frac {-5 a^2 x-2 a b x^5+3 b^2 x^9+5 a^2 x \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )}{21 b^2 \sqrt {a+b x^4}} \]

[In]

Integrate[x^8/Sqrt[a + b*x^4],x]

[Out]

(-5*a^2*x - 2*a*b*x^5 + 3*b^2*x^9 + 5*a^2*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)]
)/(21*b^2*Sqrt[a + b*x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.38 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79

method	result	size
risch	\(-\frac {x \left (-3 b \,x^{4}+5 a \right ) \sqrt {b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(103\)
default	\(\frac {x^{5} \sqrt {b \,x^{4}+a}}{7 b}-\frac {5 a x \sqrt {b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(111\)
elliptic	\(\frac {x^{5} \sqrt {b \,x^{4}+a}}{7 b}-\frac {5 a x \sqrt {b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(111\)

[In]

int(x^8/(b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/21*x*(-3*b*x^4+5*a)/b^2*(b*x^4+a)^(1/2)+5/21*a^2/b^2/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1
/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.43 \[ \int \frac {x^8}{\sqrt {a+b x^4}} \, dx=\frac {5 \, a \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (3 \, b x^{5} - 5 \, a x\right )} \sqrt {b x^{4} + a}}{21 \, b^{2}} \]

[In]

integrate(x^8/(b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

1/21*(5*a*sqrt(b)*(-a/b)^(3/4)*elliptic_f(arcsin((-a/b)^(1/4)/x), -1) + (3*b*x^5 - 5*a*x)*sqrt(b*x^4 + a))/b^2

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.28 \[ \int \frac {x^8}{\sqrt {a+b x^4}} \, dx=\frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \]

[In]

integrate(x**8/(b*x**4+a)**(1/2),x)

[Out]

x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(13/4))

Maxima [F]

\[ \int \frac {x^8}{\sqrt {a+b x^4}} \, dx=\int { \frac {x^{8}}{\sqrt {b x^{4} + a}} \,d x } \]

[In]

integrate(x^8/(b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^8/sqrt(b*x^4 + a), x)

Giac [F]

\[ \int \frac {x^8}{\sqrt {a+b x^4}} \, dx=\int { \frac {x^{8}}{\sqrt {b x^{4} + a}} \,d x } \]

[In]

integrate(x^8/(b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

integrate(x^8/sqrt(b*x^4 + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^4}} \, dx=\int \frac {x^8}{\sqrt {b\,x^4+a}} \,d x \]

[In]

int(x^8/(a + b*x^4)^(1/2),x)

[Out]

int(x^8/(a + b*x^4)^(1/2), x)