3.9.41 \(\int \frac {x^8}{\sqrt {a-b x^4}} \, dx\) [841]

Optimal. Leaf size=100 \[ -\frac {5 a x \sqrt {a-b x^4}}{21 b^2}-\frac {x^5 \sqrt {a-b x^4}}{7 b}+\frac {5 a^{9/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 b^{9/4} \sqrt {a-b x^4}} \]

[Out]

-5/21*a*x*(-b*x^4+a)^(1/2)/b^2-1/7*x^5*(-b*x^4+a)^(1/2)/b+5/21*a^(9/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*(1-b*x^4
/a)^(1/2)/b^(9/4)/(-b*x^4+a)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {327, 230, 227} \begin {gather*} \frac {5 a^{9/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 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} \end {gather*}

Antiderivative was successfully verified.

[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^(9/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[Arc
Sin[(b^(1/4)*x)/a^(1/4)], -1])/(21*b^(9/4)*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 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*} \int \frac {x^8}{\sqrt {a-b x^4}} \, dx &=-\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 {\left (5 a^2 \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{21 b^2 \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}+\frac {5 a^{9/4} \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 )}{21 b^{9/4} \sqrt {a-b x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 80, normalized size = 0.80 \begin {gather*} \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}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {b x^4}{a}\right )}{21 b^2 \sqrt {a-b x^4}} \end {gather*}

Antiderivative was successfully verified.

[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])

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Maple [A]
time = 0.16, size = 107, normalized size = 1.07

method	result	size
risch	\(-\frac {x \left (3 b \,x^{4}+5 a \right ) \sqrt {-b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(98\)
default	\(-\frac {x^{5} \sqrt {-b \,x^{4}+a}}{7 b}-\frac {5 a x \sqrt {-b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(107\)
elliptic	\(-\frac {x^{5} \sqrt {-b \,x^{4}+a}}{7 b}-\frac {5 a x \sqrt {-b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*x^5*(-b*x^4+a)^(1/2)/b-5/21*a*x*(-b*x^4+a)^(1/2)/b^2+5/21*a^2/b^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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^8/sqrt(-b*x^4 + a), x)

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Fricas [A]
time = 0.08, size = 58, normalized size = 0.58 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]
time = 0.49, size = 39, normalized size = 0.39 \begin {gather*} \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^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(2*I*pi)/a)/(4*sqrt(a)*gamma(13/4))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^8/sqrt(-b*x^4 + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^8}{\sqrt {a-b\,x^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(a - b*x^4)^(1/2),x)

[Out]

int(x^8/(a - b*x^4)^(1/2), x)

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