3.9.45 \(\int \frac {1}{x^8 \sqrt {a-b x^4}} \, dx\) [845]

Optimal. Leaf size=102 \[ -\frac {\sqrt {a-b x^4}}{7 a x^7}-\frac {5 b \sqrt {a-b x^4}}{21 a^2 x^3}+\frac {5 b^{7/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 a^{7/4} \sqrt {a-b x^4}} \]

[Out]

-1/7*(-b*x^4+a)^(1/2)/a/x^7-5/21*b*(-b*x^4+a)^(1/2)/a^2/x^3+5/21*b^(7/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*(1-b*x
^4/a)^(1/2)/a^(7/4)/(-b*x^4+a)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 102, 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 = {331, 230, 227} \begin {gather*} \frac {5 b^{7/4} \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 a^{7/4} \sqrt {a-b x^4}}-\frac {5 b \sqrt {a-b x^4}}{21 a^2 x^3}-\frac {\sqrt {a-b x^4}}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*Sqrt[a - b*x^4]),x]

[Out]

-1/7*Sqrt[a - b*x^4]/(a*x^7) - (5*b*Sqrt[a - b*x^4])/(21*a^2*x^3) + (5*b^(7/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[A
rcSin[(b^(1/4)*x)/a^(1/4)], -1])/(21*a^(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 331

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

Rubi steps

\begin {align*} \int \frac {1}{x^8 \sqrt {a-b x^4}} \, dx &=-\frac {\sqrt {a-b x^4}}{7 a x^7}+\frac {(5 b) \int \frac {1}{x^4 \sqrt {a-b x^4}} \, dx}{7 a}\\ &=-\frac {\sqrt {a-b x^4}}{7 a x^7}-\frac {5 b \sqrt {a-b x^4}}{21 a^2 x^3}+\frac {\left (5 b^2\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{21 a^2}\\ &=-\frac {\sqrt {a-b x^4}}{7 a x^7}-\frac {5 b \sqrt {a-b x^4}}{21 a^2 x^3}+\frac {\left (5 b^2 \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{21 a^2 \sqrt {a-b x^4}}\\ &=-\frac {\sqrt {a-b x^4}}{7 a x^7}-\frac {5 b \sqrt {a-b x^4}}{21 a^2 x^3}+\frac {5 b^{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 )}{21 a^{7/4} \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 = 10.01, size = 52, normalized size = 0.51 \begin {gather*} -\frac {\sqrt {1-\frac {b x^4}{a}} \, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};\frac {b x^4}{a}\right )}{7 x^7 \sqrt {a-b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*Sqrt[a - b*x^4]),x]

[Out]

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

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Maple [A]
time = 0.15, size = 109, normalized size = 1.07

method	result	size
risch	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (5 b \,x^{4}+3 a \right )}{21 a^{2} x^{7}}+\frac {5 b^{2} \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 a^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(100\)
default	\(-\frac {\sqrt {-b \,x^{4}+a}}{7 a \,x^{7}}-\frac {5 b \sqrt {-b \,x^{4}+a}}{21 a^{2} x^{3}}+\frac {5 b^{2} \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 a^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(109\)
elliptic	\(-\frac {\sqrt {-b \,x^{4}+a}}{7 a \,x^{7}}-\frac {5 b \sqrt {-b \,x^{4}+a}}{21 a^{2} x^{3}}+\frac {5 b^{2} \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 a^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(-b*x^4+a)^(1/2)/a/x^7-5/21*b*(-b*x^4+a)^(1/2)/a^2/x^3+5/21*b^2/a^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(1/x^8/(-b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-b*x^4 + a)*x^8), x)

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Fricas [A]
time = 0.08, size = 59, normalized size = 0.58 \begin {gather*} \frac {5 \, \sqrt {a} b x^{7} \left (\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (5 \, b x^{4} + 3 \, a\right )} \sqrt {-b x^{4} + a}}{21 \, a^{2} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(5*sqrt(a)*b*x^7*(b/a)^(3/4)*elliptic_f(arcsin(x*(b/a)^(1/4)), -1) - (5*b*x^4 + 3*a)*sqrt(-b*x^4 + a))/(a
^2*x^7)

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Sympy [A]
time = 0.57, size = 46, normalized size = 0.45 \begin {gather*} \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} x^{7} \Gamma \left (- \frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(1/(sqrt(-b*x^4 + a)*x^8), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(a - b*x^4)^(1/2)),x)

[Out]

int(1/(x^8*(a - b*x^4)^(1/2)), x)

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