3.9.40 \(\int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx\) [840]

Optimal. Leaf size=71 \[ -\frac {\sqrt {a-b x^4}}{10 a x^{10}}-\frac {2 b \sqrt {a-b x^4}}{15 a^2 x^6}-\frac {4 b^2 \sqrt {a-b x^4}}{15 a^3 x^2} \]

[Out]

-1/10*(-b*x^4+a)^(1/2)/a/x^10-2/15*b*(-b*x^4+a)^(1/2)/a^2/x^6-4/15*b^2*(-b*x^4+a)^(1/2)/a^3/x^2

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Rubi [A]
time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {277, 270} \begin {gather*} -\frac {4 b^2 \sqrt {a-b x^4}}{15 a^3 x^2}-\frac {2 b \sqrt {a-b x^4}}{15 a^2 x^6}-\frac {\sqrt {a-b x^4}}{10 a x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^11*Sqrt[a - b*x^4]),x]

[Out]

-1/10*Sqrt[a - b*x^4]/(a*x^10) - (2*b*Sqrt[a - b*x^4])/(15*a^2*x^6) - (4*b^2*Sqrt[a - b*x^4])/(15*a^3*x^2)

Rule 270

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] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

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

Rubi steps

\begin {align*} \int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx &=-\frac {\sqrt {a-b x^4}}{10 a x^{10}}+\frac {(4 b) \int \frac {1}{x^7 \sqrt {a-b x^4}} \, dx}{5 a}\\ &=-\frac {\sqrt {a-b x^4}}{10 a x^{10}}-\frac {2 b \sqrt {a-b x^4}}{15 a^2 x^6}+\frac {\left (8 b^2\right ) \int \frac {1}{x^3 \sqrt {a-b x^4}} \, dx}{15 a^2}\\ &=-\frac {\sqrt {a-b x^4}}{10 a x^{10}}-\frac {2 b \sqrt {a-b x^4}}{15 a^2 x^6}-\frac {4 b^2 \sqrt {a-b x^4}}{15 a^3 x^2}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 43, normalized size = 0.61 \begin {gather*} \frac {\sqrt {a-b x^4} \left (-3 a^2-4 a b x^4-8 b^2 x^8\right )}{30 a^3 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^11*Sqrt[a - b*x^4]),x]

[Out]

(Sqrt[a - b*x^4]*(-3*a^2 - 4*a*b*x^4 - 8*b^2*x^8))/(30*a^3*x^10)

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Maple [A]
time = 0.15, size = 40, normalized size = 0.56

method	result	size
gosper	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\)	\(40\)
default	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\)	\(40\)
trager	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\)	\(40\)
risch	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\)	\(40\)
elliptic	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^11/(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/30*(-b*x^4+a)^(1/2)*(8*b^2*x^8+4*a*b*x^4+3*a^2)/a^3/x^10

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Maxima [A]
time = 0.29, size = 55, normalized size = 0.77 \begin {gather*} -\frac {\frac {15 \, \sqrt {-b x^{4} + a} b^{2}}{x^{2}} + \frac {10 \, {\left (-b x^{4} + a\right )}^{\frac {3}{2}} b}{x^{6}} + \frac {3 \, {\left (-b x^{4} + a\right )}^{\frac {5}{2}}}{x^{10}}}{30 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(-b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

-1/30*(15*sqrt(-b*x^4 + a)*b^2/x^2 + 10*(-b*x^4 + a)^(3/2)*b/x^6 + 3*(-b*x^4 + a)^(5/2)/x^10)/a^3

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Fricas [A]
time = 0.36, size = 39, normalized size = 0.55 \begin {gather*} -\frac {{\left (8 \, b^{2} x^{8} + 4 \, a b x^{4} + 3 \, a^{2}\right )} \sqrt {-b x^{4} + a}}{30 \, a^{3} x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(-b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

-1/30*(8*b^2*x^8 + 4*a*b*x^4 + 3*a^2)*sqrt(-b*x^4 + a)/(a^3*x^10)

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Sympy [C] Result contains complex when optimal does not.
time = 1.38, size = 609, normalized size = 8.58 \begin {gather*} \begin {cases} - \frac {3 a^{4} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac {2 a^{3} b^{\frac {11}{2}} x^{4} \sqrt {\frac {a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac {3 a^{2} b^{\frac {13}{2}} x^{8} \sqrt {\frac {a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac {12 a b^{\frac {15}{2}} x^{12} \sqrt {\frac {a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac {8 b^{\frac {17}{2}} x^{16} \sqrt {\frac {a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {3 i a^{4} b^{\frac {9}{2}} \sqrt {- \frac {a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac {2 i a^{3} b^{\frac {11}{2}} x^{4} \sqrt {- \frac {a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac {3 i a^{2} b^{\frac {13}{2}} x^{8} \sqrt {- \frac {a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac {12 i a b^{\frac {15}{2}} x^{12} \sqrt {- \frac {a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac {8 i b^{\frac {17}{2}} x^{16} \sqrt {- \frac {a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**11/(-b*x**4+a)**(1/2),x)

[Out]

Piecewise((-3*a**4*b**(9/2)*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16)
+ 2*a**3*b**(11/2)*x**4*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16) -
3*a**2*b**(13/2)*x**8*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16) + 12*
a*b**(15/2)*x**12*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16) - 8*b**(1
7/2)*x**16*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16), Abs(a/(b*x**4))
> 1), (-3*I*a**4*b**(9/2)*sqrt(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16)
+ 2*I*a**3*b**(11/2)*x**4*sqrt(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16)
- 3*I*a**2*b**(13/2)*x**8*sqrt(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16)
+ 12*I*a*b**(15/2)*x**12*sqrt(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16)
- 8*I*b**(17/2)*x**16*sqrt(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16), Tru
e))

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Giac [A]
time = 1.76, size = 97, normalized size = 1.37 \begin {gather*} \frac {8 \, {\left (10 \, {\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{4} - 5 \, {\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{2} a + a^{2}\right )} \sqrt {-b} b^{2}}{15 \, {\left ({\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{2} - a\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(-b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

8/15*(10*(sqrt(-b)*x^2 - sqrt(-b*x^4 + a))^4 - 5*(sqrt(-b)*x^2 - sqrt(-b*x^4 + a))^2*a + a^2)*sqrt(-b)*b^2/((s
qrt(-b)*x^2 - sqrt(-b*x^4 + a))^2 - a)^5

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Mupad [B]
time = 1.31, size = 39, normalized size = 0.55 \begin {gather*} -\frac {\sqrt {a-b\,x^4}\,\left (3\,a^2+4\,a\,b\,x^4+8\,b^2\,x^8\right )}{30\,a^3\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^11*(a - b*x^4)^(1/2)),x)

[Out]

-((a - b*x^4)^(1/2)*(3*a^2 + 8*b^2*x^8 + 4*a*b*x^4))/(30*a^3*x^10)

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