3.9.20 \(\int \frac {1}{x^{11} \sqrt {a+b x^4}} \, dx\) [820]

Optimal. Leaf size=68 \[ -\frac {\sqrt {a+b x^4}}{10 a x^{10}}+\frac {2 b \sqrt {a+b x^4}}{15 a^2 x^6}-\frac {4 b^2 \sqrt {a+b x^4}}{15 a^3 x^2} \]

[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 = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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 = 0.18, size = 42, normalized size = 0.62 \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 = 0.14, size = 39, normalized size = 0.57

method	result	size
gosper	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (8 b^{2} x^{8}-4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\)	\(39\)
default	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (8 b^{2} x^{8}-4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\)	\(39\)
trager	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (8 b^{2} x^{8}-4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\)	\(39\)
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (8 b^{2} x^{8}-4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\)	\(39\)
elliptic	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (8 b^{2} x^{8}-4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\)	\(39\)

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 = 0.30, size = 52, normalized size = 0.76 \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 = 0.37, size = 38, normalized size = 0.56 \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (61) = 122\).
time = 0.78, size = 298, normalized size = 4.38 \begin {gather*} - \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}} \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]

-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**(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)

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Giac [A]
time = 2.58, size = 83, normalized size = 1.22 \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 )} b^{\frac {5}{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)*b^(5/2)/((sqrt(b)*x^
2 - sqrt(b*x^4 + a))^2 - a)^5

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Mupad [B]
time = 1.28, size = 38, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {b\,x^4+a}\,\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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