3.820 \(\int \frac{1}{x^{11} \sqrt{a+b x^4}} \, dx\)

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

[Out]

-Sqrt[a + b*x^4]/(10*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)

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Rubi [A]  time = 0.0190767, antiderivative size = 68, normalized size of antiderivative = 1., 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 = {271, 264} \[ -\frac{4 b^2 \sqrt{a+b x^4}}{15 a^3 x^2}+\frac{2 b \sqrt{a+b x^4}}{15 a^2 x^6}-\frac{\sqrt{a+b x^4}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^11*Sqrt[a + b*x^4]),x]

[Out]

-Sqrt[a + b*x^4]/(10*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 271

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]

Rule 264

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]

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*}

Mathematica [A]  time = 0.0091348, size = 42, normalized size = 0.62 \[ -\frac{\sqrt{a+b x^4} \left (3 a^2-4 a b x^4+8 b^2 x^8\right )}{30 a^3 x^{10}} \]

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 = 0.005, size = 39, normalized size = 0.6 \begin{align*} -{\frac{8\,{b}^{2}{x}^{8}-4\,ab{x}^{4}+3\,{a}^{2}}{30\,{x}^{10}{a}^{3}}\sqrt{b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^11/(b*x^4+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.993885, size = 70, normalized size = 1.03 \begin{align*} -\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{align*}

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 = 1.53857, size = 89, normalized size = 1.31 \begin{align*} -\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{align*}

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]  time = 2.62151, size = 298, normalized size = 4.38 \begin{align*} - \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{align*}

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 = 1.14136, size = 58, normalized size = 0.85 \begin{align*} -\frac{3 \,{\left (b + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} - 10 \,{\left (b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} b + 15 \, \sqrt{b + \frac{a}{x^{4}}} b^{2}}{30 \, a^{3}} \end{align*}

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]

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