3.1224 \(\int \frac {1}{x^{12} \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=71 \[ -\frac {32 b^2 \left (a-b x^4\right )^{3/4}}{231 a^3 x^3}-\frac {8 b \left (a-b x^4\right )^{3/4}}{77 a^2 x^7}-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}} \]

[Out]

-1/11*(-b*x^4+a)^(3/4)/a/x^11-8/77*b*(-b*x^4+a)^(3/4)/a^2/x^7-32/231*b^2*(-b*x^4+a)^(3/4)/a^3/x^3

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Rubi [A]  time = 0.02, 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 = {271, 264} \[ -\frac {32 b^2 \left (a-b x^4\right )^{3/4}}{231 a^3 x^3}-\frac {8 b \left (a-b x^4\right )^{3/4}}{77 a^2 x^7}-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^12*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(11*a*x^11) - (8*b*(a - b*x^4)^(3/4))/(77*a^2*x^7) - (32*b^2*(a - b*x^4)^(3/4))/(231*a^3*x^
3)

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]

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]

Rubi steps

\begin {align*} \int \frac {1}{x^{12} \sqrt [4]{a-b x^4}} \, dx &=-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}+\frac {(8 b) \int \frac {1}{x^8 \sqrt [4]{a-b x^4}} \, dx}{11 a}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}-\frac {8 b \left (a-b x^4\right )^{3/4}}{77 a^2 x^7}+\frac {\left (32 b^2\right ) \int \frac {1}{x^4 \sqrt [4]{a-b x^4}} \, dx}{77 a^2}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}-\frac {8 b \left (a-b x^4\right )^{3/4}}{77 a^2 x^7}-\frac {32 b^2 \left (a-b x^4\right )^{3/4}}{231 a^3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 43, normalized size = 0.61 \[ -\frac {\left (a-b x^4\right )^{3/4} \left (21 a^2+24 a b x^4+32 b^2 x^8\right )}{231 a^3 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^12*(a - b*x^4)^(1/4)),x]

[Out]

-1/231*((a - b*x^4)^(3/4)*(21*a^2 + 24*a*b*x^4 + 32*b^2*x^8))/(a^3*x^11)

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fricas [A]  time = 0.51, size = 39, normalized size = 0.55 \[ -\frac {{\left (32 \, b^{2} x^{8} + 24 \, a b x^{4} + 21 \, a^{2}\right )} {\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{231 \, a^{3} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(-b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

-1/231*(32*b^2*x^8 + 24*a*b*x^4 + 21*a^2)*(-b*x^4 + a)^(3/4)/(a^3*x^11)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(-b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^12), x)

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maple [A]  time = 0.00, size = 40, normalized size = 0.56 \[ -\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{8}+24 a b \,x^{4}+21 a^{2}\right )}{231 a^{3} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^12/(-b*x^4+a)^(1/4),x)

[Out]

-1/231*(-b*x^4+a)^(3/4)*(32*b^2*x^8+24*a*b*x^4+21*a^2)/a^3/x^11

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maxima [A]  time = 1.07, size = 55, normalized size = 0.77 \[ -\frac {\frac {77 \, {\left (-b x^{4} + a\right )}^{\frac {3}{4}} b^{2}}{x^{3}} + \frac {66 \, {\left (-b x^{4} + a\right )}^{\frac {7}{4}} b}{x^{7}} + \frac {21 \, {\left (-b x^{4} + a\right )}^{\frac {11}{4}}}{x^{11}}}{231 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(-b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

-1/231*(77*(-b*x^4 + a)^(3/4)*b^2/x^3 + 66*(-b*x^4 + a)^(7/4)*b/x^7 + 21*(-b*x^4 + a)^(11/4)/x^11)/a^3

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mupad [B]  time = 1.26, size = 59, normalized size = 0.83 \[ -\frac {21\,a^2\,{\left (a-b\,x^4\right )}^{3/4}+32\,b^2\,x^8\,{\left (a-b\,x^4\right )}^{3/4}+24\,a\,b\,x^4\,{\left (a-b\,x^4\right )}^{3/4}}{231\,a^3\,x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^12*(a - b*x^4)^(1/4)),x)

[Out]

-(21*a^2*(a - b*x^4)^(3/4) + 32*b^2*x^8*(a - b*x^4)^(3/4) + 24*a*b*x^4*(a - b*x^4)^(3/4))/(231*a^3*x^11)

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sympy [C]  time = 3.66, size = 1068, normalized size = 15.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**12/(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((-21*a**4*b**(19/4)*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-11/4)/(64*a**5*b**4*x**8*exp(I*pi/
4)*gamma(1/4) - 128*a**4*b**5*x**12*exp(I*pi/4)*gamma(1/4) + 64*a**3*b**6*x**16*exp(I*pi/4)*gamma(1/4)) + 18*a
**3*b**(23/4)*x**4*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-11/4)/(64*a**5*b**4*x**8*exp(I*pi/4)*gamma(1/
4) - 128*a**4*b**5*x**12*exp(I*pi/4)*gamma(1/4) + 64*a**3*b**6*x**16*exp(I*pi/4)*gamma(1/4)) - 5*a**2*b**(27/4
)*x**8*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-11/4)/(64*a**5*b**4*x**8*exp(I*pi/4)*gamma(1/4) - 128*a**
4*b**5*x**12*exp(I*pi/4)*gamma(1/4) + 64*a**3*b**6*x**16*exp(I*pi/4)*gamma(1/4)) + 40*a*b**(31/4)*x**12*(a/(b*
x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-11/4)/(64*a**5*b**4*x**8*exp(I*pi/4)*gamma(1/4) - 128*a**4*b**5*x**12*
exp(I*pi/4)*gamma(1/4) + 64*a**3*b**6*x**16*exp(I*pi/4)*gamma(1/4)) - 32*b**(35/4)*x**16*(a/(b*x**4) - 1)**(3/
4)*exp(-3*I*pi/4)*gamma(-11/4)/(64*a**5*b**4*x**8*exp(I*pi/4)*gamma(1/4) - 128*a**4*b**5*x**12*exp(I*pi/4)*gam
ma(1/4) + 64*a**3*b**6*x**16*exp(I*pi/4)*gamma(1/4)), Abs(a/(b*x**4)) > 1), (-21*a**4*b**(19/4)*(-a/(b*x**4) +
1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*exp(I*pi/4)*gamma(1/4) - 128*a**4*b**5*x**12*exp(I*pi/4)*gamma(1/4)
+ 64*a**3*b**6*x**16*exp(I*pi/4)*gamma(1/4)) + 18*a**3*b**(23/4)*x**4*(-a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(
64*a**5*b**4*x**8*exp(I*pi/4)*gamma(1/4) - 128*a**4*b**5*x**12*exp(I*pi/4)*gamma(1/4) + 64*a**3*b**6*x**16*exp
(I*pi/4)*gamma(1/4)) - 5*a**2*b**(27/4)*x**8*(-a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*exp(I*pi
/4)*gamma(1/4) - 128*a**4*b**5*x**12*exp(I*pi/4)*gamma(1/4) + 64*a**3*b**6*x**16*exp(I*pi/4)*gamma(1/4)) + 40*
a*b**(31/4)*x**12*(-a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*exp(I*pi/4)*gamma(1/4) - 128*a**4*b
**5*x**12*exp(I*pi/4)*gamma(1/4) + 64*a**3*b**6*x**16*exp(I*pi/4)*gamma(1/4)) - 32*b**(35/4)*x**16*(-a/(b*x**4
) + 1)**(3/4)*gamma(-11/4)/(64*a**5*b**4*x**8*exp(I*pi/4)*gamma(1/4) - 128*a**4*b**5*x**12*exp(I*pi/4)*gamma(1
/4) + 64*a**3*b**6*x**16*exp(I*pi/4)*gamma(1/4)), True))

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