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]

-(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)

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Rubi [A]  time = 0.018884, antiderivative size = 71, normalized size of antiderivative = 1., 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 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^{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*}

Mathematica [A]  time = 0.0154154, 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]

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

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Maple [A]  time = 0.005, size = 40, normalized size = 0.6 \begin{align*} -{\frac{32\,{b}^{2}{x}^{8}+24\,ab{x}^{4}+21\,{a}^{2}}{231\,{x}^{11}{a}^{3}} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \end{align*}

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)/x^11/a^3

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Maxima [A]  time = 1.00069, size = 74, normalized size = 1.04 \begin{align*} -\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}} \end{align*}

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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Fricas [A]  time = 1.7516, size = 99, normalized size = 1.39 \begin{align*} -\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}} \end{align*}

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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Sympy [C]  time = 3.89383, size = 1071, normalized size = 15.08 \begin{align*} \text{result too large to display} \end{align*}

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)/(Abs(b)*Abs(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)*gamm
a(-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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{12}}\,{d x} \end{align*}

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)