3.1225 \(\int \frac{1}{x^{16} \sqrt [4]{a-b x^4}} \, dx\)
Optimal. Leaf size=96 \[ -\frac{128 b^3 \left (a-b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac{32 b^2 \left (a-b x^4\right )^{3/4}}{385 a^3 x^7}-\frac{4 b \left (a-b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{\left (a-b x^4\right )^{3/4}}{15 a x^{15}} \]
[Out]
-(a - b*x^4)^(3/4)/(15*a*x^15) - (4*b*(a - b*x^4)^(3/4))/(55*a^2*x^11) - (32*b^2*(a - b*x^4)^(3/4))/(385*a^3*x
^7) - (128*b^3*(a - b*x^4)^(3/4))/(1155*a^4*x^3)
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Rubi [A] time = 0.0283588, antiderivative size = 96, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{271, 264} \[ -\frac{128 b^3 \left (a-b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac{32 b^2 \left (a-b x^4\right )^{3/4}}{385 a^3 x^7}-\frac{4 b \left (a-b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{\left (a-b x^4\right )^{3/4}}{15 a x^{15}} \]
Antiderivative was successfully verified.
[In]
Int[1/(x^16*(a - b*x^4)^(1/4)),x]
[Out]
-(a - b*x^4)^(3/4)/(15*a*x^15) - (4*b*(a - b*x^4)^(3/4))/(55*a^2*x^11) - (32*b^2*(a - b*x^4)^(3/4))/(385*a^3*x
^7) - (128*b^3*(a - b*x^4)^(3/4))/(1155*a^4*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^{16} \sqrt [4]{a-b x^4}} \, dx &=-\frac{\left (a-b x^4\right )^{3/4}}{15 a x^{15}}+\frac{(4 b) \int \frac{1}{x^{12} \sqrt [4]{a-b x^4}} \, dx}{5 a}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{15 a x^{15}}-\frac{4 b \left (a-b x^4\right )^{3/4}}{55 a^2 x^{11}}+\frac{\left (32 b^2\right ) \int \frac{1}{x^8 \sqrt [4]{a-b x^4}} \, dx}{55 a^2}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{15 a x^{15}}-\frac{4 b \left (a-b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{32 b^2 \left (a-b x^4\right )^{3/4}}{385 a^3 x^7}+\frac{\left (128 b^3\right ) \int \frac{1}{x^4 \sqrt [4]{a-b x^4}} \, dx}{385 a^3}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{15 a x^{15}}-\frac{4 b \left (a-b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{32 b^2 \left (a-b x^4\right )^{3/4}}{385 a^3 x^7}-\frac{128 b^3 \left (a-b x^4\right )^{3/4}}{1155 a^4 x^3}\\ \end{align*}
Mathematica [A] time = 0.018733, size = 54, normalized size = 0.56 \[ -\frac{\left (a-b x^4\right )^{3/4} \left (84 a^2 b x^4+77 a^3+96 a b^2 x^8+128 b^3 x^{12}\right )}{1155 a^4 x^{15}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x^16*(a - b*x^4)^(1/4)),x]
[Out]
-((a - b*x^4)^(3/4)*(77*a^3 + 84*a^2*b*x^4 + 96*a*b^2*x^8 + 128*b^3*x^12))/(1155*a^4*x^15)
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Maple [A] time = 0.006, size = 51, normalized size = 0.5 \begin{align*} -{\frac{128\,{b}^{3}{x}^{12}+96\,a{b}^{2}{x}^{8}+84\,{a}^{2}b{x}^{4}+77\,{a}^{3}}{1155\,{x}^{15}{a}^{4}} \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^16/(-b*x^4+a)^(1/4),x)
[Out]
-1/1155*(-b*x^4+a)^(3/4)*(128*b^3*x^12+96*a*b^2*x^8+84*a^2*b*x^4+77*a^3)/x^15/a^4
________________________________________________________________________________________
Maxima [A] time = 1.00758, size = 99, normalized size = 1.03 \begin{align*} -\frac{\frac{385 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} b^{3}}{x^{3}} + \frac{495 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} b^{2}}{x^{7}} + \frac{315 \,{\left (-b x^{4} + a\right )}^{\frac{11}{4}} b}{x^{11}} + \frac{77 \,{\left (-b x^{4} + a\right )}^{\frac{15}{4}}}{x^{15}}}{1155 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^16/(-b*x^4+a)^(1/4),x, algorithm="maxima")
[Out]
-1/1155*(385*(-b*x^4 + a)^(3/4)*b^3/x^3 + 495*(-b*x^4 + a)^(7/4)*b^2/x^7 + 315*(-b*x^4 + a)^(11/4)*b/x^11 + 77
*(-b*x^4 + a)^(15/4)/x^15)/a^4
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Fricas [A] time = 1.78857, size = 126, normalized size = 1.31 \begin{align*} -\frac{{\left (128 \, b^{3} x^{12} + 96 \, a b^{2} x^{8} + 84 \, a^{2} b x^{4} + 77 \, a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \, a^{4} x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^16/(-b*x^4+a)^(1/4),x, algorithm="fricas")
[Out]
-1/1155*(128*b^3*x^12 + 96*a*b^2*x^8 + 84*a^2*b*x^4 + 77*a^3)*(-b*x^4 + a)^(3/4)/(a^4*x^15)
________________________________________________________________________________________
Sympy [C] time = 7.50222, size = 1824, normalized size = 19. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**16/(-b*x**4+a)**(1/4),x)
[Out]
Piecewise((231*a**6*b**(39/4)*(a/(b*x**4) - 1)**(3/4)*exp(I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/
4)*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 25
6*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)) - 441*a**5*b**(43/4)*x**4*(a/(b*x**4) - 1)**(3/4)*exp(I*pi/4)*gamma
(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*
b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)) + 225*a**4*b**(47/4)*x**8*(a
/(b*x**4) - 1)**(3/4)*exp(I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x
**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**24*exp(I*pi/4)*g
amma(1/4)) + 45*a**3*b**(51/4)*x**12*(a/(b*x**4) - 1)**(3/4)*exp(I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*ex
p(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/
4) + 256*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)) - 540*a**2*b**(55/4)*x**16*(a/(b*x**4) - 1)**(3/4)*exp(I*pi/
4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 7
68*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)) + 864*a*b**(59/4)*x*
*20*(a/(b*x**4) - 1)**(3/4)*exp(I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 768*a**6*b
**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**24*exp(I*p
i/4)*gamma(1/4)) - 384*b**(63/4)*x**24*(a/(b*x**4) - 1)**(3/4)*exp(I*pi/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*
exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(
1/4) + 256*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)), Abs(a)/(Abs(b)*Abs(x**4)) > 1), (-231*a**6*b**(39/4)*(-a/
(b*x**4) + 1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi
/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)) +
441*a**5*b**(43/4)*x**4*(-a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 7
68*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**2
4*exp(I*pi/4)*gamma(1/4)) - 225*a**4*b**(47/4)*x**8*(-a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9*x**1
2*exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamm
a(1/4) + 256*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)) - 45*a**3*b**(51/4)*x**12*(-a/(b*x**4) + 1)**(3/4)*gamma
(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*
b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4)) + 540*a**2*b**(55/4)*x**16*(
-a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*exp(I
*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**24*exp(I*pi/4)*gamma(1/4))
- 864*a*b**(59/4)*x**20*(-a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*exp(I*pi/4)*gamma(1/4) +
768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1/4) + 256*a**4*b**12*x**
24*exp(I*pi/4)*gamma(1/4)) + 384*b**(63/4)*x**24*(-a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(-256*a**7*b**9*x**12*e
xp(I*pi/4)*gamma(1/4) + 768*a**6*b**10*x**16*exp(I*pi/4)*gamma(1/4) - 768*a**5*b**11*x**20*exp(I*pi/4)*gamma(1
/4) + 256*a**4*b**12*x**24*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^{16}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^16/(-b*x^4+a)^(1/4),x, algorithm="giac")
[Out]
integrate(1/((-b*x^4 + a)^(1/4)*x^16), x)