3.13.25 \(\int \frac {1}{x^{16} \sqrt [4]{a-b x^4}} \, dx\) [1225]
Optimal. Leaf size=96 \[ -\frac {\left (a-b x^4\right )^{3/4}}{15 a x^{15}}-\frac {4 b \left (a-b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac {32 b^2 \left (a-b x^4\right )^{3/4}}{385 a^3 x^7}-\frac {128 b^3 \left (a-b x^4\right )^{3/4}}{1155 a^4 x^3} \]
[Out]
-1/15*(-b*x^4+a)^(3/4)/a/x^15-4/55*b*(-b*x^4+a)^(3/4)/a^2/x^11-32/385*b^2*(-b*x^4+a)^(3/4)/a^3/x^7-128/1155*b^
3*(-b*x^4+a)^(3/4)/a^4/x^3
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, 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 = {277, 270}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^16*(a - b*x^4)^(1/4)),x]
[Out]
-1/15*(a - b*x^4)^(3/4)/(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 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^{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 = 0.24, size = 54, normalized size = 0.56 \begin {gather*} \frac {\left (a-b x^4\right )^{3/4} \left (-77 a^3-84 a^2 b x^4-96 a b^2 x^8-128 b^3 x^{12}\right )}{1155 a^4 x^{15}} \end {gather*}
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)
________________________________________________________________________________________
Maple [A]
time = 0.17, size = 51, normalized size = 0.53
| | |
| method |
result |
size |
| | |
| gosper |
\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (128 b^{3} x^{12}+96 a \,b^{2} x^{8}+84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) |
\(51\) |
| trager |
\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (128 b^{3} x^{12}+96 a \,b^{2} x^{8}+84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) |
\(51\) |
| risch |
\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (128 b^{3} x^{12}+96 a \,b^{2} x^{8}+84 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 x^{15} a^{4}}\) |
\(51\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^16/(-b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)
[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 = 0.29, size = 73, normalized size = 0.76 \begin {gather*} -\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 {gather*}
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
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 50, normalized size = 0.52 \begin {gather*} -\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 {gather*}
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] Result contains complex when optimal does not.
time = 1.51, size = 1821, normalized size = 18.97 \begin {gather*} \text {Too large to display} \end {gather*}
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/(b*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) + 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)*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) + 2
56*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**2
0*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)*gam
ma(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*p
i/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*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)), True))
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
________________________________________________________________________________________
Mupad [B]
time = 1.36, size = 80, normalized size = 0.83 \begin {gather*} -\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 {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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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) - (128*b^3*(a - b*x^4)^(3/4))/(1155*a^
4*x^3) - (32*b^2*(a - b*x^4)^(3/4))/(385*a^3*x^7)
________________________________________________________________________________________