3.1226 \(\int \frac{1}{x^{20} \sqrt [4]{a-b x^4}} \, dx\)
Optimal. Leaf size=121 \[ -\frac{2048 b^4 \left (a-b x^4\right )^{3/4}}{21945 a^5 x^3}-\frac{512 b^3 \left (a-b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac{64 b^2 \left (a-b x^4\right )^{3/4}}{1045 a^3 x^{11}}-\frac{16 b \left (a-b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac{\left (a-b x^4\right )^{3/4}}{19 a x^{19}} \]
[Out]
-(a - b*x^4)^(3/4)/(19*a*x^19) - (16*b*(a - b*x^4)^(3/4))/(285*a^2*x^15) - (64*b^2*(a - b*x^4)^(3/4))/(1045*a^
3*x^11) - (512*b^3*(a - b*x^4)^(3/4))/(7315*a^4*x^7) - (2048*b^4*(a - b*x^4)^(3/4))/(21945*a^5*x^3)
________________________________________________________________________________________
Rubi [A] time = 0.0411944, antiderivative size = 121, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{271, 264} \[ -\frac{2048 b^4 \left (a-b x^4\right )^{3/4}}{21945 a^5 x^3}-\frac{512 b^3 \left (a-b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac{64 b^2 \left (a-b x^4\right )^{3/4}}{1045 a^3 x^{11}}-\frac{16 b \left (a-b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac{\left (a-b x^4\right )^{3/4}}{19 a x^{19}} \]
Antiderivative was successfully verified.
[In]
Int[1/(x^20*(a - b*x^4)^(1/4)),x]
[Out]
-(a - b*x^4)^(3/4)/(19*a*x^19) - (16*b*(a - b*x^4)^(3/4))/(285*a^2*x^15) - (64*b^2*(a - b*x^4)^(3/4))/(1045*a^
3*x^11) - (512*b^3*(a - b*x^4)^(3/4))/(7315*a^4*x^7) - (2048*b^4*(a - b*x^4)^(3/4))/(21945*a^5*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^{20} \sqrt [4]{a-b x^4}} \, dx &=-\frac{\left (a-b x^4\right )^{3/4}}{19 a x^{19}}+\frac{(16 b) \int \frac{1}{x^{16} \sqrt [4]{a-b x^4}} \, dx}{19 a}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{19 a x^{19}}-\frac{16 b \left (a-b x^4\right )^{3/4}}{285 a^2 x^{15}}+\frac{\left (64 b^2\right ) \int \frac{1}{x^{12} \sqrt [4]{a-b x^4}} \, dx}{95 a^2}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{19 a x^{19}}-\frac{16 b \left (a-b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac{64 b^2 \left (a-b x^4\right )^{3/4}}{1045 a^3 x^{11}}+\frac{\left (512 b^3\right ) \int \frac{1}{x^8 \sqrt [4]{a-b x^4}} \, dx}{1045 a^3}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{19 a x^{19}}-\frac{16 b \left (a-b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac{64 b^2 \left (a-b x^4\right )^{3/4}}{1045 a^3 x^{11}}-\frac{512 b^3 \left (a-b x^4\right )^{3/4}}{7315 a^4 x^7}+\frac{\left (2048 b^4\right ) \int \frac{1}{x^4 \sqrt [4]{a-b x^4}} \, dx}{7315 a^4}\\ &=-\frac{\left (a-b x^4\right )^{3/4}}{19 a x^{19}}-\frac{16 b \left (a-b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac{64 b^2 \left (a-b x^4\right )^{3/4}}{1045 a^3 x^{11}}-\frac{512 b^3 \left (a-b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac{2048 b^4 \left (a-b x^4\right )^{3/4}}{21945 a^5 x^3}\\ \end{align*}
Mathematica [A] time = 0.0231195, size = 65, normalized size = 0.54 \[ -\frac{\left (a-b x^4\right )^{3/4} \left (1344 a^2 b^2 x^8+1232 a^3 b x^4+1155 a^4+1536 a b^3 x^{12}+2048 b^4 x^{16}\right )}{21945 a^5 x^{19}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x^20*(a - b*x^4)^(1/4)),x]
[Out]
-((a - b*x^4)^(3/4)*(1155*a^4 + 1232*a^3*b*x^4 + 1344*a^2*b^2*x^8 + 1536*a*b^3*x^12 + 2048*b^4*x^16))/(21945*a
^5*x^19)
________________________________________________________________________________________
Maple [A] time = 0.006, size = 62, normalized size = 0.5 \begin{align*} -{\frac{2048\,{b}^{4}{x}^{16}+1536\,{b}^{3}{x}^{12}a+1344\,{b}^{2}{x}^{8}{a}^{2}+1232\,b{x}^{4}{a}^{3}+1155\,{a}^{4}}{21945\,{x}^{19}{a}^{5}} \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^20/(-b*x^4+a)^(1/4),x)
[Out]
-1/21945*(-b*x^4+a)^(3/4)*(2048*b^4*x^16+1536*a*b^3*x^12+1344*a^2*b^2*x^8+1232*a^3*b*x^4+1155*a^4)/x^19/a^5
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Maxima [A] time = 1.01891, size = 123, normalized size = 1.02 \begin{align*} -\frac{\frac{7315 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} b^{4}}{x^{3}} + \frac{12540 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} b^{3}}{x^{7}} + \frac{11970 \,{\left (-b x^{4} + a\right )}^{\frac{11}{4}} b^{2}}{x^{11}} + \frac{5852 \,{\left (-b x^{4} + a\right )}^{\frac{15}{4}} b}{x^{15}} + \frac{1155 \,{\left (-b x^{4} + a\right )}^{\frac{19}{4}}}{x^{19}}}{21945 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^20/(-b*x^4+a)^(1/4),x, algorithm="maxima")
[Out]
-1/21945*(7315*(-b*x^4 + a)^(3/4)*b^4/x^3 + 12540*(-b*x^4 + a)^(7/4)*b^3/x^7 + 11970*(-b*x^4 + a)^(11/4)*b^2/x
^11 + 5852*(-b*x^4 + a)^(15/4)*b/x^15 + 1155*(-b*x^4 + a)^(19/4)/x^19)/a^5
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Fricas [A] time = 1.78608, size = 163, normalized size = 1.35 \begin{align*} -\frac{{\left (2048 \, b^{4} x^{16} + 1536 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} + 1232 \, a^{3} b x^{4} + 1155 \, a^{4}\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{21945 \, a^{5} x^{19}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^20/(-b*x^4+a)^(1/4),x, algorithm="fricas")
[Out]
-1/21945*(2048*b^4*x^16 + 1536*a*b^3*x^12 + 1344*a^2*b^2*x^8 + 1232*a^3*b*x^4 + 1155*a^4)*(-b*x^4 + a)^(3/4)/(
a^5*x^19)
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Sympy [C] time = 14.6358, size = 2791, normalized size = 23.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**20/(-b*x**4+a)**(1/4),x)
[Out]
Piecewise((-3465*a**8*b**(67/4)*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp
(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1
/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) + 10164*a**
7*b**(71/4)*x**4*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(
1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6
*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) - 10038*a**6*b**(75/4)*x**
8*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**
8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*ex
p(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) + 3204*a**5*b**(79/4)*x**12*(a/(b*x**4) -
1)**(3/4)*exp(-3*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*e
xp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma
(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) - 585*a**4*b**(83/4)*x**16*(a/(b*x**4) - 1)**(3/4)*exp(-
3*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma
(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**
5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) + 9360*a**3*b**(87/4)*x**20*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma
(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a*
*7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*e
xp(I*pi/4)*gamma(1/4)) - 22464*a**2*b**(91/4)*x**24*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-19/4)/(1024*
a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24
*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gam
ma(1/4)) + 19968*a*b**(95/4)*x**28*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*
exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamm
a(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) - 6144*b
**(99/4)*x**32*(a/(b*x**4) - 1)**(3/4)*exp(-3*I*pi/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/
4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b
**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)), Abs(a)/(Abs(b)*Abs(x**4)) >
1), (-3465*a**8*b**(67/4)*(-a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4)
- 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**
19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) + 10164*a**7*b**(71/4)*x**4*(-
a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(
I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/
4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) - 10038*a**6*b**(75/4)*x**8*(-a/(b*x**4) + 1)**(3/4)*gamma(
-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**
7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*ex
p(I*pi/4)*gamma(1/4)) + 3204*a**5*b**(79/4)*x**12*(-a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16
*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gam
ma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) - 585*a
**4*b**(83/4)*x**16*(-a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096
*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**2
8*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) + 9360*a**3*b**(87/4)*x**20*(-a/(b*x*
*4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)
*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 10
24*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) - 22464*a**2*b**(91/4)*x**24*(-a/(b*x**4) + 1)**(3/4)*gamma(-19/4)
/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**1
8*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi
/4)*gamma(1/4)) + 19968*a*b**(95/4)*x**28*(-a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*p
i/4)*gamma(1/4) - 4096*a**8*b**17*x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4)
- 4096*a**6*b**19*x**28*exp(I*pi/4)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)) - 6144*b**(99/4
)*x**32*(-a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**17*
x**20*exp(I*pi/4)*gamma(1/4) + 6144*a**7*b**18*x**24*exp(I*pi/4)*gamma(1/4) - 4096*a**6*b**19*x**28*exp(I*pi/4
)*gamma(1/4) + 1024*a**5*b**20*x**32*exp(I*pi/4)*gamma(1/4)), True))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{20}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^20/(-b*x^4+a)^(1/4),x, algorithm="giac")
[Out]
integrate(1/((-b*x^4 + a)^(1/4)*x^20), x)