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]

-1/19*(-b*x^4+a)^(3/4)/a/x^19-16/285*b*(-b*x^4+a)^(3/4)/a^2/x^15-64/1045*b^2*(-b*x^4+a)^(3/4)/a^3/x^11-512/731
5*b^3*(-b*x^4+a)^(3/4)/a^4/x^7-2048/21945*b^4*(-b*x^4+a)^(3/4)/a^5/x^3

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Rubi [A]  time = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, 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 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^{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*}

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Mathematica [A]  time = 0.03, size = 65, normalized size = 0.54 \[ -\frac {\left (a-b x^4\right )^{3/4} \left (1155 a^4+1232 a^3 b x^4+1344 a^2 b^2 x^8+1536 a b^3 x^{12}+2048 b^4 x^{16}\right )}{21945 a^5 x^{19}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^20*(a - b*x^4)^(1/4)),x]

[Out]

-1/21945*((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))/
(a^5*x^19)

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fricas [A]  time = 0.59, size = 61, normalized size = 0.50 \[ -\frac {{\left (2048 \, b^{4} x^{16} + 1536 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} + 1232 \, a^{3} b x^{4} + 1155 \, a^{4}\right )} {\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{21945 \, a^{5} x^{19}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{20}}\,{d x} \]

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)

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maple [A]  time = 0.01, size = 62, normalized size = 0.51 \[ -\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}+1536 a \,x^{12} b^{3}+1344 a^{2} x^{8} b^{2}+1232 a^{3} x^{4} b +1155 a^{4}\right )}{21945 a^{5} x^{19}} \]

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 = 1.07, size = 91, normalized size = 0.75 \[ -\frac {\frac {7315 \, {\left (-b x^{4} + a\right )}^{\frac {3}{4}} b^{4}}{x^{3}} + \frac {12540 \, {\left (-b x^{4} + a\right )}^{\frac {7}{4}} b^{3}}{x^{7}} + \frac {11970 \, {\left (-b x^{4} + a\right )}^{\frac {11}{4}} b^{2}}{x^{11}} + \frac {5852 \, {\left (-b x^{4} + a\right )}^{\frac {15}{4}} b}{x^{15}} + \frac {1155 \, {\left (-b x^{4} + a\right )}^{\frac {19}{4}}}{x^{19}}}{21945 \, a^{5}} \]

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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mupad [B]  time = 1.44, size = 101, normalized size = 0.83 \[ -\frac {{\left (a-b\,x^4\right )}^{3/4}}{19\,a\,x^{19}}-\frac {16\,b\,{\left (a-b\,x^4\right )}^{3/4}}{285\,a^2\,x^{15}}-\frac {2048\,b^4\,{\left (a-b\,x^4\right )}^{3/4}}{21945\,a^5\,x^3}-\frac {512\,b^3\,{\left (a-b\,x^4\right )}^{3/4}}{7315\,a^4\,x^7}-\frac {64\,b^2\,{\left (a-b\,x^4\right )}^{3/4}}{1045\,a^3\,x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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) - (2048*b^4*(a - b*x^4)^(3/4))/(2194
5*a^5*x^3) - (512*b^3*(a - b*x^4)^(3/4))/(7315*a^4*x^7) - (64*b^2*(a - b*x^4)^(3/4))/(1045*a^3*x^11)

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sympy [C]  time = 9.88, size = 2788, normalized size = 23.04 \[ \text {result too large to display} \]

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/(b*x**4)) > 1), (-346
5*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*e
xp(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)*ga
mma(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)/(10
24*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)) + 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)*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)*gamma(-19/4)/(1024*a**9*b**16*x**16*exp(I*pi/4)*gamma(1/4) - 4096*a**8*b**1
7*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)*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)) - 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**18*x**24*ex
p(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*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)) - 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))

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