3.1102 \(\int \frac {1}{x^{20} \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=116 \[ -\frac {2048 b^4 \left (a+b x^4\right )^{3/4}}{21945 a^5 x^3}+\frac {512 b^3 \left (a+b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac {64 b^2 \left (a+b x^4\right )^{3/4}}{1045 a^3 x^{11}}+\frac {16 b \left (a+b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}} \]

[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+512/7315*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 = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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 = 0.05, size = 64, normalized size = 0.55 \[ -\frac {\left (a+b x^4\right )^{3/4} \left (1155 a^4-1232 a^3 b x^4+1344 a^2 b^2 x^8-1536 a b^3 x^{12}+2048 b^4 x^{16}\right )}{21945 a^5 x^{19}} \]

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 = 0.81, size = 60, normalized size = 0.52 \[ -\frac {{\left (2048 \, b^{4} x^{16} - 1536 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} - 1232 \, a^{3} b x^{4} + 1155 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{21945 \, a^{5} x^{19}} \]

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 = 61, normalized size = 0.53 \[ -\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}-1536 a \,x^{12} b^{3}+1344 a^{2} x^{8} b^{2}-1232 a^{3} x^{4} b +1155 a^{4}\right )}{21945 a^{5} x^{19}} \]

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 = 1.33, size = 86, normalized size = 0.74 \[ -\frac {\frac {7315 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{4}}{x^{3}} - \frac {12540 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{3}}{x^{7}} + \frac {11970 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b^{2}}{x^{11}} - \frac {5852 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} b}{x^{15}} + \frac {1155 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}}}{x^{19}}}{21945 \, a^{5}} \]

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 = 1.49, size = 96, normalized size = 0.83 \[ \frac {16\,b\,{\left (b\,x^4+a\right )}^{3/4}}{285\,a^2\,x^{15}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{19\,a\,x^{19}}-\frac {2048\,b^4\,{\left (b\,x^4+a\right )}^{3/4}}{21945\,a^5\,x^3}+\frac {512\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{7315\,a^4\,x^7}-\frac {64\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{1045\,a^3\,x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^20*(a + b*x^4)^(1/4)),x)

[Out]

(16*b*(a + b*x^4)^(3/4))/(285*a^2*x^15) - (a + b*x^4)^(3/4)/(19*a*x^19) - (2048*b^4*(a + b*x^4)^(3/4))/(21945*
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 [B]  time = 8.48, size = 1046, normalized size = 9.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**20/(b*x**4+a)**(1/4),x)

[Out]

3465*a**8*b**(67/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) + 4096*a**8*b**17*x
**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*
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*gamma(1/4)
+ 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 10
24*a**5*b**20*x**32*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*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x
**28*gamma(1/4) + 1024*a**5*b**20*x**32*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*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4)
+ 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*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*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**
18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*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*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(
1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*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*gamma(1/4) + 4096*a*
*8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b*
*20*x**32*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*ga
mma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1
/4) + 1024*a**5*b**20*x**32*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*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**1
9*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4))

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