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]

-(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 = 0.0430372, antiderivative size = 116, normalized size of antiderivative = 1., 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 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 = 0.0264121, size = 64, normalized size = 0.55 \[ -\frac{\left (a+b x^4\right )^{3/4} \left (1344 a^2 b^2 x^8-1232 a^3 b x^4+1155 a^4-1536 a b^3 x^{12}+2048 b^4 x^{16}\right )}{21945 a^5 x^{19}} \]

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 = 0.005, size = 61, normalized size = 0.5 \begin{align*} -{\frac{2048\,{x}^{16}{b}^{4}-1536\,{b}^{3}{x}^{12}a+1344\,{a}^{2}{x}^{8}{b}^{2}-1232\,{a}^{3}{x}^{4}b+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

________________________________________________________________________________________

Maxima [A]  time = 1.02616, size = 116, normalized size = 1. \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

________________________________________________________________________________________

Fricas [A]  time = 1.49284, size = 162, normalized size = 1.4 \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)

________________________________________________________________________________________

Sympy [B]  time = 13.8855, size = 1046, normalized size = 9.02 \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]

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))

________________________________________________________________________________________

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)