3.1161 \(\int \frac{1}{x^{16} (a+b x^4)^{5/4}} \, dx\)

Optimal. Leaf size=114 \[ \frac{2048 b^4 x}{1155 a^5 \sqrt [4]{a+b x^4}}+\frac{512 b^3}{1155 a^4 x^3 \sqrt [4]{a+b x^4}}-\frac{64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}+\frac{16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac{1}{15 a x^{15} \sqrt [4]{a+b x^4}} \]

[Out]

-1/(15*a*x^15*(a + b*x^4)^(1/4)) + (16*b)/(165*a^2*x^11*(a + b*x^4)^(1/4)) - (64*b^2)/(385*a^3*x^7*(a + b*x^4)
^(1/4)) + (512*b^3)/(1155*a^4*x^3*(a + b*x^4)^(1/4)) + (2048*b^4*x)/(1155*a^5*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0395808, antiderivative size = 114, 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, 191} \[ \frac{2048 b^4 x}{1155 a^5 \sqrt [4]{a+b x^4}}+\frac{512 b^3}{1155 a^4 x^3 \sqrt [4]{a+b x^4}}-\frac{64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}+\frac{16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac{1}{15 a x^{15} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^16*(a + b*x^4)^(5/4)),x]

[Out]

-1/(15*a*x^15*(a + b*x^4)^(1/4)) + (16*b)/(165*a^2*x^11*(a + b*x^4)^(1/4)) - (64*b^2)/(385*a^3*x^7*(a + b*x^4)
^(1/4)) + (512*b^3)/(1155*a^4*x^3*(a + b*x^4)^(1/4)) + (2048*b^4*x)/(1155*a^5*(a + b*x^4)^(1/4))

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 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^{16} \left (a+b x^4\right )^{5/4}} \, dx &=-\frac{1}{15 a x^{15} \sqrt [4]{a+b x^4}}-\frac{(16 b) \int \frac{1}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx}{15 a}\\ &=-\frac{1}{15 a x^{15} \sqrt [4]{a+b x^4}}+\frac{16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}+\frac{\left (64 b^2\right ) \int \frac{1}{x^8 \left (a+b x^4\right )^{5/4}} \, dx}{55 a^2}\\ &=-\frac{1}{15 a x^{15} \sqrt [4]{a+b x^4}}+\frac{16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac{64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}-\frac{\left (512 b^3\right ) \int \frac{1}{x^4 \left (a+b x^4\right )^{5/4}} \, dx}{385 a^3}\\ &=-\frac{1}{15 a x^{15} \sqrt [4]{a+b x^4}}+\frac{16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac{64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}+\frac{512 b^3}{1155 a^4 x^3 \sqrt [4]{a+b x^4}}+\frac{\left (2048 b^4\right ) \int \frac{1}{\left (a+b x^4\right )^{5/4}} \, dx}{1155 a^4}\\ &=-\frac{1}{15 a x^{15} \sqrt [4]{a+b x^4}}+\frac{16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac{64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}+\frac{512 b^3}{1155 a^4 x^3 \sqrt [4]{a+b x^4}}+\frac{2048 b^4 x}{1155 a^5 \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0097149, size = 64, normalized size = 0.56 \[ -\frac{192 a^2 b^2 x^8-112 a^3 b x^4+77 a^4-512 a b^3 x^{12}-2048 b^4 x^{16}}{1155 a^5 x^{15} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^16*(a + b*x^4)^(5/4)),x]

[Out]

-(77*a^4 - 112*a^3*b*x^4 + 192*a^2*b^2*x^8 - 512*a*b^3*x^12 - 2048*b^4*x^16)/(1155*a^5*x^15*(a + b*x^4)^(1/4))

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Maple [A]  time = 0.006, size = 61, normalized size = 0.5 \begin{align*} -{\frac{-2048\,{x}^{16}{b}^{4}-512\,{b}^{3}{x}^{12}a+192\,{b}^{2}{x}^{8}{a}^{2}-112\,{a}^{3}{x}^{4}b+77\,{a}^{4}}{1155\,{a}^{5}{x}^{15}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^16/(b*x^4+a)^(5/4),x)

[Out]

-1/1155*(-2048*b^4*x^16-512*a*b^3*x^12+192*a^2*b^2*x^8-112*a^3*b*x^4+77*a^4)/x^15/(b*x^4+a)^(1/4)/a^5

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Maxima [A]  time = 1.05959, size = 117, normalized size = 1.03 \begin{align*} \frac{b^{4} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{5}} + \frac{\frac{1540 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} b^{3}}{x^{3}} - \frac{990 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b^{2}}{x^{7}} + \frac{420 \,{\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^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

b^4*x/((b*x^4 + a)^(1/4)*a^5) + 1/1155*(1540*(b*x^4 + a)^(3/4)*b^3/x^3 - 990*(b*x^4 + a)^(7/4)*b^2/x^7 + 420*(
b*x^4 + a)^(11/4)*b/x^11 - 77*(b*x^4 + a)^(15/4)/x^15)/a^5

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Fricas [A]  time = 1.55323, size = 170, normalized size = 1.49 \begin{align*} \frac{{\left (2048 \, b^{4} x^{16} + 512 \, a b^{3} x^{12} - 192 \, a^{2} b^{2} x^{8} + 112 \, a^{3} b x^{4} - 77 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \,{\left (a^{5} b x^{19} + a^{6} x^{15}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

1/1155*(2048*b^4*x^16 + 512*a*b^3*x^12 - 192*a^2*b^2*x^8 + 112*a^3*b*x^4 - 77*a^4)*(b*x^4 + a)^(3/4)/(a^5*b*x^
19 + a^6*x^15)

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Sympy [B]  time = 16.5563, size = 928, normalized size = 8.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**16/(b*x**4+a)**(5/4),x)

[Out]

-231*a**7*b**(67/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4) + 4096*a**8*b**17*x
**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*
gamma(5/4)) - 357*a**6*b**(71/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4) +
4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 1024
*a**5*b**20*x**28*gamma(5/4)) - 261*a**5*b**(75/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*
x**12*gamma(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24
*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 585*a**4*b**(79/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4
)/(1024*a**9*b**16*x**12*gamma(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 40
96*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 9360*a**3*b**(83/4)*x**16*(a/(b*x**4) + 1
)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x
**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 22464*a**2*b**(87/4)*
x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4) + 4096*a**8*b**17*x**16*gamma(5/4
) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) +
19968*a*b**(91/4)*x**24*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4) + 4096*a**8*b**
17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x*
*28*gamma(5/4)) + 6144*b**(95/4)*x**28*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4)
+ 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 102
4*a**5*b**20*x**28*gamma(5/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{16}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^16), x)