3.1101 \(\int \frac{1}{x^{16} \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=92 \[ \frac{128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac{32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac{4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{\left (a+b x^4\right )^{3/4}}{15 a x^{15}} \]

[Out]

-(a + b*x^4)^(3/4)/(15*a*x^15) + (4*b*(a + b*x^4)^(3/4))/(55*a^2*x^11) - (32*b^2*(a + b*x^4)^(3/4))/(385*a^3*x
^7) + (128*b^3*(a + b*x^4)^(3/4))/(1155*a^4*x^3)

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Rubi [A]  time = 0.0283792, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3}-\frac{32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac{4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{\left (a+b x^4\right )^{3/4}}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^16*(a + b*x^4)^(1/4)),x]

[Out]

-(a + b*x^4)^(3/4)/(15*a*x^15) + (4*b*(a + b*x^4)^(3/4))/(55*a^2*x^11) - (32*b^2*(a + b*x^4)^(3/4))/(385*a^3*x
^7) + (128*b^3*(a + b*x^4)^(3/4))/(1155*a^4*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^{16} \sqrt [4]{a+b x^4}} \, dx &=-\frac{\left (a+b x^4\right )^{3/4}}{15 a x^{15}}-\frac{(4 b) \int \frac{1}{x^{12} \sqrt [4]{a+b x^4}} \, dx}{5 a}\\ &=-\frac{\left (a+b x^4\right )^{3/4}}{15 a x^{15}}+\frac{4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}+\frac{\left (32 b^2\right ) \int \frac{1}{x^8 \sqrt [4]{a+b x^4}} \, dx}{55 a^2}\\ &=-\frac{\left (a+b x^4\right )^{3/4}}{15 a x^{15}}+\frac{4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}-\frac{\left (128 b^3\right ) \int \frac{1}{x^4 \sqrt [4]{a+b x^4}} \, dx}{385 a^3}\\ &=-\frac{\left (a+b x^4\right )^{3/4}}{15 a x^{15}}+\frac{4 b \left (a+b x^4\right )^{3/4}}{55 a^2 x^{11}}-\frac{32 b^2 \left (a+b x^4\right )^{3/4}}{385 a^3 x^7}+\frac{128 b^3 \left (a+b x^4\right )^{3/4}}{1155 a^4 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0202353, size = 53, normalized size = 0.58 \[ \frac{\left (a+b x^4\right )^{3/4} \left (84 a^2 b x^4-77 a^3-96 a b^2 x^8+128 b^3 x^{12}\right )}{1155 a^4 x^{15}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x^4)^(3/4)*(-77*a^3 + 84*a^2*b*x^4 - 96*a*b^2*x^8 + 128*b^3*x^12))/(1155*a^4*x^15)

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Maple [A]  time = 0.004, size = 50, normalized size = 0.5 \begin{align*} -{\frac{-128\,{b}^{3}{x}^{12}+96\,a{b}^{2}{x}^{8}-84\,{a}^{2}b{x}^{4}+77\,{a}^{3}}{1155\,{x}^{15}{a}^{4}} \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^16/(b*x^4+a)^(1/4),x)

[Out]

-1/1155*(b*x^4+a)^(3/4)*(-128*b^3*x^12+96*a*b^2*x^8-84*a^2*b*x^4+77*a^3)/x^15/a^4

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Maxima [A]  time = 0.985869, size = 93, normalized size = 1.01 \begin{align*} \frac{\frac{385 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} b^{3}}{x^{3}} - \frac{495 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b^{2}}{x^{7}} + \frac{315 \,{\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^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1155*(385*(b*x^4 + a)^(3/4)*b^3/x^3 - 495*(b*x^4 + a)^(7/4)*b^2/x^7 + 315*(b*x^4 + a)^(11/4)*b/x^11 - 77*(b*
x^4 + a)^(15/4)/x^15)/a^4

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Fricas [A]  time = 1.51217, size = 123, normalized size = 1.34 \begin{align*} \frac{{\left (128 \, b^{3} x^{12} - 96 \, a b^{2} x^{8} + 84 \, a^{2} b x^{4} - 77 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \, a^{4} x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1155*(128*b^3*x^12 - 96*a*b^2*x^8 + 84*a^2*b*x^4 - 77*a^3)*(b*x^4 + a)^(3/4)/(a^4*x^15)

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Sympy [B]  time = 7.16376, size = 692, normalized size = 7.52 \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)**(1/4),x)

[Out]

-231*a**6*b**(39/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**1
6*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) - 441*a**5*b**(43/4)*x**4*(a
/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**
5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) - 225*a**4*b**(47/4)*x**8*(a/(b*x**4) + 1)**(3/4)*
gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/
4) + 256*a**4*b**12*x**24*gamma(1/4)) + 45*a**3*b**(51/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7
*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x*
*24*gamma(1/4)) + 540*a**2*b**(55/4)*x**16*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4
) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) + 864
*a*b**(59/4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16
*gamma(1/4) + 768*a**5*b**11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4)) + 384*b**(63/4)*x**24*(a/(b*x
**4) + 1)**(3/4)*gamma(-15/4)/(256*a**7*b**9*x**12*gamma(1/4) + 768*a**6*b**10*x**16*gamma(1/4) + 768*a**5*b**
11*x**20*gamma(1/4) + 256*a**4*b**12*x**24*gamma(1/4))

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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{1}{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)^(1/4),x, algorithm="giac")

[Out]

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