3.1008 \(\int \frac{\sqrt [4]{a+b x^4}}{x^{18}} \, dx\)

Optimal. Leaf size=92 \[ \frac{128 b^3 \left (a+b x^4\right )^{5/4}}{3315 a^4 x^5}-\frac{32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}+\frac{12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac{\left (a+b x^4\right )^{5/4}}{17 a x^{17}} \]

[Out]

-(a + b*x^4)^(5/4)/(17*a*x^17) + (12*b*(a + b*x^4)^(5/4))/(221*a^2*x^13) - (32*b^2*(a + b*x^4)^(5/4))/(663*a^3
*x^9) + (128*b^3*(a + b*x^4)^(5/4))/(3315*a^4*x^5)

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Rubi [A]  time = 0.0267598, 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} \[ \frac{128 b^3 \left (a+b x^4\right )^{5/4}}{3315 a^4 x^5}-\frac{32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}+\frac{12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac{\left (a+b x^4\right )^{5/4}}{17 a x^{17}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(1/4)/x^18,x]

[Out]

-(a + b*x^4)^(5/4)/(17*a*x^17) + (12*b*(a + b*x^4)^(5/4))/(221*a^2*x^13) - (32*b^2*(a + b*x^4)^(5/4))/(663*a^3
*x^9) + (128*b^3*(a + b*x^4)^(5/4))/(3315*a^4*x^5)

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{\sqrt [4]{a+b x^4}}{x^{18}} \, dx &=-\frac{\left (a+b x^4\right )^{5/4}}{17 a x^{17}}-\frac{(12 b) \int \frac{\sqrt [4]{a+b x^4}}{x^{14}} \, dx}{17 a}\\ &=-\frac{\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac{12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}+\frac{\left (96 b^2\right ) \int \frac{\sqrt [4]{a+b x^4}}{x^{10}} \, dx}{221 a^2}\\ &=-\frac{\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac{12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac{32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}-\frac{\left (128 b^3\right ) \int \frac{\sqrt [4]{a+b x^4}}{x^6} \, dx}{663 a^3}\\ &=-\frac{\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac{12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac{32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}+\frac{128 b^3 \left (a+b x^4\right )^{5/4}}{3315 a^4 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0127608, size = 53, normalized size = 0.58 \[ \frac{\left (a+b x^4\right )^{5/4} \left (180 a^2 b x^4-195 a^3-160 a b^2 x^8+128 b^3 x^{12}\right )}{3315 a^4 x^{17}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(1/4)/x^18,x]

[Out]

((a + b*x^4)^(5/4)*(-195*a^3 + 180*a^2*b*x^4 - 160*a*b^2*x^8 + 128*b^3*x^12))/(3315*a^4*x^17)

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Maple [A]  time = 0.005, size = 50, normalized size = 0.5 \begin{align*} -{\frac{-128\,{b}^{3}{x}^{12}+160\,a{b}^{2}{x}^{8}-180\,{a}^{2}b{x}^{4}+195\,{a}^{3}}{3315\,{x}^{17}{a}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(1/4)/x^18,x)

[Out]

-1/3315*(b*x^4+a)^(5/4)*(-128*b^3*x^12+160*a*b^2*x^8-180*a^2*b*x^4+195*a^3)/x^17/a^4

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Maxima [A]  time = 1.00171, size = 93, normalized size = 1.01 \begin{align*} \frac{\frac{663 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} b^{3}}{x^{5}} - \frac{1105 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} b^{2}}{x^{9}} + \frac{765 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} b}{x^{13}} - \frac{195 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}}}{x^{17}}}{3315 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^18,x, algorithm="maxima")

[Out]

1/3315*(663*(b*x^4 + a)^(5/4)*b^3/x^5 - 1105*(b*x^4 + a)^(9/4)*b^2/x^9 + 765*(b*x^4 + a)^(13/4)*b/x^13 - 195*(
b*x^4 + a)^(17/4)/x^17)/a^4

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Fricas [A]  time = 1.73912, size = 149, normalized size = 1.62 \begin{align*} \frac{{\left (128 \, b^{4} x^{16} - 32 \, a b^{3} x^{12} + 20 \, a^{2} b^{2} x^{8} - 15 \, a^{3} b x^{4} - 195 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{3315 \, a^{4} x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^18,x, algorithm="fricas")

[Out]

1/3315*(128*b^4*x^16 - 32*a*b^3*x^12 + 20*a^2*b^2*x^8 - 15*a^3*b*x^4 - 195*a^4)*(b*x^4 + a)^(1/4)/(a^4*x^17)

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Sympy [B]  time = 7.85754, size = 847, normalized size = 9.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(1/4)/x**18,x)

[Out]

-585*a**7*b**(37/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**
20*gamma(-1/4) + 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) - 1800*a**6*b**(41/4)*x*
*4*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) +
768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) - 1830*a**5*b**(45/4)*x**8*(a/(b*x**4) +
1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) + 768*a**5*b**11*x*
*24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) - 636*a**4*b**(49/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-
17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) + 768*a**5*b**11*x**24*gamma(-1/4) +
256*a**4*b**12*x**28*gamma(-1/4)) + 231*a**3*b**(53/4)*x**16*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b
**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) + 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x
**28*gamma(-1/4)) + 924*a**2*b**(57/4)*x**20*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-
1/4) + 768*a**6*b**10*x**20*gamma(-1/4) + 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4))
+ 1056*a*b**(61/4)*x**24*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**
10*x**20*gamma(-1/4) + 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) + 384*b**(65/4)*x*
*28*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) +
768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4))

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Giac [B]  time = 1.15784, size = 220, normalized size = 2.39 \begin{align*} \frac{\frac{663 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )} b^{3}}{x} - \frac{1105 \,{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}}{x^{9}} + \frac{765 \,{\left (b^{3} x^{12} + 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} + a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x^{13}} - \frac{195 \,{\left (b^{4} x^{16} + 4 \, a b^{3} x^{12} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{4} + a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{17}}}{3315 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^18,x, algorithm="giac")

[Out]

1/3315*(663*(b*x^4 + a)^(1/4)*(b + a/x^4)*b^3/x - 1105*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)*b^2/x^9 +
765*(b^3*x^12 + 3*a*b^2*x^8 + 3*a^2*b*x^4 + a^3)*(b*x^4 + a)^(1/4)*b/x^13 - 195*(b^4*x^16 + 4*a*b^3*x^12 + 6*
a^2*b^2*x^8 + 4*a^3*b*x^4 + a^4)*(b*x^4 + a)^(1/4)/x^17)/a^4