\(\int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx\) [1195]

Optimal result

Integrand size = 16, antiderivative size = 96 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx=-\frac {\left (a-b x^4\right )^{5/4}}{17 a x^{17}}-\frac {12 b \left (a-b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{663 a^3 x^9}-\frac {128 b^3 \left (a-b x^4\right )^{5/4}}{3315 a^4 x^5} \]

[Out]

-1/17*(-b*x^4+a)^(5/4)/a/x^17-12/221*b*(-b*x^4+a)^(5/4)/a^2/x^13-32/663*b^2*(-b*x^4+a)^(5/4)/a^3/x^9-128/3315*
b^3*(-b*x^4+a)^(5/4)/a^4/x^5

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {277, 270} \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx=-\frac {128 b^3 \left (a-b x^4\right )^{5/4}}{3315 a^4 x^5}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{663 a^3 x^9}-\frac {12 b \left (a-b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {\left (a-b x^4\right )^{5/4}}{17 a x^{17}} \]

[In]

Int[(a - b*x^4)^(1/4)/x^18,x]

[Out]

-1/17*(a - b*x^4)^(5/4)/(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 270

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 277

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*} \text {integral}& = -\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] (verified)

Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx=\frac {\left (a-b x^4\right )^{5/4} \left (-195 a^3-180 a^2 b x^4-160 a b^2 x^8-128 b^3 x^{12}\right )}{3315 a^4 x^{17}} \]

[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)

Maple [A] (verified)

Time = 4.54 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.53

method	result	size
gosper	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (128 b^{3} x^{12}+160 a \,b^{2} x^{8}+180 a^{2} b \,x^{4}+195 a^{3}\right )}{3315 x^{17} a^{4}}\)	\(51\)
pseudoelliptic	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (128 b^{3} x^{12}+160 a \,b^{2} x^{8}+180 a^{2} b \,x^{4}+195 a^{3}\right )}{3315 x^{17} a^{4}}\)	\(51\)
trager	\(-\frac {\left (-128 x^{16} b^{4}-32 a \,b^{3} x^{12}-20 a^{2} b^{2} x^{8}-15 a^{3} b \,x^{4}+195 a^{4}\right ) \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{3315 x^{17} a^{4}}\)	\(62\)
risch	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} {\left (\left (-b \,x^{4}+a \right )^{3}\right )}^{\frac {1}{4}} \left (-128 x^{16} b^{4}-32 a \,b^{3} x^{12}-20 a^{2} b^{2} x^{8}-15 a^{3} b \,x^{4}+195 a^{4}\right )}{3315 x^{17} {\left (-\left (b \,x^{4}-a \right )^{3}\right )}^{\frac {1}{4}} a^{4}}\)	\(89\)

[In]

int((-b*x^4+a)^(1/4)/x^18,x,method=_RETURNVERBOSE)

[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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx=\frac {{\left (128 \, b^{4} x^{16} + 32 \, a b^{3} x^{12} + 20 \, a^{2} b^{2} x^{8} + 15 \, a^{3} b x^{4} - 195 \, a^{4}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{3315 \, a^{4} x^{17}} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.52 (sec) , antiderivative size = 1756, normalized size of antiderivative = 18.29 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((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*ga
mma(-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) + 2
56*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)), Abs(a/(b*x**4)) > 1),
(585*a**7*b**(37/4)*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/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)*exp(I*pi/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)*exp(I*pi/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)*exp(I*pi/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)*exp(I*pi/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)*exp(I*pi/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)*exp(I*pi/4)*gamma(-17/4)/(-256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*ga
mma(-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)*exp(I*pi/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)), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx=-\frac {\frac {663 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} b^{3}}{x^{5}} + \frac {1105 \, {\left (-b x^{4} + a\right )}^{\frac {9}{4}} b^{2}}{x^{9}} + \frac {765 \, {\left (-b x^{4} + a\right )}^{\frac {13}{4}} b}{x^{13}} + \frac {195 \, {\left (-b x^{4} + a\right )}^{\frac {17}{4}}}{x^{17}}}{3315 \, a^{4}} \]

[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 + 1
95*(-b*x^4 + a)^(17/4)/x^17)/a^4

Giac [F]

\[ \int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{18}} \,d x } \]

[In]

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

[Out]

integrate((-b*x^4 + a)^(1/4)/x^18, x)

Mupad [B] (verification not implemented)

Time = 6.39 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx=\frac {b\,{\left (a-b\,x^4\right )}^{1/4}}{221\,a\,x^{13}}-\frac {{\left (a-b\,x^4\right )}^{1/4}}{17\,x^{17}}+\frac {128\,b^4\,{\left (a-b\,x^4\right )}^{1/4}}{3315\,a^4\,x}+\frac {32\,b^3\,{\left (a-b\,x^4\right )}^{1/4}}{3315\,a^3\,x^5}+\frac {4\,b^2\,{\left (a-b\,x^4\right )}^{1/4}}{663\,a^2\,x^9} \]

[In]

int((a - b*x^4)^(1/4)/x^18,x)

[Out]

(b*(a - b*x^4)^(1/4))/(221*a*x^13) - (a - b*x^4)^(1/4)/(17*x^17) + (128*b^4*(a - b*x^4)^(1/4))/(3315*a^4*x) +
(32*b^3*(a - b*x^4)^(1/4))/(3315*a^3*x^5) + (4*b^2*(a - b*x^4)^(1/4))/(663*a^2*x^9)