3.1195 \(\int \frac {\sqrt [4]{a-b x^4}}{x^{18}} \, dx\)
Optimal. Leaf size=96 \[ -\frac {128 b^3 \left (a-b x^4\right )^{5/4}}{3315 a^4 x^5}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{663 a^3 x^9}-\frac {12 b \left (a-b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {\left (a-b x^4\right )^{5/4}}{17 a x^{17}} \]
[Out]
-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
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Rubi [A] time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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 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]
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]
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*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 0.56 \[ -\frac {\left (a-b x^4\right )^{5/4} \left (195 a^3+180 a^2 b x^4+160 a b^2 x^8+128 b^3 x^{12}\right )}{3315 a^4 x^{17}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a - b*x^4)^(1/4)/x^18,x]
[Out]
-1/3315*((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))/(a^4*x^17)
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fricas [A] time = 1.19, size = 61, normalized size = 0.64 \[ \frac {{\left (128 \, b^{4} x^{16} + 32 \, a b^{3} x^{12} + 20 \, a^{2} b^{2} x^{8} + 15 \, a^{3} b x^{4} - 195 \, a^{4}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{3315 \, a^{4} x^{17}} \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{18}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-b*x^4+a)^(1/4)/x^18,x, algorithm="giac")
[Out]
integrate((-b*x^4 + a)^(1/4)/x^18, x)
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maple [A] time = 0.01, size = 51, normalized size = 0.53 \[ -\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (128 b^{3} x^{12}+160 a \,b^{2} x^{8}+180 a^{2} b \,x^{4}+195 a^{3}\right )}{3315 a^{4} x^{17}} \]
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 = 1.32, size = 73, normalized size = 0.76 \[ -\frac {\frac {663 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} b^{3}}{x^{5}} + \frac {1105 \, {\left (-b x^{4} + a\right )}^{\frac {9}{4}} b^{2}}{x^{9}} + \frac {765 \, {\left (-b x^{4} + a\right )}^{\frac {13}{4}} b}{x^{13}} + \frac {195 \, {\left (-b x^{4} + a\right )}^{\frac {17}{4}}}{x^{17}}}{3315 \, a^{4}} \]
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 + 1
95*(-b*x^4 + a)^(17/4)/x^17)/a^4
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mupad [B] time = 1.76, size = 98, normalized size = 1.02 \[ \frac {b\,{\left (a-b\,x^4\right )}^{1/4}}{221\,a\,x^{13}}-\frac {{\left (a-b\,x^4\right )}^{1/4}}{17\,x^{17}}+\frac {128\,b^4\,{\left (a-b\,x^4\right )}^{1/4}}{3315\,a^4\,x}+\frac {32\,b^3\,{\left (a-b\,x^4\right )}^{1/4}}{3315\,a^3\,x^5}+\frac {4\,b^2\,{\left (a-b\,x^4\right )}^{1/4}}{663\,a^2\,x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [B] time = 4.92, size = 1756, normalized size = 18.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[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))
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