3.1255 \(\int \frac{1}{x^{14} (a-b x^4)^{3/4}} \, dx\)
Optimal. Leaf size=96 \[ -\frac{128 b^3 \sqrt [4]{a-b x^4}}{195 a^4 x}-\frac{32 b^2 \sqrt [4]{a-b x^4}}{195 a^3 x^5}-\frac{4 b \sqrt [4]{a-b x^4}}{39 a^2 x^9}-\frac{\sqrt [4]{a-b x^4}}{13 a x^{13}} \]
[Out]
-(a - b*x^4)^(1/4)/(13*a*x^13) - (4*b*(a - b*x^4)^(1/4))/(39*a^2*x^9) - (32*b^2*(a - b*x^4)^(1/4))/(195*a^3*x^
5) - (128*b^3*(a - b*x^4)^(1/4))/(195*a^4*x)
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Rubi [A] time = 0.0295717, antiderivative size = 96, normalized size of antiderivative = 1.,
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} \[ -\frac{128 b^3 \sqrt [4]{a-b x^4}}{195 a^4 x}-\frac{32 b^2 \sqrt [4]{a-b x^4}}{195 a^3 x^5}-\frac{4 b \sqrt [4]{a-b x^4}}{39 a^2 x^9}-\frac{\sqrt [4]{a-b x^4}}{13 a x^{13}} \]
Antiderivative was successfully verified.
[In]
Int[1/(x^14*(a - b*x^4)^(3/4)),x]
[Out]
-(a - b*x^4)^(1/4)/(13*a*x^13) - (4*b*(a - b*x^4)^(1/4))/(39*a^2*x^9) - (32*b^2*(a - b*x^4)^(1/4))/(195*a^3*x^
5) - (128*b^3*(a - b*x^4)^(1/4))/(195*a^4*x)
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^{14} \left (a-b x^4\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a-b x^4}}{13 a x^{13}}+\frac{(12 b) \int \frac{1}{x^{10} \left (a-b x^4\right )^{3/4}} \, dx}{13 a}\\ &=-\frac{\sqrt [4]{a-b x^4}}{13 a x^{13}}-\frac{4 b \sqrt [4]{a-b x^4}}{39 a^2 x^9}+\frac{\left (32 b^2\right ) \int \frac{1}{x^6 \left (a-b x^4\right )^{3/4}} \, dx}{39 a^2}\\ &=-\frac{\sqrt [4]{a-b x^4}}{13 a x^{13}}-\frac{4 b \sqrt [4]{a-b x^4}}{39 a^2 x^9}-\frac{32 b^2 \sqrt [4]{a-b x^4}}{195 a^3 x^5}+\frac{\left (128 b^3\right ) \int \frac{1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx}{195 a^3}\\ &=-\frac{\sqrt [4]{a-b x^4}}{13 a x^{13}}-\frac{4 b \sqrt [4]{a-b x^4}}{39 a^2 x^9}-\frac{32 b^2 \sqrt [4]{a-b x^4}}{195 a^3 x^5}-\frac{128 b^3 \sqrt [4]{a-b x^4}}{195 a^4 x}\\ \end{align*}
Mathematica [A] time = 0.018778, size = 54, normalized size = 0.56 \[ -\frac{\sqrt [4]{a-b x^4} \left (20 a^2 b x^4+15 a^3+32 a b^2 x^8+128 b^3 x^{12}\right )}{195 a^4 x^{13}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x^14*(a - b*x^4)^(3/4)),x]
[Out]
-((a - b*x^4)^(1/4)*(15*a^3 + 20*a^2*b*x^4 + 32*a*b^2*x^8 + 128*b^3*x^12))/(195*a^4*x^13)
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Maple [A] time = 0.007, size = 51, normalized size = 0.5 \begin{align*} -{\frac{128\,{b}^{3}{x}^{12}+32\,a{b}^{2}{x}^{8}+20\,{a}^{2}b{x}^{4}+15\,{a}^{3}}{195\,{x}^{13}{a}^{4}}\sqrt [4]{-b{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^14/(-b*x^4+a)^(3/4),x)
[Out]
-1/195*(-b*x^4+a)^(1/4)*(128*b^3*x^12+32*a*b^2*x^8+20*a^2*b*x^4+15*a^3)/x^13/a^4
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Maxima [A] time = 1.19611, size = 99, normalized size = 1.03 \begin{align*} -\frac{\frac{195 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{3}}{x} + \frac{117 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} b^{2}}{x^{5}} + \frac{65 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}} b}{x^{9}} + \frac{15 \,{\left (-b x^{4} + a\right )}^{\frac{13}{4}}}{x^{13}}}{195 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^14/(-b*x^4+a)^(3/4),x, algorithm="maxima")
[Out]
-1/195*(195*(-b*x^4 + a)^(1/4)*b^3/x + 117*(-b*x^4 + a)^(5/4)*b^2/x^5 + 65*(-b*x^4 + a)^(9/4)*b/x^9 + 15*(-b*x
^4 + a)^(13/4)/x^13)/a^4
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Fricas [A] time = 1.69329, size = 124, normalized size = 1.29 \begin{align*} -\frac{{\left (128 \, b^{3} x^{12} + 32 \, a b^{2} x^{8} + 20 \, a^{2} b x^{4} + 15 \, a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{195 \, a^{4} x^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^14/(-b*x^4+a)^(3/4),x, algorithm="fricas")
[Out]
-1/195*(128*b^3*x^12 + 32*a*b^2*x^8 + 20*a^2*b*x^4 + 15*a^3)*(-b*x^4 + a)^(1/4)/(a^4*x^13)
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Sympy [C] time = 6.44405, size = 1931, normalized size = 20.11 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**14/(-b*x**4+a)**(3/4),x)
[Out]
Piecewise((45*a**6*b**(37/4)*(a/(b*x**4) - 1)**(1/4)*exp(3*I*pi/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*exp(3*I*
pi/4)*gamma(3/4) + 768*a**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/4) - 768*a**5*b**11*x**20*exp(3*I*pi/4)*gamma(3/
4) + 256*a**4*b**12*x**24*exp(3*I*pi/4)*gamma(3/4)) - 75*a**5*b**(41/4)*x**4*(a/(b*x**4) - 1)**(1/4)*exp(3*I*p
i/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*exp(3*I*pi/4)*gamma(3/4) + 768*a**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/
4) - 768*a**5*b**11*x**20*exp(3*I*pi/4)*gamma(3/4) + 256*a**4*b**12*x**24*exp(3*I*pi/4)*gamma(3/4)) + 51*a**4*
b**(45/4)*x**8*(a/(b*x**4) - 1)**(1/4)*exp(3*I*pi/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*exp(3*I*pi/4)*gamma(3/
4) + 768*a**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/4) - 768*a**5*b**11*x**20*exp(3*I*pi/4)*gamma(3/4) + 256*a**4*
b**12*x**24*exp(3*I*pi/4)*gamma(3/4)) + 231*a**3*b**(49/4)*x**12*(a/(b*x**4) - 1)**(1/4)*exp(3*I*pi/4)*gamma(-
13/4)/(-256*a**7*b**9*x**12*exp(3*I*pi/4)*gamma(3/4) + 768*a**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/4) - 768*a**
5*b**11*x**20*exp(3*I*pi/4)*gamma(3/4) + 256*a**4*b**12*x**24*exp(3*I*pi/4)*gamma(3/4)) - 924*a**2*b**(53/4)*x
**16*(a/(b*x**4) - 1)**(1/4)*exp(3*I*pi/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*exp(3*I*pi/4)*gamma(3/4) + 768*a
**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/4) - 768*a**5*b**11*x**20*exp(3*I*pi/4)*gamma(3/4) + 256*a**4*b**12*x**2
4*exp(3*I*pi/4)*gamma(3/4)) + 1056*a*b**(57/4)*x**20*(a/(b*x**4) - 1)**(1/4)*exp(3*I*pi/4)*gamma(-13/4)/(-256*
a**7*b**9*x**12*exp(3*I*pi/4)*gamma(3/4) + 768*a**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/4) - 768*a**5*b**11*x**2
0*exp(3*I*pi/4)*gamma(3/4) + 256*a**4*b**12*x**24*exp(3*I*pi/4)*gamma(3/4)) - 384*b**(61/4)*x**24*(a/(b*x**4)
- 1)**(1/4)*exp(3*I*pi/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*exp(3*I*pi/4)*gamma(3/4) + 768*a**6*b**10*x**16*e
xp(3*I*pi/4)*gamma(3/4) - 768*a**5*b**11*x**20*exp(3*I*pi/4)*gamma(3/4) + 256*a**4*b**12*x**24*exp(3*I*pi/4)*g
amma(3/4)), Abs(a)/(Abs(b)*Abs(x**4)) > 1), (-45*a**6*b**(37/4)*(-a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(-256*a*
*7*b**9*x**12*exp(3*I*pi/4)*gamma(3/4) + 768*a**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/4) - 768*a**5*b**11*x**20*
exp(3*I*pi/4)*gamma(3/4) + 256*a**4*b**12*x**24*exp(3*I*pi/4)*gamma(3/4)) + 75*a**5*b**(41/4)*x**4*(-a/(b*x**4
) + 1)**(1/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*exp(3*I*pi/4)*gamma(3/4) + 768*a**6*b**10*x**16*exp(3*I*pi/4)
*gamma(3/4) - 768*a**5*b**11*x**20*exp(3*I*pi/4)*gamma(3/4) + 256*a**4*b**12*x**24*exp(3*I*pi/4)*gamma(3/4)) -
51*a**4*b**(45/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*exp(3*I*pi/4)*gamma(3/4) +
768*a**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/4) - 768*a**5*b**11*x**20*exp(3*I*pi/4)*gamma(3/4) + 256*a**4*b**1
2*x**24*exp(3*I*pi/4)*gamma(3/4)) - 231*a**3*b**(49/4)*x**12*(-a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(-256*a**7*
b**9*x**12*exp(3*I*pi/4)*gamma(3/4) + 768*a**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/4) - 768*a**5*b**11*x**20*exp
(3*I*pi/4)*gamma(3/4) + 256*a**4*b**12*x**24*exp(3*I*pi/4)*gamma(3/4)) + 924*a**2*b**(53/4)*x**16*(-a/(b*x**4)
+ 1)**(1/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*exp(3*I*pi/4)*gamma(3/4) + 768*a**6*b**10*x**16*exp(3*I*pi/4)*
gamma(3/4) - 768*a**5*b**11*x**20*exp(3*I*pi/4)*gamma(3/4) + 256*a**4*b**12*x**24*exp(3*I*pi/4)*gamma(3/4)) -
1056*a*b**(57/4)*x**20*(-a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(-256*a**7*b**9*x**12*exp(3*I*pi/4)*gamma(3/4) +
768*a**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/4) - 768*a**5*b**11*x**20*exp(3*I*pi/4)*gamma(3/4) + 256*a**4*b**12
*x**24*exp(3*I*pi/4)*gamma(3/4)) + 384*b**(61/4)*x**24*(-a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(-256*a**7*b**9*x
**12*exp(3*I*pi/4)*gamma(3/4) + 768*a**6*b**10*x**16*exp(3*I*pi/4)*gamma(3/4) - 768*a**5*b**11*x**20*exp(3*I*p
i/4)*gamma(3/4) + 256*a**4*b**12*x**24*exp(3*I*pi/4)*gamma(3/4)), True))
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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{3}{4}} x^{14}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^14/(-b*x^4+a)^(3/4),x, algorithm="giac")
[Out]
integrate(1/((-b*x^4 + a)^(3/4)*x^14), x)