\(\int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx\) [1194]

Optimal result

Integrand size = 16, antiderivative size = 71 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx=-\frac {\left (a-b x^4\right )^{5/4}}{13 a x^{13}}-\frac {8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5} \]

[Out]

-1/13*(-b*x^4+a)^(5/4)/a/x^13-8/117*b*(-b*x^4+a)^(5/4)/a^2/x^9-32/585*b^2*(-b*x^4+a)^(5/4)/a^3/x^5

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {277, 270} \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx=-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5}-\frac {8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac {\left (a-b x^4\right )^{5/4}}{13 a x^{13}} \]

[In]

Int[(a - b*x^4)^(1/4)/x^14,x]

[Out]

-1/13*(a - b*x^4)^(5/4)/(a*x^13) - (8*b*(a - b*x^4)^(5/4))/(117*a^2*x^9) - (32*b^2*(a - b*x^4)^(5/4))/(585*a^3
*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}}{13 a x^{13}}+\frac {(8 b) \int \frac {\sqrt [4]{a-b x^4}}{x^{10}} \, dx}{13 a} \\ & = -\frac {\left (a-b x^4\right )^{5/4}}{13 a x^{13}}-\frac {8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}+\frac {\left (32 b^2\right ) \int \frac {\sqrt [4]{a-b x^4}}{x^6} \, dx}{117 a^2} \\ & = -\frac {\left (a-b x^4\right )^{5/4}}{13 a x^{13}}-\frac {8 b \left (a-b x^4\right )^{5/4}}{117 a^2 x^9}-\frac {32 b^2 \left (a-b x^4\right )^{5/4}}{585 a^3 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx=\frac {\left (a-b x^4\right )^{5/4} \left (-45 a^2-40 a b x^4-32 b^2 x^8\right )}{585 a^3 x^{13}} \]

[In]

Integrate[(a - b*x^4)^(1/4)/x^14,x]

[Out]

((a - b*x^4)^(5/4)*(-45*a^2 - 40*a*b*x^4 - 32*b^2*x^8))/(585*a^3*x^13)

Maple [A] (verified)

Time = 4.50 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.56

method	result	size
gosper	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (32 b^{2} x^{8}+40 a b \,x^{4}+45 a^{2}\right )}{585 x^{13} a^{3}}\)	\(40\)
pseudoelliptic	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (32 b^{2} x^{8}+40 a b \,x^{4}+45 a^{2}\right )}{585 x^{13} a^{3}}\)	\(40\)
trager	\(-\frac {\left (-32 b^{3} x^{12}-8 a \,b^{2} x^{8}-5 a^{2} b \,x^{4}+45 a^{3}\right ) \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{585 x^{13} a^{3}}\)	\(51\)
risch	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} {\left (\left (-b \,x^{4}+a \right )^{3}\right )}^{\frac {1}{4}} \left (-32 b^{3} x^{12}-8 a \,b^{2} x^{8}-5 a^{2} b \,x^{4}+45 a^{3}\right )}{585 x^{13} {\left (-\left (b \,x^{4}-a \right )^{3}\right )}^{\frac {1}{4}} a^{3}}\)	\(78\)

[In]

int((-b*x^4+a)^(1/4)/x^14,x,method=_RETURNVERBOSE)

[Out]

-1/585*(-b*x^4+a)^(5/4)*(32*b^2*x^8+40*a*b*x^4+45*a^2)/x^13/a^3

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx=\frac {{\left (32 \, b^{3} x^{12} + 8 \, a b^{2} x^{8} + 5 \, a^{2} b x^{4} - 45 \, a^{3}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{585 \, a^{3} x^{13}} \]

[In]

integrate((-b*x^4+a)^(1/4)/x^14,x, algorithm="fricas")

[Out]

1/585*(32*b^3*x^12 + 8*a*b^2*x^8 + 5*a^2*b*x^4 - 45*a^3)*(-b*x^4 + a)^(1/4)/(a^3*x^13)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.06 (sec) , antiderivative size = 1090, normalized size of antiderivative = 15.35 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx=\text {Too large to display} \]

[In]

integrate((-b*x**4+a)**(1/4)/x**14,x)

[Out]

Piecewise((45*a**5*b**(17/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b
**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 95*a**4*b**(21/4)*x**4*(a/(b*x**4) - 1)**(1/4)*gamma
(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) +
47*a**3*b**(25/4)*x**8*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x*
*16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 21*a**2*b**(29/4)*x**12*(a/(b*x**4) - 1)**(1/4)*gamma(-13/
4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 56*a*
b**(33/4)*x**16*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gam
ma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 32*b**(37/4)*x**20*(a/(b*x**4) - 1)**(1/4)*gamma(-13/4)/(64*a**5*
b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)), Abs(a/(b*x**4)) >
1), (45*a**5*b**(17/4)*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128
*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 95*a**4*b**(21/4)*x**4*(-a/(b*x**4) + 1)**(1/
4)*exp(I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x
**20*gamma(-1/4)) + 47*a**3*b**(25/4)*x**8*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**
12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 21*a**2*b**(29/4)*x**12*(
-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-
1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 56*a*b**(33/4)*x**16*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-13/4
)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) - 32*b**
(37/4)*x**20*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma(-1/4) - 128*a**4*b**5
*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx=-\frac {\frac {117 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} b^{2}}{x^{5}} + \frac {130 \, {\left (-b x^{4} + a\right )}^{\frac {9}{4}} b}{x^{9}} + \frac {45 \, {\left (-b x^{4} + a\right )}^{\frac {13}{4}}}{x^{13}}}{585 \, a^{3}} \]

[In]

integrate((-b*x^4+a)^(1/4)/x^14,x, algorithm="maxima")

[Out]

-1/585*(117*(-b*x^4 + a)^(5/4)*b^2/x^5 + 130*(-b*x^4 + a)^(9/4)*b/x^9 + 45*(-b*x^4 + a)^(13/4)/x^13)/a^3

Giac [F]

\[ \int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{14}} \,d x } \]

[In]

integrate((-b*x^4+a)^(1/4)/x^14,x, algorithm="giac")

[Out]

integrate((-b*x^4 + a)^(1/4)/x^14, x)

Mupad [B] (verification not implemented)

Time = 6.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^{14}} \, dx=\frac {b\,{\left (a-b\,x^4\right )}^{1/4}}{117\,a\,x^9}-\frac {{\left (a-b\,x^4\right )}^{1/4}}{13\,x^{13}}+\frac {32\,b^3\,{\left (a-b\,x^4\right )}^{1/4}}{585\,a^3\,x}+\frac {8\,b^2\,{\left (a-b\,x^4\right )}^{1/4}}{585\,a^2\,x^5} \]

[In]

int((a - b*x^4)^(1/4)/x^14,x)

[Out]

(b*(a - b*x^4)^(1/4))/(117*a*x^9) - (a - b*x^4)^(1/4)/(13*x^13) + (32*b^3*(a - b*x^4)^(1/4))/(585*a^3*x) + (8*
b^2*(a - b*x^4)^(1/4))/(585*a^2*x^5)