Integrand size = 16, antiderivative size = 99 \[ \int \frac {\sqrt {a-b x^4}}{x^8} \, dx=-\frac {\sqrt {a-b x^4}}{7 x^7}+\frac {2 b \sqrt {a-b x^4}}{21 a x^3}-\frac {2 b^{7/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{21 a^{3/4} \sqrt {a-b x^4}} \] Output:
-1/7*(-b*x^4+a)^(1/2)/x^7+2/21*b*(-b*x^4+a)^(1/2)/a/x^3-2/21*b^(7/4)*(1-b* x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/a^(3/4)/(-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {a-b x^4}}{x^8} \, dx=-\frac {\sqrt {a-b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {1}{2},-\frac {3}{4},\frac {b x^4}{a}\right )}{7 x^7 \sqrt {1-\frac {b x^4}{a}}} \] Input:
Integrate[Sqrt[a - b*x^4]/x^8,x]
Output:
-1/7*(Sqrt[a - b*x^4]*Hypergeometric2F1[-7/4, -1/2, -3/4, (b*x^4)/a])/(x^7 *Sqrt[1 - (b*x^4)/a])
Time = 0.36 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {809, 847, 765, 762}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {a-b x^4}}{x^8} \, dx\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\frac {2}{7} b \int \frac {1}{x^4 \sqrt {a-b x^4}}dx-\frac {\sqrt {a-b x^4}}{7 x^7}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {2}{7} b \left (\frac {b \int \frac {1}{\sqrt {a-b x^4}}dx}{3 a}-\frac {\sqrt {a-b x^4}}{3 a x^3}\right )-\frac {\sqrt {a-b x^4}}{7 x^7}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle -\frac {2}{7} b \left (\frac {b \sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{3 a \sqrt {a-b x^4}}-\frac {\sqrt {a-b x^4}}{3 a x^3}\right )-\frac {\sqrt {a-b x^4}}{7 x^7}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -\frac {2}{7} b \left (\frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 a^{3/4} \sqrt {a-b x^4}}-\frac {\sqrt {a-b x^4}}{3 a x^3}\right )-\frac {\sqrt {a-b x^4}}{7 x^7}\) |
Input:
Int[Sqrt[a - b*x^4]/x^8,x]
Output:
-1/7*Sqrt[a - b*x^4]/x^7 - (2*b*(-1/3*Sqrt[a - b*x^4]/(a*x^3) + (b^(3/4)*S qrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(3*a^(3/4)* Sqrt[a - b*x^4])))/7
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
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] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Time = 0.89 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.01
method | result | size |
risch | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (-2 b \,x^{4}+3 a \right )}{21 x^{7} a}-\frac {2 b^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(100\) |
default | \(-\frac {\sqrt {-b \,x^{4}+a}}{7 x^{7}}+\frac {2 b \sqrt {-b \,x^{4}+a}}{21 a \,x^{3}}-\frac {2 b^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(106\) |
elliptic | \(-\frac {\sqrt {-b \,x^{4}+a}}{7 x^{7}}+\frac {2 b \sqrt {-b \,x^{4}+a}}{21 a \,x^{3}}-\frac {2 b^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(106\) |
Input:
int((-b*x^4+a)^(1/2)/x^8,x,method=_RETURNVERBOSE)
Output:
-1/21*(-b*x^4+a)^(1/2)*(-2*b*x^4+3*a)/x^7/a-2/21/a*b^2/(1/a^(1/2)*b^(1/2)) ^(1/2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4 +a)^(1/2)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {a-b x^4}}{x^8} \, dx=-\frac {2 \, \sqrt {a} b x^{7} \left (\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (2 \, b x^{4} - 3 \, a\right )} \sqrt {-b x^{4} + a}}{21 \, a x^{7}} \] Input:
integrate((-b*x^4+a)^(1/2)/x^8,x, algorithm="fricas")
Output:
-1/21*(2*sqrt(a)*b*x^7*(b/a)^(3/4)*elliptic_f(arcsin(x*(b/a)^(1/4)), -1) - (2*b*x^4 - 3*a)*sqrt(-b*x^4 + a))/(a*x^7)
Time = 0.70 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {a-b x^4}}{x^8} \, dx=\frac {\sqrt {a} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} \] Input:
integrate((-b*x**4+a)**(1/2)/x**8,x)
Output:
sqrt(a)*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), b*x**4*exp_polar(2*I*pi)/ a)/(4*x**7*gamma(-3/4))
\[ \int \frac {\sqrt {a-b x^4}}{x^8} \, dx=\int { \frac {\sqrt {-b x^{4} + a}}{x^{8}} \,d x } \] Input:
integrate((-b*x^4+a)^(1/2)/x^8,x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^4 + a)/x^8, x)
\[ \int \frac {\sqrt {a-b x^4}}{x^8} \, dx=\int { \frac {\sqrt {-b x^{4} + a}}{x^{8}} \,d x } \] Input:
integrate((-b*x^4+a)^(1/2)/x^8,x, algorithm="giac")
Output:
integrate(sqrt(-b*x^4 + a)/x^8, x)
Timed out. \[ \int \frac {\sqrt {a-b x^4}}{x^8} \, dx=\int \frac {\sqrt {a-b\,x^4}}{x^8} \,d x \] Input:
int((a - b*x^4)^(1/2)/x^8,x)
Output:
int((a - b*x^4)^(1/2)/x^8, x)
\[ \int \frac {\sqrt {a-b x^4}}{x^8} \, dx=\frac {-\sqrt {-b \,x^{4}+a}-2 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{12}+a \,x^{8}}d x \right ) a \,x^{7}}{5 x^{7}} \] Input:
int((-b*x^4+a)^(1/2)/x^8,x)
Output:
( - sqrt(a - b*x**4) - 2*int(sqrt(a - b*x**4)/(a*x**8 - b*x**12),x)*a*x**7 )/(5*x**7)