Integrand size = 16, antiderivative size = 97 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^8} \, dx=-\frac {a \sqrt {a-b x^4}}{7 x^7}+\frac {3 b \sqrt {a-b x^4}}{7 x^3}+\frac {4 \sqrt [4]{a} b^{7/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{7 \sqrt {a-b x^4}} \] Output:
-1/7*a*(-b*x^4+a)^(1/2)/x^7+3/7*b*(-b*x^4+a)^(1/2)/x^3+4/7*a^(1/4)*b^(7/4) *(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/(-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 = 53, normalized size of antiderivative = 0.55 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^8} \, dx=-\frac {a \sqrt {a-b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {3}{2},-\frac {3}{4},\frac {b x^4}{a}\right )}{7 x^7 \sqrt {1-\frac {b x^4}{a}}} \] Input:
Integrate[(a - b*x^4)^(3/2)/x^8,x]
Output:
-1/7*(a*Sqrt[a - b*x^4]*Hypergeometric2F1[-7/4, -3/2, -3/4, (b*x^4)/a])/(x ^7*Sqrt[1 - (b*x^4)/a])
Time = 0.35 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {809, 809, 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.
\(\displaystyle \int \frac {\left (a-b x^4\right )^{3/2}}{x^8} \, dx\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\frac {6}{7} b \int \frac {\sqrt {a-b x^4}}{x^4}dx-\frac {\left (a-b x^4\right )^{3/2}}{7 x^7}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\frac {6}{7} b \left (-\frac {2}{3} b \int \frac {1}{\sqrt {a-b x^4}}dx-\frac {\sqrt {a-b x^4}}{3 x^3}\right )-\frac {\left (a-b x^4\right )^{3/2}}{7 x^7}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle -\frac {6}{7} b \left (-\frac {2 b \sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{3 \sqrt {a-b x^4}}-\frac {\sqrt {a-b x^4}}{3 x^3}\right )-\frac {\left (a-b x^4\right )^{3/2}}{7 x^7}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -\frac {6}{7} b \left (-\frac {2 \sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 \sqrt {a-b x^4}}-\frac {\sqrt {a-b x^4}}{3 x^3}\right )-\frac {\left (a-b x^4\right )^{3/2}}{7 x^7}\) |
Input:
Int[(a - b*x^4)^(3/2)/x^8,x]
Output:
-1/7*(a - b*x^4)^(3/2)/x^7 - (6*b*(-1/3*Sqrt[a - b*x^4]/x^3 - (2*a^(1/4)*b ^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(3* 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]
Time = 0.93 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95
method | result | size |
risch | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (-3 b \,x^{4}+a \right )}{7 x^{7}}+\frac {4 b^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(92\) |
default | \(-\frac {a \sqrt {-b \,x^{4}+a}}{7 x^{7}}+\frac {3 b \sqrt {-b \,x^{4}+a}}{7 x^{3}}+\frac {4 b^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(101\) |
elliptic | \(-\frac {a \sqrt {-b \,x^{4}+a}}{7 x^{7}}+\frac {3 b \sqrt {-b \,x^{4}+a}}{7 x^{3}}+\frac {4 b^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(101\) |
Input:
int((-b*x^4+a)^(3/2)/x^8,x,method=_RETURNVERBOSE)
Output:
-1/7*(-b*x^4+a)^(1/2)*(-3*b*x^4+a)/x^7+4/7*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 = 55, normalized size of antiderivative = 0.57 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^8} \, dx=\frac {4 \, \sqrt {a} b x^{7} \left (\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (3 \, b x^{4} - a\right )} \sqrt {-b x^{4} + a}}{7 \, x^{7}} \] Input:
integrate((-b*x^4+a)^(3/2)/x^8,x, algorithm="fricas")
Output:
1/7*(4*sqrt(a)*b*x^7*(b/a)^(3/4)*elliptic_f(arcsin(x*(b/a)^(1/4)), -1) + ( 3*b*x^4 - a)*sqrt(-b*x^4 + a))/x^7
Time = 0.67 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.38 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^8} \, dx=- \frac {i b^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \] Input:
integrate((-b*x**4+a)**(3/2)/x**8,x)
Output:
-I*b**(3/2)*gamma(-1/4)*hyper((-3/2, 1/4), (5/4,), a/(b*x**4))/(4*x*gamma( 3/4))
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \] Input:
integrate((-b*x^4+a)^(3/2)/x^8,x, algorithm="maxima")
Output:
integrate((-b*x^4 + a)^(3/2)/x^8, x)
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \] Input:
integrate((-b*x^4+a)^(3/2)/x^8,x, algorithm="giac")
Output:
integrate((-b*x^4 + a)^(3/2)/x^8, x)
Timed out. \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^8} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{3/2}}{x^8} \,d x \] Input:
int((a - b*x^4)^(3/2)/x^8,x)
Output:
int((a - b*x^4)^(3/2)/x^8, x)
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^8} \, dx=\frac {\sqrt {-b \,x^{4}+a}\, a +5 \sqrt {-b \,x^{4}+a}\, b \,x^{4}+12 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{12}+a \,x^{8}}d x \right ) a^{2} x^{7}}{5 x^{7}} \] Input:
int((-b*x^4+a)^(3/2)/x^8,x)
Output:
(sqrt(a - b*x**4)*a + 5*sqrt(a - b*x**4)*b*x**4 + 12*int(sqrt(a - b*x**4)/ (a*x**8 - b*x**12),x)*a**2*x**7)/(5*x**7)