\(\int \frac {1}{x^8 (a-b x^4)^{3/2}} \, dx\) [239]

Optimal result

Integrand size = 16, antiderivative size = 124 \[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/2}} \, dx=\frac {1}{2 a x^7 \sqrt {a-b x^4}}-\frac {9 \sqrt {a-b x^4}}{14 a^2 x^7}-\frac {15 b \sqrt {a-b x^4}}{14 a^3 x^3}+\frac {15 b^{7/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{14 a^{11/4} \sqrt {a-b x^4}} \] Output:

1/2/a/x^7/(-b*x^4+a)^(1/2)-9/14*(-b*x^4+a)^(1/2)/a^2/x^7-15/14*b*(-b*x^4+a 
)^(1/2)/a^3/x^3+15/14*b^(7/4)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4 
),I)/a^(11/4)/(-b*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/2}} \, dx=-\frac {\sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {3}{2},-\frac {3}{4},\frac {b x^4}{a}\right )}{7 a x^7 \sqrt {a-b x^4}} \] Input:

Integrate[1/(x^8*(a - b*x^4)^(3/2)),x]
 

Output:

-1/7*(Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[-7/4, 3/2, -3/4, (b*x^4)/a])/( 
a*x^7*Sqrt[a - b*x^4])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {819, 847, 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.

\(\displaystyle \int \frac {1}{x^8 \left (a-b x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {9 \int \frac {1}{x^8 \sqrt {a-b x^4}}dx}{2 a}+\frac {1}{2 a x^7 \sqrt {a-b x^4}}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {9 \left (\frac {5 b \int \frac {1}{x^4 \sqrt {a-b x^4}}dx}{7 a}-\frac {\sqrt {a-b x^4}}{7 a x^7}\right )}{2 a}+\frac {1}{2 a x^7 \sqrt {a-b x^4}}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {9 \left (\frac {5 b \left (\frac {b \int \frac {1}{\sqrt {a-b x^4}}dx}{3 a}-\frac {\sqrt {a-b x^4}}{3 a x^3}\right )}{7 a}-\frac {\sqrt {a-b x^4}}{7 a x^7}\right )}{2 a}+\frac {1}{2 a x^7 \sqrt {a-b x^4}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {9 \left (\frac {5 b \left (\frac {b \sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{3 a \sqrt {a-b x^4}}-\frac {\sqrt {a-b x^4}}{3 a x^3}\right )}{7 a}-\frac {\sqrt {a-b x^4}}{7 a x^7}\right )}{2 a}+\frac {1}{2 a x^7 \sqrt {a-b x^4}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {9 \left (\frac {5 b \left (\frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 a^{3/4} \sqrt {a-b x^4}}-\frac {\sqrt {a-b x^4}}{3 a x^3}\right )}{7 a}-\frac {\sqrt {a-b x^4}}{7 a x^7}\right )}{2 a}+\frac {1}{2 a x^7 \sqrt {a-b x^4}}\)

Input:

Int[1/(x^8*(a - b*x^4)^(3/2)),x]
 

Output:

1/(2*a*x^7*Sqrt[a - b*x^4]) + (9*(-1/7*Sqrt[a - b*x^4]/(a*x^7) + (5*b*(-1/ 
3*Sqrt[a - b*x^4]/(a*x^3) + (b^(3/4)*Sqrt[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*a)))/(2*a)
 

Defintions of rubi rules used

rule 762

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]
 

rule 765

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]
 

rule 819

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( 
c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 
1) + 1)/(a*n*(p + 1))   Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a 
, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 847

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]
 

Maple [A] (verified)

Time = 1.66 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07

method	result	size
default	\(-\frac {\sqrt {-b \,x^{4}+a}}{7 a^{2} x^{7}}-\frac {4 b \sqrt {-b \,x^{4}+a}}{7 a^{3} x^{3}}+\frac {b^{2} x}{2 a^{3} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {15 b^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{14 a^{3} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(133\)
elliptic	\(-\frac {\sqrt {-b \,x^{4}+a}}{7 a^{2} x^{7}}-\frac {4 b \sqrt {-b \,x^{4}+a}}{7 a^{3} x^{3}}+\frac {b^{2} x}{2 a^{3} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {15 b^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{14 a^{3} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(133\)
risch	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (4 b \,x^{4}+a \right )}{7 a^{3} x^{7}}+\frac {b^{2} \left (\frac {4 \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-7 a \left (-\frac {x}{2 a \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {\sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\right )}{7 a^{3}}\)	\(193\)

Input:

int(1/x^8/(-b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/7*(-b*x^4+a)^(1/2)/a^2/x^7-4/7*b*(-b*x^4+a)^(1/2)/a^3/x^3+1/2*b^2/a^3*x 
/(-(x^4-a/b)*b)^(1/2)+15/14*b^2/a^3/(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)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/2}} \, dx=\frac {15 \, {\left (b^{2} x^{11} - a b x^{7}\right )} \sqrt {a} \left (\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (15 \, b^{2} x^{8} - 6 \, a b x^{4} - 2 \, a^{2}\right )} \sqrt {-b x^{4} + a}}{14 \, {\left (a^{3} b x^{11} - a^{4} x^{7}\right )}} \] Input:

integrate(1/x^8/(-b*x^4+a)^(3/2),x, algorithm="fricas")
 

Output:

1/14*(15*(b^2*x^11 - a*b*x^7)*sqrt(a)*(b/a)^(3/4)*elliptic_f(arcsin(x*(b/a 
)^(1/4)), -1) - (15*b^2*x^8 - 6*a*b*x^4 - 2*a^2)*sqrt(-b*x^4 + a))/(a^3*b* 
x^11 - a^4*x^7)
 

Sympy [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.37 \[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {3}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x^{7} \Gamma \left (- \frac {3}{4}\right )} \] Input:

integrate(1/x**8/(-b*x**4+a)**(3/2),x)
 

Output:

gamma(-7/4)*hyper((-7/4, 3/2), (-3/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a** 
(3/2)*x**7*gamma(-3/4))
 

Maxima [F]

\[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} x^{8}} \,d x } \] Input:

integrate(1/x^8/(-b*x^4+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((-b*x^4 + a)^(3/2)*x^8), x)
 

Giac [F]

\[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} x^{8}} \,d x } \] Input:

integrate(1/x^8/(-b*x^4+a)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((-b*x^4 + a)^(3/2)*x^8), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/2}} \, dx=\int \frac {1}{x^8\,{\left (a-b\,x^4\right )}^{3/2}} \,d x \] Input:

int(1/(x^8*(a - b*x^4)^(3/2)),x)
 

Output:

int(1/(x^8*(a - b*x^4)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^8 \left (a-b x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{16}-2 a b \,x^{12}+a^{2} x^{8}}d x \] Input:

int(1/x^8/(-b*x^4+a)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int(sqrt(a - b*x**4)/(a**2*x**8 - 2*a*b*x**12 + b**2*x**16),x)