\(\int \frac {1}{x^6 (a-b x^4)^{3/2}} \, dx\) [244]

Optimal result

Integrand size = 16, antiderivative size = 180 \[ \int \frac {1}{x^6 \left (a-b x^4\right )^{3/2}} \, dx=\frac {1}{2 a x^5 \sqrt {a-b x^4}}-\frac {7 \sqrt {a-b x^4}}{10 a^2 x^5}-\frac {21 b \sqrt {a-b x^4}}{10 a^3 x}-\frac {21 b^{5/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{10 a^{9/4} \sqrt {a-b x^4}}+\frac {21 b^{5/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{10 a^{9/4} \sqrt {a-b x^4}} \] Output:

1/2/a/x^5/(-b*x^4+a)^(1/2)-7/10*(-b*x^4+a)^(1/2)/a^2/x^5-21/10*b*(-b*x^4+a 
)^(1/2)/a^3/x-21/10*b^(5/4)*(1-b*x^4/a)^(1/2)*EllipticE(b^(1/4)*x/a^(1/4), 
I)/a^(9/4)/(-b*x^4+a)^(1/2)+21/10*b^(5/4)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1 
/4)*x/a^(1/4),I)/a^(9/4)/(-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 = 10.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.31 \[ \int \frac {1}{x^6 \left (a-b x^4\right )^{3/2}} \, dx=-\frac {\sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{2},-\frac {1}{4},\frac {b x^4}{a}\right )}{5 a x^5 \sqrt {a-b x^4}} \] Input:

Integrate[1/(x^6*(a - b*x^4)^(3/2)),x]
 

Output:

-1/5*(Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[-5/4, 3/2, -1/4, (b*x^4)/a])/( 
a*x^5*Sqrt[a - b*x^4])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {819, 847, 847, 836, 27, 765, 762, 1390, 1389, 327}

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.

\(\displaystyle \int \frac {1}{x^6 \left (a-b x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {7 \int \frac {1}{x^6 \sqrt {a-b x^4}}dx}{2 a}+\frac {1}{2 a x^5 \sqrt {a-b x^4}}\)

\(\Big \downarrow \) 847

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

\(\Big \downarrow \) 847

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

Int[1/(x^6*(a - b*x^4)^(3/2)),x]
 

Output:

1/(2*a*x^5*Sqrt[a - b*x^4]) + (7*(-1/5*Sqrt[a - b*x^4]/(a*x^5) + (3*b*(-(S 
qrt[a - b*x^4]/(a*x)) - (b*((a^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticE[ArcSin[ 
(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b*x^4]) - (a^(3/4)*Sqrt[1 - ( 
b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b* 
x^4])))/a))/(5*a)))/(2*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

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 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

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]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

Maple [A] (verified)

Time = 1.79 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {\sqrt {-b \,x^{4}+a}}{5 a^{2} x^{5}}-\frac {8 b \sqrt {-b \,x^{4}+a}}{5 a^{3} x}+\frac {b^{2} x^{3}}{2 a^{3} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {21 b^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{10 a^{\frac {5}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(152\)
elliptic	\(-\frac {\sqrt {-b \,x^{4}+a}}{5 a^{2} x^{5}}-\frac {8 b \sqrt {-b \,x^{4}+a}}{5 a^{3} x}+\frac {b^{2} x^{3}}{2 a^{3} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {21 b^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{10 a^{\frac {5}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(152\)
risch	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b \,x^{4}+a \right )}{5 a^{3} x^{5}}-\frac {b^{2} \left (-\frac {8 \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}+5 a \left (-\frac {x^{3}}{2 a \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {\sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\right )\right )}{5 a^{3}}\)	\(238\)

Input:

int(1/x^6/(-b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/5*(-b*x^4+a)^(1/2)/a^2/x^5-8/5*b*(-b*x^4+a)^(1/2)/a^3/x+1/2*b^2/a^3*x^3 
/(-(x^4-a/b)*b)^(1/2)+21/10*b^(3/2)/a^(5/2)/(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)-EllipticE(x*(1/a^(1/2)*b^(1/2))^( 
1/2),I))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^6 \left (a-b x^4\right )^{3/2}} \, dx=-\frac {21 \, {\left (b^{2} x^{9} - a b x^{5}\right )} \sqrt {a} \left (\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 21 \, {\left (b^{2} x^{9} - a b x^{5}\right )} \sqrt {a} \left (\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (21 \, b^{2} x^{8} - 14 \, a b x^{4} - 2 \, a^{2}\right )} \sqrt {-b x^{4} + a}}{10 \, {\left (a^{3} b x^{9} - a^{4} x^{5}\right )}} \] Input:

integrate(1/x^6/(-b*x^4+a)^(3/2),x, algorithm="fricas")
 

Output:

-1/10*(21*(b^2*x^9 - a*b*x^5)*sqrt(a)*(b/a)^(3/4)*elliptic_e(arcsin(x*(b/a 
)^(1/4)), -1) - 21*(b^2*x^9 - a*b*x^5)*sqrt(a)*(b/a)^(3/4)*elliptic_f(arcs 
in(x*(b/a)^(1/4)), -1) + (21*b^2*x^8 - 14*a*b*x^4 - 2*a^2)*sqrt(-b*x^4 + a 
))/(a^3*b*x^9 - a^4*x^5)
 

Sympy [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^6 \left (a-b x^4\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x^{5} \Gamma \left (- \frac {1}{4}\right )} \] Input:

integrate(1/x**6/(-b*x**4+a)**(3/2),x)
 

Output:

gamma(-5/4)*hyper((-5/4, 3/2), (-1/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a** 
(3/2)*x**5*gamma(-1/4))
 

Maxima [F]

\[ \int \frac {1}{x^6 \left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:

integrate(1/x^6/(-b*x^4+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((-b*x^4 + a)^(3/2)*x^6), x)
 

Giac [F]

\[ \int \frac {1}{x^6 \left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:

integrate(1/x^6/(-b*x^4+a)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((-b*x^4 + a)^(3/2)*x^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \left (a-b x^4\right )^{3/2}} \, dx=\int \frac {1}{x^6\,{\left (a-b\,x^4\right )}^{3/2}} \,d x \] Input:

int(1/(x^6*(a - b*x^4)^(3/2)),x)
 

Output:

int(1/(x^6*(a - b*x^4)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^6 \left (a-b x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{14}-2 a b \,x^{10}+a^{2} x^{6}}d x \] Input:

int(1/x^6/(-b*x^4+a)^(3/2),x)
 

Output:

int(sqrt(a - b*x**4)/(a**2*x**6 - 2*a*b*x**10 + b**2*x**14),x)