\(\int \frac {1}{x^6 (a-b x^2)^{5/4}} \, dx\) [912]

Optimal result

Integrand size = 16, antiderivative size = 151 \[ \int \frac {1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx=\frac {2}{a x^5 \sqrt [4]{a-b x^2}}-\frac {11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac {77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac {77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac {77 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a-b x^2}} \] Output:

2/a/x^5/(-b*x^2+a)^(1/4)-11/5*(-b*x^2+a)^(3/4)/a^2/x^5-77/30*b*(-b*x^2+a)^ 
(3/4)/a^3/x^3-77/20*b^2*(-b*x^2+a)^(3/4)/a^4/x-77/20*b^(5/2)*(1-b*x^2/a)^( 
1/4)*EllipticE(sin(1/2*arcsin(b^(1/2)*x/a^(1/2))),2^(1/2))/a^(7/2)/(-b*x^2 
+a)^(1/4)
 

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 = 0.36 \[ \int \frac {1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx=-\frac {\sqrt [4]{1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {5}{4},-\frac {3}{2},\frac {b x^2}{a}\right )}{5 a x^5 \sqrt [4]{a-b x^2}} \] Input:

Integrate[1/(x^6*(a - b*x^2)^(5/4)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {253, 264, 264, 264, 227, 226}

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.

\(\displaystyle \int \frac {1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {11 \int \frac {1}{x^6 \sqrt [4]{a-b x^2}}dx}{a}+\frac {2}{a x^5 \sqrt [4]{a-b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {7 b \int \frac {1}{x^4 \sqrt [4]{a-b x^2}}dx}{10 a}-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}\right )}{a}+\frac {2}{a x^5 \sqrt [4]{a-b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {7 b \left (\frac {b \int \frac {1}{x^2 \sqrt [4]{a-b x^2}}dx}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{3 a x^3}\right )}{10 a}-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}\right )}{a}+\frac {2}{a x^5 \sqrt [4]{a-b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {11 \left (\frac {7 b \left (\frac {b \left (-\frac {b \int \frac {1}{\sqrt [4]{a-b x^2}}dx}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{a x}\right )}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{3 a x^3}\right )}{10 a}-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}\right )}{a}+\frac {2}{a x^5 \sqrt [4]{a-b x^2}}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {11 \left (\frac {7 b \left (\frac {b \left (-\frac {b \sqrt [4]{1-\frac {b x^2}{a}} \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{2 a \sqrt [4]{a-b x^2}}-\frac {\left (a-b x^2\right )^{3/4}}{a x}\right )}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{3 a x^3}\right )}{10 a}-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}\right )}{a}+\frac {2}{a x^5 \sqrt [4]{a-b x^2}}\)

\(\Big \downarrow \) 226

\(\displaystyle \frac {11 \left (\frac {7 b \left (\frac {b \left (-\frac {\sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a-b x^2}}-\frac {\left (a-b x^2\right )^{3/4}}{a x}\right )}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{3 a x^3}\right )}{10 a}-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}\right )}{a}+\frac {2}{a x^5 \sqrt [4]{a-b x^2}}\)

Input:

Int[1/(x^6*(a - b*x^2)^(5/4)),x]
 

Output:

2/(a*x^5*(a - b*x^2)^(1/4)) + (11*(-1/5*(a - b*x^2)^(3/4)/(a*x^5) + (7*b*( 
-1/3*(a - b*x^2)^(3/4)/(a*x^3) + (b*(-((a - b*x^2)^(3/4)/(a*x)) - (Sqrt[b] 
*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(Sqrt[ 
a]*(a - b*x^2)^(1/4))))/(2*a)))/(10*a)))/a
 

Defintions of rubi rules used

rule 226

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] 
))*EllipticE[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ 
[a, 0] && NegQ[b/a]
 

rule 227

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( 
a + b*x^2)^(1/4)   Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] 
&& PosQ[a]
 

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

Maple [F]

\[\int \frac {1}{x^{6} \left (-b \,x^{2}+a \right )^{\frac {5}{4}}}d x\]

Input:

int(1/x^6/(-b*x^2+a)^(5/4),x)
 

Output:

int(1/x^6/(-b*x^2+a)^(5/4),x)
 

Fricas [F]

\[ \int \frac {1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}} x^{6}} \,d x } \] Input:

integrate(1/x^6/(-b*x^2+a)^(5/4),x, algorithm="fricas")
 

Output:

integral((-b*x^2 + a)^(3/4)/(b^2*x^10 - 2*a*b*x^8 + a^2*x^6), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {5}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5 a^{\frac {5}{4}} x^{5}} \] Input:

integrate(1/x**6/(-b*x**2+a)**(5/4),x)
 

Output:

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

Maxima [F]

\[ \int \frac {1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}} x^{6}} \,d x } \] Input:

integrate(1/x^6/(-b*x^2+a)^(5/4),x, algorithm="maxima")
 

Output:

integrate(1/((-b*x^2 + a)^(5/4)*x^6), x)
 

Giac [F]

\[ \int \frac {1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}} x^{6}} \,d x } \] Input:

integrate(1/x^6/(-b*x^2+a)^(5/4),x, algorithm="giac")
 

Output:

integrate(1/((-b*x^2 + a)^(5/4)*x^6), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx=\int \frac {1}{x^6\,{\left (a-b\,x^2\right )}^{5/4}} \,d x \] Input:

int(1/(x^6*(a - b*x^2)^(5/4)),x)
 

Output:

int(1/(x^6*(a - b*x^2)^(5/4)), x)
 

Reduce [F]

\[ \int \frac {1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx=\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}} a \,x^{6}-\left (-b \,x^{2}+a \right )^{\frac {1}{4}} b \,x^{8}}d x \] Input:

int(1/x^6/(-b*x^2+a)^(5/4),x)
 

Output:

int(1/((a - b*x**2)**(1/4)*a*x**6 - (a - b*x**2)**(1/4)*b*x**8),x)