Integrand size = 15, antiderivative size = 131 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^2} \, dx=-\frac {1}{6 a \left (a+\frac {b}{x^4}\right )^{3/2} x}-\frac {5}{12 a^2 \sqrt {a+\frac {b}{x^4}} x}-\frac {5 \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{24 a^{9/4} \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}} \] Output:
-1/6/a/(a+b/x^4)^(3/2)/x-5/12/a^2/(a+b/x^4)^(1/2)/x-5/24*((a+b/x^4)/(a^(1/ 2)+b^(1/2)/x^2)^2)^(1/2)*(a^(1/2)+b^(1/2)/x^2)*InverseJacobiAM(2*arccot(a^ (1/4)*x/b^(1/4)),1/2*2^(1/2))/a^(9/4)/b^(1/4)/(a+b/x^4)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^2} \, dx=\frac {-5 b-7 a x^4+5 \left (b+a x^4\right ) \sqrt {1+\frac {a x^4}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {a x^4}{b}\right )}{12 a^2 \sqrt {a+\frac {b}{x^4}} x \left (b+a x^4\right )} \] Input:
Integrate[1/((a + b/x^4)^(5/2)*x^2),x]
Output:
(-5*b - 7*a*x^4 + 5*(b + a*x^4)*Sqrt[1 + (a*x^4)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((a*x^4)/b)])/(12*a^2*Sqrt[a + b/x^4]*x*(b + a*x^4))
Time = 0.38 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {858, 749, 749, 761}
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.
\(\displaystyle \int \frac {1}{x^2 \left (a+\frac {b}{x^4}\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2}}d\frac {1}{x}\) |
\(\Big \downarrow \) 749 |
\(\displaystyle -\frac {5 \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{3/2}}d\frac {1}{x}}{6 a}-\frac {1}{6 a x \left (a+\frac {b}{x^4}\right )^{3/2}}\) |
\(\Big \downarrow \) 749 |
\(\displaystyle -\frac {5 \left (\frac {\int \frac {1}{\sqrt {a+\frac {b}{x^4}}}d\frac {1}{x}}{2 a}+\frac {1}{2 a x \sqrt {a+\frac {b}{x^4}}}\right )}{6 a}-\frac {1}{6 a x \left (a+\frac {b}{x^4}\right )^{3/2}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle -\frac {5 \left (\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b}}{\sqrt [4]{a} x}\right ),\frac {1}{2}\right )}{4 a^{5/4} \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}}+\frac {1}{2 a x \sqrt {a+\frac {b}{x^4}}}\right )}{6 a}-\frac {1}{6 a x \left (a+\frac {b}{x^4}\right )^{3/2}}\) |
Input:
Int[1/((a + b/x^4)^(5/2)*x^2),x]
Output:
-1/6*1/(a*(a + b/x^4)^(3/2)*x) - (5*(1/(2*a*Sqrt[a + b/x^4]*x) + (Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*Ar cTan[b^(1/4)/(a^(1/4)*x)], 1/2])/(4*a^(5/4)*b^(1/4)*Sqrt[a + b/x^4])))/(6* a)
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.13
method | result | size |
default | \(-\frac {-5 \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) a^{2} x^{8}+7 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{2} x^{9}-10 \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) a b \,x^{4}+12 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a b \,x^{5}-5 \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) b^{2}+5 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{2} x}{12 a^{2} \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} x^{10} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}\) | \(279\) |
Input:
int(1/(a+b/x^4)^(5/2)/x^2,x,method=_RETURNVERBOSE)
Output:
-1/12*(-5*(-(I*a^(1/2)*x^2-b^(1/2))/b^(1/2))^(1/2)*((I*a^(1/2)*x^2+b^(1/2) )/b^(1/2))^(1/2)*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*a^2*x^8+7*(I*a^( 1/2)/b^(1/2))^(1/2)*a^2*x^9-10*(-(I*a^(1/2)*x^2-b^(1/2))/b^(1/2))^(1/2)*(( I*a^(1/2)*x^2+b^(1/2))/b^(1/2))^(1/2)*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2 ),I)*a*b*x^4+12*(I*a^(1/2)/b^(1/2))^(1/2)*a*b*x^5-5*(-(I*a^(1/2)*x^2-b^(1/ 2))/b^(1/2))^(1/2)*((I*a^(1/2)*x^2+b^(1/2))/b^(1/2))^(1/2)*EllipticF(x*(I* a^(1/2)/b^(1/2))^(1/2),I)*b^2+5*(I*a^(1/2)/b^(1/2))^(1/2)*b^2*x)/a^2/((a*x ^4+b)/x^4)^(5/2)/x^10/(I*a^(1/2)/b^(1/2))^(1/2)
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^2} \, dx=-\frac {5 \, {\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (7 \, a^{2} x^{7} + 5 \, a b x^{3}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{12 \, {\left (a^{5} x^{8} + 2 \, a^{4} b x^{4} + a^{3} b^{2}\right )}} \] Input:
integrate(1/(a+b/x^4)^(5/2)/x^2,x, algorithm="fricas")
Output:
-1/12*(5*(a^2*x^8 + 2*a*b*x^4 + b^2)*sqrt(b)*(-a/b)^(3/4)*elliptic_f(arcsi n(x*(-a/b)^(1/4)), -1) + (7*a^2*x^7 + 5*a*b*x^3)*sqrt((a*x^4 + b)/x^4))/(a ^5*x^8 + 2*a^4*b*x^4 + a^3*b^2)
Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^2} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac {5}{2}} x \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate(1/(a+b/x**4)**(5/2)/x**2,x)
Output:
-gamma(1/4)*hyper((1/4, 5/2), (5/4,), b*exp_polar(I*pi)/(a*x**4))/(4*a**(5 /2)*x*gamma(5/4))
\[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^2} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate(1/(a+b/x^4)^(5/2)/x^2,x, algorithm="maxima")
Output:
integrate(1/((a + b/x^4)^(5/2)*x^2), x)
\[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^2} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate(1/(a+b/x^4)^(5/2)/x^2,x, algorithm="giac")
Output:
integrate(1/((a + b/x^4)^(5/2)*x^2), x)
Time = 0.69 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^2} \, dx=-\frac {{\left (\frac {b}{a}+x^4\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {5}{2};\ \frac {5}{4};\ -\frac {b}{a\,x^4}\right )}{x\,{\left (a\,x^4+b\right )}^{5/2}} \] Input:
int(1/(x^2*(a + b/x^4)^(5/2)),x)
Output:
-((b/a + x^4)^(5/2)*hypergeom([1/4, 5/2], 5/4, -b/(a*x^4)))/(x*(b + a*x^4) ^(5/2))
\[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^2} \, dx=\frac {-\sqrt {a \,x^{4}+b}\, x +\left (\int \frac {\sqrt {a \,x^{4}+b}}{a^{3} x^{12}+3 a^{2} b \,x^{8}+3 a \,b^{2} x^{4}+b^{3}}d x \right ) a \,b^{2} x^{4}+\left (\int \frac {\sqrt {a \,x^{4}+b}}{a^{3} x^{12}+3 a^{2} b \,x^{8}+3 a \,b^{2} x^{4}+b^{3}}d x \right ) b^{3}}{a^{2} \left (a \,x^{4}+b \right )} \] Input:
int(1/(a+b/x^4)^(5/2)/x^2,x)
Output:
( - sqrt(a*x**4 + b)*x + int(sqrt(a*x**4 + b)/(a**3*x**12 + 3*a**2*b*x**8 + 3*a*b**2*x**4 + b**3),x)*a*b**2*x**4 + int(sqrt(a*x**4 + b)/(a**3*x**12 + 3*a**2*b*x**8 + 3*a*b**2*x**4 + b**3),x)*b**3)/(a**2*(a*x**4 + b))