Integrand size = 28, antiderivative size = 94 \[ \int \frac {-2 b+a x^4}{x^4 \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {4 \left (-3 b+4 a x^2\right ) \left (b x^2+a x^4\right )^{3/4}}{21 b x^5}+a^{3/4} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )+a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right ) \] Output:
-4/21*(4*a*x^2-3*b)*(a*x^4+b*x^2)^(3/4)/b/x^5+a^(3/4)*arctan(a^(1/4)*x/(a* x^4+b*x^2)^(1/4))+a^(3/4)*arctanh(a^(1/4)*x/(a*x^4+b*x^2)^(1/4))
Time = 0.39 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.51 \[ \int \frac {-2 b+a x^4}{x^4 \sqrt [4]{b x^2+a x^4}} \, dx=\frac {-4 \left (-3 b^2+a b x^2+4 a^2 x^4\right )+21 a^{3/4} b x^{7/2} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+21 a^{3/4} b x^{7/2} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{21 b x^3 \sqrt [4]{x^2 \left (b+a x^2\right )}} \] Input:
Integrate[(-2*b + a*x^4)/(x^4*(b*x^2 + a*x^4)^(1/4)),x]
Output:
(-4*(-3*b^2 + a*b*x^2 + 4*a^2*x^4) + 21*a^(3/4)*b*x^(7/2)*(b + a*x^2)^(1/4 )*ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)] + 21*a^(3/4)*b*x^(7/2)*(b + a*x^2)^(1/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(21*b*x^3*(x^2* (b + a*x^2))^(1/4))
Time = 0.39 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2449, 2009}
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.
\(\displaystyle \int \frac {a x^4-2 b}{x^4 \sqrt [4]{a x^4+b x^2}} \, dx\) |
\(\Big \downarrow \) 2449 |
\(\displaystyle \int \left (\frac {a}{\sqrt [4]{a x^4+b x^2}}-\frac {2 b}{x^4 \sqrt [4]{a x^4+b x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^{3/4} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a x^4+b x^2}}+\frac {a^{3/4} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a x^4+b x^2}}+\frac {4 \left (a x^4+b x^2\right )^{3/4}}{7 x^5}-\frac {16 a \left (a x^4+b x^2\right )^{3/4}}{21 b x^3}\) |
Input:
Int[(-2*b + a*x^4)/(x^4*(b*x^2 + a*x^4)^(1/4)),x]
Output:
(4*(b*x^2 + a*x^4)^(3/4))/(7*x^5) - (16*a*(b*x^2 + a*x^4)^(3/4))/(21*b*x^3 ) + (a^(3/4)*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2 )^(1/4)])/(b*x^2 + a*x^4)^(1/4) + (a^(3/4)*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTa nh[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(b*x^2 + a*x^4)^(1/4)
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_S ymbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ [{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !Integer Q[p] && NeQ[n, j]
Time = 0.37 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.39
method | result | size |
pseudoelliptic | \(\frac {-42 a^{\frac {3}{4}} \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b \,x^{5}+21 a^{\frac {3}{4}} \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right ) b \,x^{5}-32 \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {3}{4}} a \,x^{2}+24 b \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {3}{4}}}{42 b \,x^{5}}\) | \(131\) |
Input:
int((a*x^4-2*b)/x^4/(a*x^4+b*x^2)^(1/4),x,method=_RETURNVERBOSE)
Output:
1/42*(-42*a^(3/4)*arctan(1/a^(1/4)/x*(x^2*(a*x^2+b))^(1/4))*b*x^5+21*a^(3/ 4)*ln((-a^(1/4)*x-(x^2*(a*x^2+b))^(1/4))/(a^(1/4)*x-(x^2*(a*x^2+b))^(1/4)) )*b*x^5-32*(x^2*(a*x^2+b))^(3/4)*a*x^2+24*b*(x^2*(a*x^2+b))^(3/4))/b/x^5
Timed out. \[ \int \frac {-2 b+a x^4}{x^4 \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \] Input:
integrate((a*x^4-2*b)/x^4/(a*x^4+b*x^2)^(1/4),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {-2 b+a x^4}{x^4 \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a x^{4} - 2 b}{x^{4} \sqrt [4]{x^{2} \left (a x^{2} + b\right )}}\, dx \] Input:
integrate((a*x**4-2*b)/x**4/(a*x**4+b*x**2)**(1/4),x)
Output:
Integral((a*x**4 - 2*b)/(x**4*(x**2*(a*x**2 + b))**(1/4)), x)
\[ \int \frac {-2 b+a x^4}{x^4 \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{4} - 2 \, b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} x^{4}} \,d x } \] Input:
integrate((a*x^4-2*b)/x^4/(a*x^4+b*x^2)^(1/4),x, algorithm="maxima")
Output:
integrate((a*x^4 - 2*b)/((a*x^4 + b*x^2)^(1/4)*x^4), x)
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (78) = 156\).
Time = 0.33 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.22 \[ \int \frac {-2 b+a x^4}{x^4 \sqrt [4]{b x^2+a x^4}} \, dx=\frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right ) + \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right ) + \frac {4 \, {\left (3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {7}{4}} b^{6} - 7 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{4}} a b^{6}\right )}}{21 \, b^{7}} \] Input:
integrate((a*x^4-2*b)/x^4/(a*x^4+b*x^2)^(1/4),x, algorithm="giac")
Output:
1/2*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x ^2)^(1/4))/(-a)^(1/4)) + 1/2*sqrt(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt( 2)*(-a)^(1/4) - 2*(a + b/x^2)^(1/4))/(-a)^(1/4)) - 1/4*sqrt(2)*(-a)^(3/4)* log(sqrt(2)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt(-a) + sqrt(a + b/x^2)) + 1 /4*sqrt(2)*(-a)^(3/4)*log(-sqrt(2)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt(-a) + sqrt(a + b/x^2)) + 4/21*(3*(a + b/x^2)^(7/4)*b^6 - 7*(a + b/x^2)^(3/4)* a*b^6)/b^7
Time = 8.54 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.80 \[ \int \frac {-2 b+a x^4}{x^4 \sqrt [4]{b x^2+a x^4}} \, dx=\frac {4\,{\left (a\,x^4+b\,x^2\right )}^{3/4}\,\left (3\,b-4\,a\,x^2\right )}{21\,b\,x^5}+\frac {2\,a\,x\,{\left (\frac {a\,x^2}{b}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -\frac {a\,x^2}{b}\right )}{{\left (a\,x^4+b\,x^2\right )}^{1/4}} \] Input:
int(-(2*b - a*x^4)/(x^4*(a*x^4 + b*x^2)^(1/4)),x)
Output:
(4*(a*x^4 + b*x^2)^(3/4)*(3*b - 4*a*x^2))/(21*b*x^5) + (2*a*x*((a*x^2)/b + 1)^(1/4)*hypergeom([1/4, 1/4], 5/4, -(a*x^2)/b))/(a*x^4 + b*x^2)^(1/4)
Time = 0.26 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.85 \[ \int \frac {-2 b+a x^4}{x^4 \sqrt [4]{b x^2+a x^4}} \, dx=\frac {256 \left (a \,x^{2}+b \right ) a^{3} x^{4}+672 \left (a \,x^{2}+b \right ) a^{2} b \,x^{4}+64 \left (a \,x^{2}+b \right ) a^{2} b \,x^{2}+168 \left (a \,x^{2}+b \right ) a \,b^{2} x^{2}-24 \left (a \,x^{2}+b \right ) a \,b^{2}-63 \left (a \,x^{2}+b \right ) b^{3}+252 a^{3} b \,x^{6}+126 a^{2} b^{2} x^{4}+168 a^{2} b^{2} x^{2}-63 a \,b^{3} x^{2}+168 a \,b^{3}+63 b^{4}}{252 \left (a \,x^{2}+b \right )^{\frac {3}{4}} \sqrt {x}\, \sqrt {a \,x^{2}+b}\, a b \,x^{3}} \] Input:
int((a*x^4-2*b)/x^4/(a*x^4+b*x^2)^(1/4),x)
Output:
((a*x**2 + b)**(1/4)*(256*(a*x**2 + b)*a**3*x**4 + 672*(a*x**2 + b)*a**2*b *x**4 + 64*(a*x**2 + b)*a**2*b*x**2 + 168*(a*x**2 + b)*a*b**2*x**2 - 24*(a *x**2 + b)*a*b**2 - 63*(a*x**2 + b)*b**3 + 252*a**3*b*x**6 + 126*a**2*b**2 *x**4 + 168*a**2*b**2*x**2 - 63*a*b**3*x**2 + 168*a*b**3 + 63*b**4))/(252* sqrt(x)*sqrt(a*x**2 + b)*a*b*x**3*(a*x**2 + b))