Integrand size = 17, antiderivative size = 138 \[ \int \frac {\sqrt [4]{-b+a x^2}}{x} \, dx=2 \sqrt [4]{-b+a x^2}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{-\sqrt {b}+\sqrt {-b+a x^2}}\right )}{\sqrt {2}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {2}} \]
2*(a*x^2-b)^(1/4)+1/2*b^(1/4)*arctan(2^(1/2)*b^(1/4)*(a*x^2-b)^(1/4)/(-b^( 1/2)+(a*x^2-b)^(1/2)))*2^(1/2)-1/2*b^(1/4)*arctanh((1/2*b^(1/4)*2^(1/2)+1/ 2*(a*x^2-b)^(1/2)*2^(1/2)/b^(1/4))/(a*x^2-b)^(1/4))*2^(1/2)
Time = 0.18 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [4]{-b+a x^2}}{x} \, dx=2 \sqrt [4]{-b+a x^2}-\frac {\sqrt [4]{b} \arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {b}+\sqrt {-b+a x^2}}\right )}{\sqrt {2}} \]
2*(-b + a*x^2)^(1/4) - (b^(1/4)*ArcTan[(-Sqrt[b] + Sqrt[-b + a*x^2])/(Sqrt [2]*b^(1/4)*(-b + a*x^2)^(1/4))])/Sqrt[2] - (b^(1/4)*ArcTanh[(Sqrt[2]*b^(1 /4)*(-b + a*x^2)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^2])])/Sqrt[2]
Time = 0.37 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.62, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {243, 60, 73, 755, 27, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {\sqrt [4]{a x^2-b}}{x} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt [4]{a x^2-b}}{x^2}dx^2\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-b \int \frac {1}{x^2 \left (a x^2-b\right )^{3/4}}dx^2\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-\frac {4 b \int \frac {1}{\frac {x^8}{a}+\frac {b}{a}}d\sqrt [4]{a x^2-b}}{a}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-\frac {4 b \left (\frac {\int \frac {a \left (\sqrt {b}-x^4\right )}{x^8+b}d\sqrt [4]{a x^2-b}}{2 \sqrt {b}}+\frac {\int \frac {a \left (x^4+\sqrt {b}\right )}{x^8+b}d\sqrt [4]{a x^2-b}}{2 \sqrt {b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-\frac {4 b \left (\frac {a \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}}{2 \sqrt {b}}+\frac {a \int \frac {x^4+\sqrt {b}}{x^8+b}d\sqrt [4]{a x^2-b}}{2 \sqrt {b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-\frac {4 b \left (\frac {a \left (\frac {1}{2} \int \frac {1}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}+\frac {1}{2} \int \frac {1}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}\right )}{2 \sqrt {b}}+\frac {a \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}}{2 \sqrt {b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-\frac {4 b \left (\frac {a \left (\frac {\int \frac {1}{-x^4-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^4-1}d\left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}}{2 \sqrt {b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-\frac {4 b \left (\frac {a \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-\frac {4 b \left (\frac {a \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^2-b}\right )}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-\frac {4 b \left (\frac {a \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^2-b}\right )}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-\frac {4 b \left (\frac {a \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int \frac {\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt [4]{a x^2-b}-\frac {4 b \left (\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {b}+x^4\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {b}+x^4\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )}{a}\right )\) |
(4*(-b + a*x^2)^(1/4) - (4*b*((a*(-(ArcTan[1 - (Sqrt[2]*(-b + a*x^2)^(1/4) )/b^(1/4)]/(Sqrt[2]*b^(1/4))) + ArcTan[1 + (Sqrt[2]*(-b + a*x^2)^(1/4))/b^ (1/4)]/(Sqrt[2]*b^(1/4))))/(2*Sqrt[b]) + (a*(-1/2*Log[Sqrt[b] + x^4 - Sqrt [2]*b^(1/4)*(-b + a*x^2)^(1/4)]/(Sqrt[2]*b^(1/4)) + Log[Sqrt[b] + x^4 + Sq rt[2]*b^(1/4)*(-b + a*x^2)^(1/4)]/(2*Sqrt[2]*b^(1/4))))/(2*Sqrt[b])))/a)/2
3.20.54.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.52 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.21
| method | result | size |
| pseudoelliptic | \(2 \left (a \,x^{2}-b \right )^{\frac {1}{4}}-\frac {\ln \left (\frac {-b^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} \sqrt {2}-\sqrt {a \,x^{2}-b}-\sqrt {b}}{b^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} \sqrt {2}-\sqrt {a \,x^{2}-b}-\sqrt {b}}\right ) b^{\frac {1}{4}} \sqrt {2}}{4}-\frac {\arctan \left (\frac {\sqrt {2}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) b^{\frac {1}{4}} \sqrt {2}}{2}+\frac {\arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) b^{\frac {1}{4}} \sqrt {2}}{2}\) | \(167\) |
2*(a*x^2-b)^(1/4)-1/4*ln((-b^(1/4)*(a*x^2-b)^(1/4)*2^(1/2)-(a*x^2-b)^(1/2) -b^(1/2))/(b^(1/4)*(a*x^2-b)^(1/4)*2^(1/2)-(a*x^2-b)^(1/2)-b^(1/2)))*b^(1/ 4)*2^(1/2)-1/2*arctan((2^(1/2)*(a*x^2-b)^(1/4)+b^(1/4))/b^(1/4))*b^(1/4)*2 ^(1/2)+1/2*arctan((-2^(1/2)*(a*x^2-b)^(1/4)+b^(1/4))/b^(1/4))*b^(1/4)*2^(1 /2)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [4]{-b+a x^2}}{x} \, dx=-\frac {1}{2} \, \left (-b\right )^{\frac {1}{4}} \log \left ({\left (a x^{2} - b\right )}^{\frac {1}{4}} + \left (-b\right )^{\frac {1}{4}}\right ) - \frac {1}{2} i \, \left (-b\right )^{\frac {1}{4}} \log \left ({\left (a x^{2} - b\right )}^{\frac {1}{4}} + i \, \left (-b\right )^{\frac {1}{4}}\right ) + \frac {1}{2} i \, \left (-b\right )^{\frac {1}{4}} \log \left ({\left (a x^{2} - b\right )}^{\frac {1}{4}} - i \, \left (-b\right )^{\frac {1}{4}}\right ) + \frac {1}{2} \, \left (-b\right )^{\frac {1}{4}} \log \left ({\left (a x^{2} - b\right )}^{\frac {1}{4}} - \left (-b\right )^{\frac {1}{4}}\right ) + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} \]
-1/2*(-b)^(1/4)*log((a*x^2 - b)^(1/4) + (-b)^(1/4)) - 1/2*I*(-b)^(1/4)*log ((a*x^2 - b)^(1/4) + I*(-b)^(1/4)) + 1/2*I*(-b)^(1/4)*log((a*x^2 - b)^(1/4 ) - I*(-b)^(1/4)) + 1/2*(-b)^(1/4)*log((a*x^2 - b)^(1/4) - (-b)^(1/4)) + 2 *(a*x^2 - b)^(1/4)
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt [4]{-b+a x^2}}{x} \, dx=- \frac {\sqrt [4]{a} \sqrt {x} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{2}}} \right )}}{2 \Gamma \left (\frac {3}{4}\right )} \]
-a**(1/4)*sqrt(x)*gamma(-1/4)*hyper((-1/4, -1/4), (3/4,), b*exp_polar(2*I* pi)/(a*x**2))/(2*gamma(3/4))
Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [4]{-b+a x^2}}{x} \, dx=-\frac {1}{2} \, \sqrt {2} b^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) - \frac {1}{2} \, \sqrt {2} b^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right ) + \frac {1}{4} \, \sqrt {2} b^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right ) + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} \]
-1/2*sqrt(2)*b^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^2 - b)^( 1/4))/b^(1/4)) - 1/2*sqrt(2)*b^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^2 - b)^(1/4))/b^(1/4)) - 1/4*sqrt(2)*b^(1/4)*log(sqrt(2)*(a*x^2 - b)^(1/4)*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b)) + 1/4*sqrt(2)*b^(1/4)*log(- sqrt(2)*(a*x^2 - b)^(1/4)*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b)) + 2*(a*x^2 - b)^(1/4)
Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [4]{-b+a x^2}}{x} \, dx=-\frac {1}{2} \, \sqrt {2} b^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) - \frac {1}{2} \, \sqrt {2} b^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right ) + \frac {1}{4} \, \sqrt {2} b^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right ) + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} \]
-1/2*sqrt(2)*b^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^2 - b)^( 1/4))/b^(1/4)) - 1/2*sqrt(2)*b^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^2 - b)^(1/4))/b^(1/4)) - 1/4*sqrt(2)*b^(1/4)*log(sqrt(2)*(a*x^2 - b)^(1/4)*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b)) + 1/4*sqrt(2)*b^(1/4)*log(- sqrt(2)*(a*x^2 - b)^(1/4)*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b)) + 2*(a*x^2 - b)^(1/4)
Time = 5.50 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt [4]{-b+a x^2}}{x} \, dx=2\,{\left (a\,x^2-b\right )}^{1/4}-{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )-{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right ) \]