Integrand size = 17, antiderivative size = 150 \[ \int \frac {\left (-b+a x^2\right )^{3/4}}{x^3} \, dx=-\frac {\left (-b+a x^2\right )^{3/4}}{2 x^2}-\frac {3 a \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{-\sqrt {b}+\sqrt {-b+a x^2}}\right )}{4 \sqrt {2} \sqrt [4]{b}}-\frac {3 a \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt {2} \sqrt [4]{b}} \]
-1/2*(a*x^2-b)^(3/4)/x^2-3/8*a*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)/b^(1/4)-3/8*a*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)/b^(1/4)
Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-b+a x^2\right )^{3/4}}{x^3} \, dx=\frac {1}{8} \left (-\frac {4 \left (-b+a x^2\right )^{3/4}}{x^2}+\frac {3 \sqrt {2} a \arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{b}}-\frac {3 \sqrt {2} a \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {b}+\sqrt {-b+a x^2}}\right )}{\sqrt [4]{b}}\right ) \]
((-4*(-b + a*x^2)^(3/4))/x^2 + (3*Sqrt[2]*a*ArcTan[(-Sqrt[b] + Sqrt[-b + a *x^2])/(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))])/b^(1/4) - (3*Sqrt[2]*a*ArcTa nh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^2])])/b^( 1/4))/8
Time = 0.37 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.41, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {243, 51, 73, 27, 826, 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.
| \(\displaystyle \int \frac {\left (a x^2-b\right )^{3/4}}{x^3} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (a x^2-b\right )^{3/4}}{x^4}dx^2\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} a \int \frac {1}{x^2 \sqrt [4]{a x^2-b}}dx^2-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (3 \int \frac {a x^4}{x^8+b}d\sqrt [4]{a x^2-b}-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (3 a \int \frac {x^4}{x^8+b}d\sqrt [4]{a x^2-b}-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {1}{2} \left (3 a \left (\frac {1}{2} \int \frac {x^4+\sqrt {b}}{x^8+b}d\sqrt [4]{a x^2-b}-\frac {1}{2} \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}\right )-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{2} \left (3 a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}+\frac {1}{2} \int \frac {1}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}\right )-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (3 a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x^4-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^4-1}d\left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}\right )-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (3 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}\right )-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{2} \left (3 a \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^2-b}\right )}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (3 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^2-b}\right )}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (3 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (3 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {b}+x^4\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {b}+x^4\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )\right )-\frac {\left (a x^2-b\right )^{3/4}}{x^2}\right )\) |
(-((-b + a*x^2)^(3/4)/x^2) + 3*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 + (Log[Sqrt[b] + x^4 - Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4)]/(2*Sqrt[2]*b^(1/4)) - Log[Sqrt[b] + x^4 + Sqrt[2]*b^(1/4)*(- b + a*x^2)^(1/4)]/(2*Sqrt[2]*b^(1/4)))/2))/2
3.21.70.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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 = 0.40 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.15
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (\arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,x^{2}-\arctan \left (\frac {\sqrt {2}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,x^{2}-\frac {\ln \left (\frac {\sqrt {a \,x^{2}-b}-b^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}{\sqrt {a \,x^{2}-b}+b^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}\right ) \sqrt {2}\, a \,x^{2}}{2}+\frac {4 \left (a \,x^{2}-b \right )^{\frac {3}{4}} b^{\frac {1}{4}}}{3}\right )}{8 b^{\frac {1}{4}} x^{2}}\) | \(172\) |
-3/8/b^(1/4)*(arctan((-2^(1/2)*(a*x^2-b)^(1/4)+b^(1/4))/b^(1/4))*2^(1/2)*a *x^2-arctan((2^(1/2)*(a*x^2-b)^(1/4)+b^(1/4))/b^(1/4))*2^(1/2)*a*x^2-1/2*l n(((a*x^2-b)^(1/2)-b^(1/4)*(a*x^2-b)^(1/4)*2^(1/2)+b^(1/2))/((a*x^2-b)^(1/ 2)+b^(1/4)*(a*x^2-b)^(1/4)*2^(1/2)+b^(1/2)))*2^(1/2)*a*x^2+4/3*(a*x^2-b)^( 3/4)*b^(1/4))/x^2
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.35 \[ \int \frac {\left (-b+a x^2\right )^{3/4}}{x^3} \, dx=\frac {3 \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{2} \log \left (27 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} a^{3} + 27 \, \left (-\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) - 3 i \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{2} \log \left (27 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} a^{3} + 27 i \, \left (-\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) + 3 i \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{2} \log \left (27 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} a^{3} - 27 i \, \left (-\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) - 3 \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{2} \log \left (27 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} a^{3} - 27 \, \left (-\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) - 4 \, {\left (a x^{2} - b\right )}^{\frac {3}{4}}}{8 \, x^{2}} \]
1/8*(3*(-a^4/b)^(1/4)*x^2*log(27*(a*x^2 - b)^(1/4)*a^3 + 27*(-a^4/b)^(3/4) *b) - 3*I*(-a^4/b)^(1/4)*x^2*log(27*(a*x^2 - b)^(1/4)*a^3 + 27*I*(-a^4/b)^ (3/4)*b) + 3*I*(-a^4/b)^(1/4)*x^2*log(27*(a*x^2 - b)^(1/4)*a^3 - 27*I*(-a^ 4/b)^(3/4)*b) - 3*(-a^4/b)^(1/4)*x^2*log(27*(a*x^2 - b)^(1/4)*a^3 - 27*(-a ^4/b)^(3/4)*b) - 4*(a*x^2 - b)^(3/4))/x^2
Result contains complex when optimal does not.
Time = 0.84 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.29 \[ \int \frac {\left (-b+a x^2\right )^{3/4}}{x^3} \, dx=- \frac {a^{\frac {3}{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{2}}} \right )}}{2 \sqrt {x} \Gamma \left (\frac {5}{4}\right )} \]
-a**(3/4)*gamma(1/4)*hyper((-3/4, 1/4), (5/4,), b*exp_polar(2*I*pi)/(a*x** 2))/(2*sqrt(x)*gamma(5/4))
Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-b+a x^2\right )^{3/4}}{x^3} \, dx=\frac {3}{16} \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}}\right )} a - \frac {{\left (a x^{2} - b\right )}^{\frac {3}{4}}}{2 \, x^{2}} \]
3/16*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^2 - b)^(1/4)) /b^(1/4))/b^(1/4) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a* x^2 - b)^(1/4))/b^(1/4))/b^(1/4) - sqrt(2)*log(sqrt(2)*(a*x^2 - b)^(1/4)*b ^(1/4) + sqrt(a*x^2 - b) + sqrt(b))/b^(1/4) + sqrt(2)*log(-sqrt(2)*(a*x^2 - b)^(1/4)*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b))/b^(1/4))*a - 1/2*(a*x^2 - b)^(3/4)/x^2
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-b+a x^2\right )^{3/4}}{x^3} \, dx=\frac {\frac {6 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {6 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {3 \, \sqrt {2} a^{2} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {3 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} - \frac {8 \, {\left (a x^{2} - b\right )}^{\frac {3}{4}} a}{x^{2}}}{16 \, a} \]
1/16*(6*sqrt(2)*a^2*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^2 - b)^(1 /4))/b^(1/4))/b^(1/4) + 6*sqrt(2)*a^2*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^2 - b)^(1/4))/b^(1/4))/b^(1/4) - 3*sqrt(2)*a^2*log(sqrt(2)*(a*x^ 2 - b)^(1/4)*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b))/b^(1/4) + 3*sqrt(2)*a^2* log(-sqrt(2)*(a*x^2 - b)^(1/4)*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b))/b^(1/4 ) - 8*(a*x^2 - b)^(3/4)*a/x^2)/a
Time = 6.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.46 \[ \int \frac {\left (-b+a x^2\right )^{3/4}}{x^3} \, dx=\frac {3\,a\,\mathrm {atan}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{4\,{\left (-b\right )}^{1/4}}-\frac {{\left (a\,x^2-b\right )}^{3/4}}{2\,x^2}-\frac {3\,a\,\mathrm {atanh}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{4\,{\left (-b\right )}^{1/4}} \]