Integrand size = 17, antiderivative size = 128 \[ \int \frac {1}{x \left (-b+a x^5\right )^{3/4}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}}{-\sqrt {b}+\sqrt {-b+a x^5}}\right )}{5 b^{3/4}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^5}}\right )}{5 b^{3/4}} \]
-1/5*2^(1/2)*arctan(2^(1/2)*b^(1/4)*(a*x^5-b)^(1/4)/(-b^(1/2)+(a*x^5-b)^(1 /2)))/b^(3/4)+1/5*2^(1/2)*arctanh((1/2*b^(1/4)*2^(1/2)+1/2*(a*x^5-b)^(1/2) *2^(1/2)/b^(1/4))/(a*x^5-b)^(1/4))/b^(3/4)
Time = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x \left (-b+a x^5\right )^{3/4}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}}{\sqrt {b}+\sqrt {-b+a x^5}}\right )\right )}{5 b^{3/4}} \]
(Sqrt[2]*(ArcTan[(-Sqrt[b] + Sqrt[-b + a*x^5])/(Sqrt[2]*b^(1/4)*(-b + a*x^ 5)^(1/4))] + ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^5)^(1/4))/(Sqrt[b] + Sqrt[ -b + a*x^5])]))/(5*b^(3/4))
Time = 0.38 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.60, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {798, 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.
| \(\displaystyle \int \frac {1}{x \left (a x^5-b\right )^{3/4}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{5} \int \frac {1}{x^5 \left (a x^5-b\right )^{3/4}}dx^5\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {4 \int \frac {1}{\frac {x^{20}}{a}+\frac {b}{a}}d\sqrt [4]{a x^5-b}}{5 a}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {4 \left (\frac {\int \frac {a \left (\sqrt {b}-x^{10}\right )}{x^{20}+b}d\sqrt [4]{a x^5-b}}{2 \sqrt {b}}+\frac {\int \frac {a \left (x^{10}+\sqrt {b}\right )}{x^{20}+b}d\sqrt [4]{a x^5-b}}{2 \sqrt {b}}\right )}{5 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \left (\frac {a \int \frac {\sqrt {b}-x^{10}}{x^{20}+b}d\sqrt [4]{a x^5-b}}{2 \sqrt {b}}+\frac {a \int \frac {x^{10}+\sqrt {b}}{x^{20}+b}d\sqrt [4]{a x^5-b}}{2 \sqrt {b}}\right )}{5 a}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {4 \left (\frac {a \left (\frac {1}{2} \int \frac {1}{x^{10}+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}d\sqrt [4]{a x^5-b}+\frac {1}{2} \int \frac {1}{x^{10}+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}d\sqrt [4]{a x^5-b}\right )}{2 \sqrt {b}}+\frac {a \int \frac {\sqrt {b}-x^{10}}{x^{20}+b}d\sqrt [4]{a x^5-b}}{2 \sqrt {b}}\right )}{5 a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {4 \left (\frac {a \left (\frac {\int \frac {1}{-x^{10}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^{10}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \int \frac {\sqrt {b}-x^{10}}{x^{20}+b}d\sqrt [4]{a x^5-b}}{2 \sqrt {b}}\right )}{5 a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4 \left (\frac {a \int \frac {\sqrt {b}-x^{10}}{x^{20}+b}d\sqrt [4]{a x^5-b}}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )}{5 a}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {4 \left (\frac {a \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^5-b}}{x^{10}+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}d\sqrt [4]{a x^5-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^5-b}\right )}{x^{10}+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}d\sqrt [4]{a x^5-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )}{5 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \left (\frac {a \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^5-b}}{x^{10}+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}d\sqrt [4]{a x^5-b}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^5-b}\right )}{x^{10}+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}d\sqrt [4]{a x^5-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )}{5 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \left (\frac {a \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^5-b}}{x^{10}+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}d\sqrt [4]{a x^5-b}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int \frac {\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^5-b}}{x^{10}+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}d\sqrt [4]{a x^5-b}}{2 \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )}{5 a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4 \left (\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}+\sqrt {b}+x^{10}\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}+\sqrt {b}+x^{10}\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )}{5 a}\) |
(4*((a*(-(ArcTan[1 - (Sqrt[2]*(-b + a*x^5)^(1/4))/b^(1/4)]/(Sqrt[2]*b^(1/4 ))) + ArcTan[1 + (Sqrt[2]*(-b + a*x^5)^(1/4))/b^(1/4)]/(Sqrt[2]*b^(1/4)))) /(2*Sqrt[b]) + (a*(-1/2*Log[Sqrt[b] + x^10 - Sqrt[2]*b^(1/4)*(-b + a*x^5)^ (1/4)]/(Sqrt[2]*b^(1/4)) + Log[Sqrt[b] + x^10 + Sqrt[2]*b^(1/4)*(-b + a*x^ 5)^(1/4)]/(2*Sqrt[2]*b^(1/4))))/(2*Sqrt[b])))/(5*a)
3.19.63.3.1 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] :> 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[((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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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 = 1.06 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.11
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {-b^{\frac {1}{4}} \left (a \,x^{5}-b \right )^{\frac {1}{4}} \sqrt {2}-\sqrt {a \,x^{5}-b}-\sqrt {b}}{b^{\frac {1}{4}} \left (a \,x^{5}-b \right )^{\frac {1}{4}} \sqrt {2}-\sqrt {a \,x^{5}-b}-\sqrt {b}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )\right )}{10 b^{\frac {3}{4}}}\) | \(142\) |
1/10/b^(3/4)*2^(1/2)*(ln((-b^(1/4)*(a*x^5-b)^(1/4)*2^(1/2)-(a*x^5-b)^(1/2) -b^(1/2))/(b^(1/4)*(a*x^5-b)^(1/4)*2^(1/2)-(a*x^5-b)^(1/2)-b^(1/2)))+2*arc tan((2^(1/2)*(a*x^5-b)^(1/4)+b^(1/4))/b^(1/4))-2*arctan((-2^(1/2)*(a*x^5-b )^(1/4)+b^(1/4))/b^(1/4)))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (-b+a x^5\right )^{3/4}} \, dx=\frac {1}{5} \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \log \left (b \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right ) + \frac {1}{5} i \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \log \left (i \, b \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right ) - \frac {1}{5} i \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \log \left (-i \, b \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right ) - \frac {1}{5} \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \log \left (-b \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right ) \]
1/5*(-1/b^3)^(1/4)*log(b*(-1/b^3)^(1/4) + (a*x^5 - b)^(1/4)) + 1/5*I*(-1/b ^3)^(1/4)*log(I*b*(-1/b^3)^(1/4) + (a*x^5 - b)^(1/4)) - 1/5*I*(-1/b^3)^(1/ 4)*log(-I*b*(-1/b^3)^(1/4) + (a*x^5 - b)^(1/4)) - 1/5*(-1/b^3)^(1/4)*log(- b*(-1/b^3)^(1/4) + (a*x^5 - b)^(1/4))
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.33 \[ \int \frac {1}{x \left (-b+a x^5\right )^{3/4}} \, dx=- \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{5}}} \right )}}{5 a^{\frac {3}{4}} x^{\frac {15}{4}} \Gamma \left (\frac {7}{4}\right )} \]
-gamma(3/4)*hyper((3/4, 3/4), (7/4,), b*exp_polar(2*I*pi)/(a*x**5))/(5*a** (3/4)*x**(15/4)*gamma(7/4))
Time = 0.29 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \left (-b+a x^5\right )^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{10 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{10 \, b^{\frac {3}{4}}} \]
1/5*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^5 - b)^(1/4))/b^( 1/4))/b^(3/4) + 1/5*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^ 5 - b)^(1/4))/b^(1/4))/b^(3/4) + 1/10*sqrt(2)*log(sqrt(2)*(a*x^5 - b)^(1/4 )*b^(1/4) + sqrt(a*x^5 - b) + sqrt(b))/b^(3/4) - 1/10*sqrt(2)*log(-sqrt(2) *(a*x^5 - b)^(1/4)*b^(1/4) + sqrt(a*x^5 - b) + sqrt(b))/b^(3/4)
Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \left (-b+a x^5\right )^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{10 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{10 \, b^{\frac {3}{4}}} \]
1/5*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^5 - b)^(1/4))/b^( 1/4))/b^(3/4) + 1/5*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^ 5 - b)^(1/4))/b^(1/4))/b^(3/4) + 1/10*sqrt(2)*log(sqrt(2)*(a*x^5 - b)^(1/4 )*b^(1/4) + sqrt(a*x^5 - b) + sqrt(b))/b^(3/4) - 1/10*sqrt(2)*log(-sqrt(2) *(a*x^5 - b)^(1/4)*b^(1/4) + sqrt(a*x^5 - b) + sqrt(b))/b^(3/4)
Time = 5.58 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x \left (-b+a x^5\right )^{3/4}} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\left (a\,x^5-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{5\,{\left (-b\right )}^{3/4}}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (a\,x^5-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{5\,{\left (-b\right )}^{3/4}} \]