Integrand size = 17, antiderivative size = 150 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=-\frac {\sqrt [4]{-b+a x^3}}{3 x^3}-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}{-\sqrt {b}+\sqrt {-b+a x^3}}\right )}{6 \sqrt {2} b^{3/4}}+\frac {a \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^3}}\right )}{6 \sqrt {2} b^{3/4}} \]
-1/3*(a*x^3-b)^(1/4)/x^3-1/12*a*arctan(2^(1/2)*b^(1/4)*(a*x^3-b)^(1/4)/(-b ^(1/2)+(a*x^3-b)^(1/2)))*2^(1/2)/b^(3/4)+1/12*a*arctanh((1/2*b^(1/4)*2^(1/ 2)+1/2*(a*x^3-b)^(1/2)*2^(1/2)/b^(1/4))/(a*x^3-b)^(1/4))*2^(1/2)/b^(3/4)
Time = 0.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=\frac {-4 b^{3/4} \sqrt [4]{-b+a x^3}+\sqrt {2} a x^3 \arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^3}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}\right )+\sqrt {2} a x^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}{\sqrt {b}+\sqrt {-b+a x^3}}\right )}{12 b^{3/4} x^3} \]
(-4*b^(3/4)*(-b + a*x^3)^(1/4) + Sqrt[2]*a*x^3*ArcTan[(-Sqrt[b] + Sqrt[-b + a*x^3])/(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))] + Sqrt[2]*a*x^3*ArcTanh[(S qrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(12*b^(3 /4)*x^3)
Time = 0.37 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.47, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {798, 51, 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.
| \(\displaystyle \int \frac {\sqrt [4]{a x^3-b}}{x^4} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{3} \int \frac {\sqrt [4]{a x^3-b}}{x^6}dx^3\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} a \int \frac {1}{x^3 \left (a x^3-b\right )^{3/4}}dx^3-\frac {\sqrt [4]{a x^3-b}}{x^3}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {1}{\frac {x^{12}}{a}+\frac {b}{a}}d\sqrt [4]{a x^3-b}-\frac {\sqrt [4]{a x^3-b}}{x^3}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {1}{3} \left (\frac {\int \frac {a \left (\sqrt {b}-x^6\right )}{x^{12}+b}d\sqrt [4]{a x^3-b}}{2 \sqrt {b}}+\frac {\int \frac {a \left (x^6+\sqrt {b}\right )}{x^{12}+b}d\sqrt [4]{a x^3-b}}{2 \sqrt {b}}-\frac {\sqrt [4]{a x^3-b}}{x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \int \frac {\sqrt {b}-x^6}{x^{12}+b}d\sqrt [4]{a x^3-b}}{2 \sqrt {b}}+\frac {a \int \frac {x^6+\sqrt {b}}{x^{12}+b}d\sqrt [4]{a x^3-b}}{2 \sqrt {b}}-\frac {\sqrt [4]{a x^3-b}}{x^3}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {1}{2} \int \frac {1}{x^6+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}+\frac {1}{2} \int \frac {1}{x^6+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}\right )}{2 \sqrt {b}}+\frac {a \int \frac {\sqrt {b}-x^6}{x^{12}+b}d\sqrt [4]{a x^3-b}}{2 \sqrt {b}}-\frac {\sqrt [4]{a x^3-b}}{x^3}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {\int \frac {1}{-x^6-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^6-1}d\left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \int \frac {\sqrt {b}-x^6}{x^{12}+b}d\sqrt [4]{a x^3-b}}{2 \sqrt {b}}-\frac {\sqrt [4]{a x^3-b}}{x^3}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \int \frac {\sqrt {b}-x^6}{x^{12}+b}d\sqrt [4]{a x^3-b}}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}-\frac {\sqrt [4]{a x^3-b}}{x^3}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^3-b}}{x^6+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^3-b}\right )}{x^6+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}-\frac {\sqrt [4]{a x^3-b}}{x^3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^3-b}}{x^6+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^3-b}\right )}{x^6+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}-\frac {\sqrt [4]{a x^3-b}}{x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^3-b}}{x^6+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int \frac {\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^3-b}}{x^6+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}-\frac {\sqrt [4]{a x^3-b}}{x^3}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}-\frac {\sqrt [4]{a x^3-b}}{x^3}+\frac {a \left (\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {b}+x^6\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {b}+x^6\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )}{2 \sqrt {b}}\right )\) |
(-((-b + a*x^3)^(1/4)/x^3) + (a*(-(ArcTan[1 - (Sqrt[2]*(-b + a*x^3)^(1/4)) /b^(1/4)]/(Sqrt[2]*b^(1/4))) + ArcTan[1 + (Sqrt[2]*(-b + a*x^3)^(1/4))/b^( 1/4)]/(Sqrt[2]*b^(1/4))))/(2*Sqrt[b]) + (a*(-1/2*Log[Sqrt[b] + x^6 - Sqrt[ 2]*b^(1/4)*(-b + a*x^3)^(1/4)]/(Sqrt[2]*b^(1/4)) + Log[Sqrt[b] + x^6 + Sqr t[2]*b^(1/4)*(-b + a*x^3)^(1/4)]/(2*Sqrt[2]*b^(1/4))))/(2*Sqrt[b]))/3
3.21.73.3.1 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[((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 = 0.46 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.20
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {-b^{\frac {1}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}} \sqrt {2}-\sqrt {a \,x^{3}-b}-\sqrt {b}}{b^{\frac {1}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}} \sqrt {2}-\sqrt {a \,x^{3}-b}-\sqrt {b}}\right ) \sqrt {2}\, a \,x^{3}+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{3}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,x^{3}-2 \arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{3}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,x^{3}-8 \left (a \,x^{3}-b \right )^{\frac {1}{4}} b^{\frac {3}{4}}}{24 x^{3} b^{\frac {3}{4}}}\) | \(180\) |
1/24*(ln((-b^(1/4)*(a*x^3-b)^(1/4)*2^(1/2)-(a*x^3-b)^(1/2)-b^(1/2))/(b^(1/ 4)*(a*x^3-b)^(1/4)*2^(1/2)-(a*x^3-b)^(1/2)-b^(1/2)))*2^(1/2)*a*x^3+2*arcta n((2^(1/2)*(a*x^3-b)^(1/4)+b^(1/4))/b^(1/4))*2^(1/2)*a*x^3-2*arctan((-2^(1 /2)*(a*x^3-b)^(1/4)+b^(1/4))/b^(1/4))*2^(1/2)*a*x^3-8*(a*x^3-b)^(1/4)*b^(3 /4))/x^3/b^(3/4)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=\frac {\left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a + \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b\right ) + i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a + i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b\right ) - i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a - i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b\right ) - \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a - \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b\right ) - 4 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}}{12 \, x^{3}} \]
1/12*((-a^4/b^3)^(1/4)*x^3*log((a*x^3 - b)^(1/4)*a + (-a^4/b^3)^(1/4)*b) + I*(-a^4/b^3)^(1/4)*x^3*log((a*x^3 - b)^(1/4)*a + I*(-a^4/b^3)^(1/4)*b) - I*(-a^4/b^3)^(1/4)*x^3*log((a*x^3 - b)^(1/4)*a - I*(-a^4/b^3)^(1/4)*b) - ( -a^4/b^3)^(1/4)*x^3*log((a*x^3 - b)^(1/4)*a - (-a^4/b^3)^(1/4)*b) - 4*(a*x ^3 - b)^(1/4))/x^3
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=- \frac {\sqrt [4]{a} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 x^{\frac {9}{4}} \Gamma \left (\frac {7}{4}\right )} \]
-a**(1/4)*gamma(3/4)*hyper((-1/4, 3/4), (7/4,), b*exp_polar(2*I*pi)/(a*x** 3))/(3*x**(9/4)*gamma(7/4))
Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=\frac {\sqrt {2} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{12 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{12 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} a \log \left (\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{24 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} a \log \left (-\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{24 \, b^{\frac {3}{4}}} - \frac {{\left (a x^{3} - b\right )}^{\frac {1}{4}}}{3 \, x^{3}} \]
1/12*sqrt(2)*a*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^3 - b)^(1/4))/ b^(1/4))/b^(3/4) + 1/12*sqrt(2)*a*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2 *(a*x^3 - b)^(1/4))/b^(1/4))/b^(3/4) + 1/24*sqrt(2)*a*log(sqrt(2)*(a*x^3 - b)^(1/4)*b^(1/4) + sqrt(a*x^3 - b) + sqrt(b))/b^(3/4) - 1/24*sqrt(2)*a*lo g(-sqrt(2)*(a*x^3 - b)^(1/4)*b^(1/4) + sqrt(a*x^3 - b) + sqrt(b))/b^(3/4) - 1/3*(a*x^3 - b)^(1/4)/x^3
Time = 0.27 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=\frac {\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} + \frac {\sqrt {2} a^{2} \log \left (\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {3}{4}}} - \frac {\sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {3}{4}}} - \frac {8 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}} a}{x^{3}}}{24 \, a} \]
1/24*(2*sqrt(2)*a^2*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^3 - b)^(1 /4))/b^(1/4))/b^(3/4) + 2*sqrt(2)*a^2*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^3 - b)^(1/4))/b^(1/4))/b^(3/4) + sqrt(2)*a^2*log(sqrt(2)*(a*x^3 - b)^(1/4)*b^(1/4) + sqrt(a*x^3 - b) + sqrt(b))/b^(3/4) - sqrt(2)*a^2*log( -sqrt(2)*(a*x^3 - b)^(1/4)*b^(1/4) + sqrt(a*x^3 - b) + sqrt(b))/b^(3/4) - 8*(a*x^3 - b)^(1/4)*a/x^3)/a
Time = 6.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx=-\frac {{\left (a\,x^3-b\right )}^{1/4}}{3\,x^3}-\frac {a\,\mathrm {atan}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{6\,{\left (-b\right )}^{3/4}}-\frac {a\,\mathrm {atanh}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{6\,{\left (-b\right )}^{3/4}} \]