Integrand size = 17, antiderivative size = 153 \[ \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx=\frac {\left (-b+a x^3\right )^{3/4}}{3 b x^3}-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}{-\sqrt {b}+\sqrt {-b+a x^3}}\right )}{6 \sqrt {2} b^{5/4}}-\frac {a \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^3}}\right )}{6 \sqrt {2} b^{5/4}} \]
1/3*(a*x^3-b)^(3/4)/b/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^(5/4)-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^(5/4)
Time = 0.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx=\frac {4 \sqrt [4]{b} \left (-b+a x^3\right )^{3/4}+\sqrt {2} a x^3 \arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^3}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}\right )-\sqrt {2} a x^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}{\sqrt {b}+\sqrt {-b+a x^3}}\right )}{12 b^{5/4} x^3} \]
(4*b^(1/4)*(-b + a*x^3)^(3/4) + 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[(Sq rt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(12*b^(5/ 4)*x^3)
Time = 0.37 (sec) , antiderivative size = 216, 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 = {798, 52, 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.
| \(\displaystyle \int \frac {1}{x^4 \sqrt [4]{a x^3-b}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{x^6 \sqrt [4]{a x^3-b}}dx^3\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \int \frac {1}{x^3 \sqrt [4]{a x^3-b}}dx^3}{4 b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (\frac {\int \frac {a x^6}{x^{12}+b}d\sqrt [4]{a x^3-b}}{b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \int \frac {x^6}{x^{12}+b}d\sqrt [4]{a x^3-b}}{b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {1}{2} \int \frac {x^6+\sqrt {b}}{x^{12}+b}d\sqrt [4]{a x^3-b}-\frac {1}{2} \int \frac {\sqrt {b}-x^6}{x^{12}+b}d\sqrt [4]{a x^3-b}\right )}{b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^6+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}+\frac {1}{2} \int \frac {1}{x^6+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^6}{x^{12}+b}d\sqrt [4]{a x^3-b}\right )}{b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x^6-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^6-1}d\left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^6}{x^{12}+b}d\sqrt [4]{a x^3-b}\right )}{b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^6}{x^{12}+b}d\sqrt [4]{a x^3-b}\right )}{b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^3-b}}{x^6+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^3-b}\right )}{x^6+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )}{b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^3-b}}{x^6+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^3-b}\right )}{x^6+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )}{b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^3-b}}{x^6+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^3-b}}{x^6+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}d\sqrt [4]{a x^3-b}}{2 \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )}{b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{3} \left (\frac {a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {b}+x^6\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {b}+x^6\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )\right )}{b}+\frac {\left (a x^3-b\right )^{3/4}}{b x^3}\right )\) |
((-b + a*x^3)^(3/4)/(b*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 + (Log[Sqrt[b] + x^6 - Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4)]/(2*Sqrt[2]*b^(1/4)) - Log[Sqrt[b] + x^6 + Sqrt[2]*b^(1/4)*(- b + a*x^3)^(1/4)]/(2*Sqrt[2]*b^(1/4)))/2))/b)/3
3.22.6.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[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_)^(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[(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 = 0.32 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.12
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {\sqrt {a \,x^{3}-b}-b^{\frac {1}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}{\sqrt {a \,x^{3}-b}+b^{\frac {1}{4}} \left (a \,x^{3}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}\right ) \sqrt {2}\, a \,x^{3}+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{3}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,x^{3}-2 \arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{3}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,x^{3}+8 \left (a \,x^{3}-b \right )^{\frac {3}{4}} b^{\frac {1}{4}}}{24 b^{\frac {5}{4}} x^{3}}\) | \(172\) |
1/24/b^(5/4)*(ln(((a*x^3-b)^(1/2)-b^(1/4)*(a*x^3-b)^(1/4)*2^(1/2)+b^(1/2)) /((a*x^3-b)^(1/2)+b^(1/4)*(a*x^3-b)^(1/4)*2^(1/2)+b^(1/2)))*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^3-2*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)^(3/ 4)*b^(1/4))/x^3
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx=\frac {b x^{3} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (b^{4} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} + {\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{3}\right ) - i \, b x^{3} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (i \, b^{4} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} + {\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{3}\right ) + i \, b x^{3} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (-i \, b^{4} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} + {\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{3}\right ) - b x^{3} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (-b^{4} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} + {\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{3}\right ) + 4 \, {\left (a x^{3} - b\right )}^{\frac {3}{4}}}{12 \, b x^{3}} \]
1/12*(b*x^3*(-a^4/b^5)^(1/4)*log(b^4*(-a^4/b^5)^(3/4) + (a*x^3 - b)^(1/4)* a^3) - I*b*x^3*(-a^4/b^5)^(1/4)*log(I*b^4*(-a^4/b^5)^(3/4) + (a*x^3 - b)^( 1/4)*a^3) + I*b*x^3*(-a^4/b^5)^(1/4)*log(-I*b^4*(-a^4/b^5)^(3/4) + (a*x^3 - b)^(1/4)*a^3) - b*x^3*(-a^4/b^5)^(1/4)*log(-b^4*(-a^4/b^5)^(3/4) + (a*x^ 3 - b)^(1/4)*a^3) + 4*(a*x^3 - b)^(3/4))/(b*x^3)
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.27 \[ \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx=- \frac {\Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [4]{a} x^{\frac {15}{4}} \Gamma \left (\frac {9}{4}\right )} \]
-gamma(5/4)*hyper((1/4, 5/4), (9/4,), b*exp_polar(2*I*pi)/(a*x**3))/(3*a** (1/4)*x**(15/4)*gamma(9/4))
Time = 0.30 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx=\frac {{\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}}\right )} a}{24 \, b} + \frac {{\left (a x^{3} - b\right )}^{\frac {3}{4}} a}{3 \, {\left ({\left (a x^{3} - b\right )} b + b^{2}\right )}} \]
1/24*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^3 - 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^3 - b)^(1/4))/b^(1/4))/b^(1/4) - sqrt(2)*log(sqrt(2)*(a*x^3 - b)^(1/4)*b ^(1/4) + sqrt(a*x^3 - b) + sqrt(b))/b^(1/4) + sqrt(2)*log(-sqrt(2)*(a*x^3 - b)^(1/4)*b^(1/4) + sqrt(a*x^3 - b) + sqrt(b))/b^(1/4))*a/b + 1/3*(a*x^3 - b)^(3/4)*a/((a*x^3 - b)*b + b^2)
Time = 0.26 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx=\frac {\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {5}{4}}} + \frac {2 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {5}{4}}} - \frac {\sqrt {2} a^{2} \log \left (\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {5}{4}}} + \frac {\sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {5}{4}}} + \frac {8 \, {\left (a x^{3} - b\right )}^{\frac {3}{4}} a}{b x^{3}}}{24 \, a} \]
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^(5/4) + 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^(5/4) - sqrt(2)*a^2*log(sqrt(2)*(a*x^3 - b)^(1/4)*b^(1/4) + sqrt(a*x^3 - b) + sqrt(b))/b^(5/4) + sqrt(2)*a^2*log( -sqrt(2)*(a*x^3 - b)^(1/4)*b^(1/4) + sqrt(a*x^3 - b) + sqrt(b))/b^(5/4) + 8*(a*x^3 - b)^(3/4)*a/(b*x^3))/a
Time = 5.77 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx=\frac {{\left (a\,x^3-b\right )}^{3/4}}{3\,b\,x^3}-\frac {a\,\mathrm {atan}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{6\,{\left (-b\right )}^{5/4}}+\frac {a\,\mathrm {atanh}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{6\,{\left (-b\right )}^{5/4}} \]