Integrand size = 23, antiderivative size = 188 \[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} x \sqrt [4]{-x^3+x^4}}{\sqrt {a-b} x^2-\sqrt {b} \sqrt {-x^3+x^4}}\right )}{\sqrt [4]{a-b} b^{3/4}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\frac {\sqrt [4]{a-b} x^2}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {-x^3+x^4}}{\sqrt {2} \sqrt [4]{a-b}}}{x \sqrt [4]{-x^3+x^4}}\right )}{\sqrt [4]{a-b} b^{3/4}} \]
2^(1/2)*arctan(2^(1/2)*(a-b)^(1/4)*b^(1/4)*x*(x^4-x^3)^(1/4)/((a-b)^(1/2)* x^2-b^(1/2)*(x^4-x^3)^(1/2)))/(a-b)^(1/4)/b^(3/4)-2^(1/2)*arctanh((1/2*(a- b)^(1/4)*x^2*2^(1/2)/b^(1/4)+1/2*b^(1/4)*(x^4-x^3)^(1/2)*2^(1/2)/(a-b)^(1/ 4))/x/(x^4-x^3)^(1/4))/(a-b)^(1/4)/b^(3/4)
Time = 0.57 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=\frac {\sqrt {2} \sqrt [4]{-1+x} x^{3/4} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} \sqrt [4]{-1+x} \sqrt [4]{x}}{-\sqrt {b} \sqrt {-1+x}+\sqrt {a-b} \sqrt {x}}\right )-\text {arctanh}\left (\frac {\sqrt {b} \sqrt {-1+x}+\sqrt {a-b} \sqrt {x}}{\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} \sqrt [4]{-1+x} \sqrt [4]{x}}\right )\right )}{\sqrt [4]{a-b} b^{3/4} \sqrt [4]{(-1+x) x^3}} \]
(Sqrt[2]*(-1 + x)^(1/4)*x^(3/4)*(ArcTan[(Sqrt[2]*(a - b)^(1/4)*b^(1/4)*(-1 + x)^(1/4)*x^(1/4))/(-(Sqrt[b]*Sqrt[-1 + x]) + Sqrt[a - b]*Sqrt[x])] - Ar cTanh[(Sqrt[b]*Sqrt[-1 + x] + Sqrt[a - b]*Sqrt[x])/(Sqrt[2]*(a - b)^(1/4)* b^(1/4)*(-1 + x)^(1/4)*x^(1/4))]))/((a - b)^(1/4)*b^(3/4)*((-1 + x)*x^3)^( 1/4))
Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.64, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2467, 25, 104, 756, 218, 221}
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.
| \(\displaystyle \int \frac {1}{\sqrt [4]{x^4-x^3} (a x-b)} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x-1} x^{3/4} \int -\frac {1}{\sqrt [4]{x-1} x^{3/4} (b-a x)}dx}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x-1} x^{3/4} \int \frac {1}{\sqrt [4]{x-1} x^{3/4} (b-a x)}dx}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {4 \sqrt [4]{x-1} x^{3/4} \int \frac {1}{b+\frac {(a-b) x}{x-1}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {4 \sqrt [4]{x-1} x^{3/4} \left (\frac {\int \frac {1}{\sqrt {b}-\frac {\sqrt {b-a} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\sqrt {b}+\frac {\sqrt {b-a} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {b}}\right )}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {4 \sqrt [4]{x-1} x^{3/4} \left (\frac {\int \frac {1}{\sqrt {b}-\frac {\sqrt {b-a} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{2 b^{3/4} \sqrt [4]{b-a}}\right )}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {4 \sqrt [4]{x-1} x^{3/4} \left (\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{2 b^{3/4} \sqrt [4]{b-a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{2 b^{3/4} \sqrt [4]{b-a}}\right )}{\sqrt [4]{x^4-x^3}}\) |
(-4*(-1 + x)^(1/4)*x^(3/4)*(ArcTan[((-a + b)^(1/4)*x^(1/4))/(b^(1/4)*(-1 + x)^(1/4))]/(2*b^(3/4)*(-a + b)^(1/4)) + ArcTanh[((-a + b)^(1/4)*x^(1/4))/ (b^(1/4)*(-1 + x)^(1/4))]/(2*b^(3/4)*(-a + b)^(1/4))))/(-x^3 + x^4)^(1/4)
3.24.66.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 1.33 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.15
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (\frac {a -b}{b}\right )^{\frac {1}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a -b}{b}}\, x^{2}+\sqrt {x^{3} \left (-1+x \right )}}{\left (\frac {a -b}{b}\right )^{\frac {1}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a -b}{b}}\, x^{2}+\sqrt {x^{3} \left (-1+x \right )}}\right )+2 \arctan \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x +\sqrt {2}\, \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x -\sqrt {2}\, \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}\right )\right )}{2 \left (\frac {a -b}{b}\right )^{\frac {1}{4}} b}\) | \(217\) |
1/2/((a-b)/b)^(1/4)*2^(1/2)*(ln((-((a-b)/b)^(1/4)*(x^3*(-1+x))^(1/4)*2^(1/ 2)*x+((a-b)/b)^(1/2)*x^2+(x^3*(-1+x))^(1/2))/(((a-b)/b)^(1/4)*(x^3*(-1+x)) ^(1/4)*2^(1/2)*x+((a-b)/b)^(1/2)*x^2+(x^3*(-1+x))^(1/2)))+2*arctan((((a-b) /b)^(1/4)*x+2^(1/2)*(x^3*(-1+x))^(1/4))/((a-b)/b)^(1/4)/x)-2*arctan((((a-b )/b)^(1/4)*x-2^(1/2)*(x^3*(-1+x))^(1/4))/((a-b)/b)^(1/4)/x))/b
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=\left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a b^{2} - b^{3}\right )} x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {3}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a b^{2} - b^{3}\right )} x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {3}{4}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, {\left (a b^{2} - b^{3}\right )} x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {3}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, {\left (a b^{2} - b^{3}\right )} x \left (-\frac {1}{a b^{3} - b^{4}}\right )^{\frac {3}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
(-1/(a*b^3 - b^4))^(1/4)*log(((a*b^2 - b^3)*x*(-1/(a*b^3 - b^4))^(3/4) + ( x^4 - x^3)^(1/4))/x) - (-1/(a*b^3 - b^4))^(1/4)*log(-((a*b^2 - b^3)*x*(-1/ (a*b^3 - b^4))^(3/4) - (x^4 - x^3)^(1/4))/x) - I*(-1/(a*b^3 - b^4))^(1/4)* log((I*(a*b^2 - b^3)*x*(-1/(a*b^3 - b^4))^(3/4) + (x^4 - x^3)^(1/4))/x) + I*(-1/(a*b^3 - b^4))^(1/4)*log((-I*(a*b^2 - b^3)*x*(-1/(a*b^3 - b^4))^(3/4 ) + (x^4 - x^3)^(1/4))/x)
\[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{3} \left (x - 1\right )} \left (a x - b\right )}\, dx \]
\[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (a x - b\right )}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (147) = 294\).
Time = 0.29 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.69 \[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=\frac {2 \, {\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a - b}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} + \frac {2 \, {\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a - b}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} - \frac {{\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \log \left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {a - b}{b}} + \sqrt {-\frac {1}{x} + 1}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} + \frac {{\left (a b^{3} - b^{4}\right )}^{\frac {3}{4}} \log \left (-\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {a - b}{b}} + \sqrt {-\frac {1}{x} + 1}\right )}{\sqrt {2} a b^{3} - \sqrt {2} b^{4}} \]
2*(a*b^3 - b^4)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*((a - b)/b)^(1/4) + 2*(- 1/x + 1)^(1/4))/((a - b)/b)^(1/4))/(sqrt(2)*a*b^3 - sqrt(2)*b^4) + 2*(a*b^ 3 - b^4)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*((a - b)/b)^(1/4) - 2*(-1/x + 1)^(1/4))/((a - b)/b)^(1/4))/(sqrt(2)*a*b^3 - sqrt(2)*b^4) - (a*b^3 - b^4) ^(3/4)*log(sqrt(2)*((a - b)/b)^(1/4)*(-1/x + 1)^(1/4) + sqrt((a - b)/b) + sqrt(-1/x + 1))/(sqrt(2)*a*b^3 - sqrt(2)*b^4) + (a*b^3 - b^4)^(3/4)*log(-s qrt(2)*((a - b)/b)^(1/4)*(-1/x + 1)^(1/4) + sqrt((a - b)/b) + sqrt(-1/x + 1))/(sqrt(2)*a*b^3 - sqrt(2)*b^4)
Timed out. \[ \int \frac {1}{(-b+a x) \sqrt [4]{-x^3+x^4}} \, dx=-\int \frac {1}{{\left (x^4-x^3\right )}^{1/4}\,\left (b-a\,x\right )} \,d x \]