Integrand size = 26, antiderivative size = 237 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=-\frac {2 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{a}-\frac {\sqrt {2} \sqrt [4]{a-b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} x \sqrt [4]{-x^3+x^4}}{\sqrt {a-b} x^2-\sqrt {b} \sqrt {-x^3+x^4}}\right )}{a \sqrt [4]{b}}+\frac {2 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{a}-\frac {\sqrt {2} \sqrt [4]{a-b} \text {arctanh}\left (\frac {\frac {\sqrt [4]{a-b} x^2}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {-x^3+x^4}}{\sqrt {2} \sqrt [4]{a-b}}}{x \sqrt [4]{-x^3+x^4}}\right )}{a \sqrt [4]{b}} \]
-2*arctan(x/(x^4-x^3)^(1/4))/a-2^(1/2)*(a-b)^(1/4)*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)+2*arctanh(x/(x^4-x^3)^(1/4))/a-2^(1/2)*(a-b)^(1/4)*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)
Time = 0.69 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=-\frac {(-1+x)^{3/4} x^{9/4} \left (2 \sqrt [4]{b} \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+\sqrt {2} \sqrt [4]{a-b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} \sqrt [4]{-1+x} \sqrt [4]{x}}{-\sqrt {b} \sqrt {-1+x}+\sqrt {a-b} \sqrt {x}}\right )-2 \sqrt [4]{b} \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+\sqrt {2} \sqrt [4]{a-b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {-1+x}+\sqrt {a-b} \sqrt {x}}{\sqrt {2} \sqrt [4]{a-b} \sqrt [4]{b} \sqrt [4]{-1+x} \sqrt [4]{x}}\right )\right )}{a \sqrt [4]{b} \left ((-1+x) x^3\right )^{3/4}} \]
-(((-1 + x)^(3/4)*x^(9/4)*(2*b^(1/4)*ArcTan[((-1 + x)/x)^(-1/4)] + Sqrt[2] *(a - b)^(1/4)*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])] - 2*b^(1/4)*ArcTanh[(( -1 + x)/x)^(-1/4)] + Sqrt[2]*(a - b)^(1/4)*ArcTanh[(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)*((-1 + x)*x^3)^(3/4)))
Time = 0.34 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.73, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1948, 25, 140, 27, 73, 104, 770, 756, 216, 219, 827, 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.
| \(\displaystyle \int \frac {\sqrt [4]{x^4-x^3}}{x (a x-b)} \, dx\) |
| \(\Big \downarrow \) 1948 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \int -\frac {\sqrt [4]{x-1}}{\sqrt [4]{x} (b-a x)}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1}}{\sqrt [4]{x} (b-a x)}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 140 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (\int \frac {\frac {b}{a}-1}{(x-1)^{3/4} \sqrt [4]{x} (b-a x)}dx-\frac {\int \frac {1}{(x-1)^{3/4} \sqrt [4]{x}}dx}{a}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-\left (1-\frac {b}{a}\right ) \int \frac {1}{(x-1)^{3/4} \sqrt [4]{x} (b-a x)}dx-\frac {\int \frac {1}{(x-1)^{3/4} \sqrt [4]{x}}dx}{a}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-\left (1-\frac {b}{a}\right ) \int \frac {1}{(x-1)^{3/4} \sqrt [4]{x} (b-a x)}dx-\frac {4 \int \frac {1}{\sqrt [4]{x}}d\sqrt [4]{x-1}}{a}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-4 \left (1-\frac {b}{a}\right ) \int \frac {\sqrt {x}}{\sqrt {x-1} \left (b+\frac {(a-b) x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}-\frac {4 \int \frac {1}{\sqrt [4]{x}}d\sqrt [4]{x-1}}{a}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-4 \left (1-\frac {b}{a}\right ) \int \frac {\sqrt {x}}{\sqrt {x-1} \left (b+\frac {(a-b) x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}-\frac {4 \int \frac {1}{2-x}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}}{a}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-4 \left (1-\frac {b}{a}\right ) \int \frac {\sqrt {x}}{\sqrt {x-1} \left (b+\frac {(a-b) x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}-\frac {4 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x-1}}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}+\frac {1}{2} \int \frac {1}{\sqrt {x-1}+1}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{a}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-\frac {4 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x-1}}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}+\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )}{a}-4 \left (1-\frac {b}{a}\right ) \int \frac {\sqrt {x}}{\sqrt {x-1} \left (b+\frac {(a-b) x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-4 \left (1-\frac {b}{a}\right ) \int \frac {\sqrt {x}}{\sqrt {x-1} \left (b+\frac {(a-b) x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}-\frac {4 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )}{a}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-4 \left (1-\frac {b}{a}\right ) \left (\frac {\int \frac {1}{\sqrt {b}-\frac {\sqrt {b-a} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\sqrt {b}+\frac {\sqrt {b-a} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {b-a}}\right )-\frac {4 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )}{a}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-4 \left (1-\frac {b}{a}\right ) \left (\frac {\int \frac {1}{\sqrt {b}-\frac {\sqrt {b-a} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {b-a}}-\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{2 \sqrt [4]{b} (b-a)^{3/4}}\right )-\frac {4 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )}{a}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\sqrt [4]{x^4-x^3} \left (-4 \left (1-\frac {b}{a}\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{2 \sqrt [4]{b} (b-a)^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{b-a}}{\sqrt [4]{b} \sqrt [4]{x-1}}\right )}{2 \sqrt [4]{b} (b-a)^{3/4}}\right )-\frac {4 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )}{a}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
-(((-x^3 + x^4)^(1/4)*((-4*(ArcTan[(-1 + x)^(1/4)/x^(1/4)]/2 + ArcTanh[(-1 + x)^(1/4)/x^(1/4)]/2))/a - 4*(1 - b/a)*(-1/2*ArcTan[((-a + b)^(1/4)*x^(1 /4))/(b^(1/4)*(-1 + x)^(1/4))]/(b^(1/4)*(-a + b)^(3/4)) + ArcTanh[((-a + b )^(1/4)*x^(1/4))/(b^(1/4)*(-1 + x)^(1/4))]/(2*b^(1/4)*(-a + b)^(3/4)))))/( (-1 + x)^(1/4)*x^(3/4)))
3.27.56.3.1 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_))^(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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 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[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
Time = 1.08 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.27
| method | result | size |
| pseudoelliptic | \(\frac {-\ln \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {1}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a -b}{b}}\, x^{2}+\sqrt {x^{3} \left (-1+x \right )}}{\sqrt {x^{3} \left (-1+x \right )}-\left (\frac {a -b}{b}\right )^{\frac {1}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a -b}{b}}\, x^{2}}\right ) \left (\frac {a -b}{b}\right )^{\frac {1}{4}} \sqrt {2}-2 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}+\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}{\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}\right ) \left (\frac {a -b}{b}\right )^{\frac {1}{4}} \sqrt {2}-2 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}-\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}{\left (\frac {a -b}{b}\right )^{\frac {1}{4}} x}\right ) \left (\frac {a -b}{b}\right )^{\frac {1}{4}} \sqrt {2}+2 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )-2 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )+4 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{2 a}\) | \(301\) |
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))/((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))*((a-b)/b)^(1/4)*2^(1/2)-2*arctan(((x^3*( -1+x))^(1/4)*2^(1/2)+((a-b)/b)^(1/4)*x)/((a-b)/b)^(1/4)/x)*((a-b)/b)^(1/4) *2^(1/2)-2*arctan(((x^3*(-1+x))^(1/4)*2^(1/2)-((a-b)/b)^(1/4)*x)/((a-b)/b) ^(1/4)/x)*((a-b)/b)^(1/4)*2^(1/2)+2*ln((x+(x^3*(-1+x))^(1/4))/x)-2*ln(((x^ 3*(-1+x))^(1/4)-x)/x)+4*arctan((x^3*(-1+x))^(1/4)/x))/a
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=-\frac {a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \log \left (\frac {a x \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \log \left (-\frac {a x \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \log \left (\frac {i \, a x \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, a \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} \log \left (\frac {-i \, a x \left (-\frac {a - b}{a^{4} b}\right )^{\frac {1}{4}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 2 \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right )}{a} \]
-(a*(-(a - b)/(a^4*b))^(1/4)*log((a*x*(-(a - b)/(a^4*b))^(1/4) + (x^4 - x^ 3)^(1/4))/x) - a*(-(a - b)/(a^4*b))^(1/4)*log(-(a*x*(-(a - b)/(a^4*b))^(1/ 4) - (x^4 - x^3)^(1/4))/x) + I*a*(-(a - b)/(a^4*b))^(1/4)*log((I*a*x*(-(a - b)/(a^4*b))^(1/4) + (x^4 - x^3)^(1/4))/x) - I*a*(-(a - b)/(a^4*b))^(1/4) *log((-I*a*x*(-(a - b)/(a^4*b))^(1/4) + (x^4 - x^3)^(1/4))/x) - 2*arctan(( x^4 - x^3)^(1/4)/x) - log((x + (x^4 - x^3)^(1/4))/x) + log(-(x - (x^4 - x^ 3)^(1/4))/x))/a
\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )}}{x \left (a x - b\right )}\, dx \]
\[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{{\left (a x - b\right )} x} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=\frac {2 \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}{a} + \frac {\log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right )}{a} - \frac {\log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right )}{a} - \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a - b}{b}\right )^{\frac {1}{4}}}\right )}{a b} - \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a - b}{b}\right )^{\frac {1}{4}}}\right )}{a b} - \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{4}} \log \left (\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {a - b}{b}} + \sqrt {-\frac {1}{x} + 1}\right )}{2 \, a b} + \frac {\sqrt {2} {\left (a b^{3} - b^{4}\right )}^{\frac {1}{4}} \log \left (-\sqrt {2} \left (\frac {a - b}{b}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {a - b}{b}} + \sqrt {-\frac {1}{x} + 1}\right )}{2 \, a b} \]
2*arctan((-1/x + 1)^(1/4))/a + log((-1/x + 1)^(1/4) + 1)/a - log(abs((-1/x + 1)^(1/4) - 1))/a - sqrt(2)*(a*b^3 - b^4)^(1/4)*arctan(1/2*sqrt(2)*(sqrt (2)*((a - b)/b)^(1/4) + 2*(-1/x + 1)^(1/4))/((a - b)/b)^(1/4))/(a*b) - sqr t(2)*(a*b^3 - b^4)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*((a - b)/b)^(1/4) - 2*(-1/x + 1)^(1/4))/((a - b)/b)^(1/4))/(a*b) - 1/2*sqrt(2)*(a*b^3 - b^4)^( 1/4)*log(sqrt(2)*((a - b)/b)^(1/4)*(-1/x + 1)^(1/4) + sqrt((a - b)/b) + sq rt(-1/x + 1))/(a*b) + 1/2*sqrt(2)*(a*b^3 - b^4)^(1/4)*log(-sqrt(2)*((a - b )/b)^(1/4)*(-1/x + 1)^(1/4) + sqrt((a - b)/b) + sqrt(-1/x + 1))/(a*b)
Timed out. \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x (-b+a x)} \, dx=\int -\frac {{\left (x^4-x^3\right )}^{1/4}}{x\,\left (b-a\,x\right )} \,d x \]