Integrand size = 29, antiderivative size = 291 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=-\frac {2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {(1-i) \sqrt [4]{-b c+a d} \arctan \left (\frac {(1+i) \sqrt [4]{d} \sqrt [4]{-b c+a d} x \sqrt [4]{-b x^3+a x^4}}{\sqrt {-b c+a d} x^2-i \sqrt {d} \sqrt {-b x^3+a x^4}}\right )}{c \sqrt [4]{d}}+\frac {2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}-\frac {(1-i) \sqrt [4]{-b c+a d} \text {arctanh}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{-b c+a d} x^2}{\sqrt [4]{d}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{d} \sqrt {-b x^3+a x^4}}{\sqrt [4]{-b c+a d}}}{x \sqrt [4]{-b x^3+a x^4}}\right )}{c \sqrt [4]{d}} \]
-2*a^(1/4)*arctan(a^(1/4)*x/(a*x^4-b*x^3)^(1/4))/c+(-1+I)*(a*d-b*c)^(1/4)* arctan((1+I)*d^(1/4)*(a*d-b*c)^(1/4)*x*(a*x^4-b*x^3)^(1/4)/((a*d-b*c)^(1/2 )*x^2-I*d^(1/2)*(a*x^4-b*x^3)^(1/2)))/c/d^(1/4)+2*a^(1/4)*arctanh(a^(1/4)* x/(a*x^4-b*x^3)^(1/4))/c+(-1+I)*(a*d-b*c)^(1/4)*arctanh(((1/2-1/2*I)*(a*d- b*c)^(1/4)*x^2/d^(1/4)+(1/2+1/2*I)*d^(1/4)*(a*x^4-b*x^3)^(1/2)/(a*d-b*c)^( 1/4))/x/(a*x^4-b*x^3)^(1/4))/c/d^(1/4)
Time = 1.04 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=-\frac {x^{9/4} (-b+a x)^{3/4} \left (2 \sqrt [4]{a} \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )+\sqrt {2} \sqrt [4]{b c-a d} \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{b c-a d} \sqrt [4]{x} \sqrt [4]{-b+a x}}{\sqrt {b c-a d} \sqrt {x}-\sqrt {d} \sqrt {-b+a x}}\right )-2 \sqrt [4]{a} \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )+\sqrt {2} \sqrt [4]{b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} \sqrt {x}+\sqrt {d} \sqrt {-b+a x}}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{b c-a d} \sqrt [4]{x} \sqrt [4]{-b+a x}}\right )\right )}{c \sqrt [4]{d} \left (x^3 (-b+a x)\right )^{3/4}} \]
-((x^(9/4)*(-b + a*x)^(3/4)*(2*a^(1/4)*d^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(- b + a*x)^(1/4)] + Sqrt[2]*(b*c - a*d)^(1/4)*ArcTan[(Sqrt[2]*d^(1/4)*(b*c - a*d)^(1/4)*x^(1/4)*(-b + a*x)^(1/4))/(Sqrt[b*c - a*d]*Sqrt[x] - Sqrt[d]*S qrt[-b + a*x])] - 2*a^(1/4)*d^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^( 1/4)] + Sqrt[2]*(b*c - a*d)^(1/4)*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x] + Sqrt[ d]*Sqrt[-b + a*x])/(Sqrt[2]*d^(1/4)*(b*c - a*d)^(1/4)*x^(1/4)*(-b + a*x)^( 1/4))]))/(c*d^(1/4)*(x^3*(-b + a*x))^(3/4)))
Time = 0.42 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.79, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {1948, 25, 140, 27, 73, 104, 827, 218, 221, 854, 827, 216, 219}
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.
| \(\displaystyle \int \frac {\sqrt [4]{a x^4-b x^3}}{x (c x-d)} \, dx\) |
| \(\Big \downarrow \) 1948 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \int -\frac {\sqrt [4]{a x-b}}{\sqrt [4]{x} (d-c x)}dx}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \int \frac {\sqrt [4]{a x-b}}{\sqrt [4]{x} (d-c x)}dx}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 140 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\int \frac {\frac {a d}{c}-b}{\sqrt [4]{x} (a x-b)^{3/4} (d-c x)}dx-\frac {a \int \frac {1}{\sqrt [4]{x} (a x-b)^{3/4}}dx}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-\left (b-\frac {a d}{c}\right ) \int \frac {1}{\sqrt [4]{x} (a x-b)^{3/4} (d-c x)}dx-\frac {a \int \frac {1}{\sqrt [4]{x} (a x-b)^{3/4}}dx}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-\left (b-\frac {a d}{c}\right ) \int \frac {1}{\sqrt [4]{x} (a x-b)^{3/4} (d-c x)}dx-\frac {4 a \int \frac {\sqrt {x}}{(a x-b)^{3/4}}d\sqrt [4]{x}}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-4 \left (b-\frac {a d}{c}\right ) \int \frac {\sqrt {x}}{\sqrt {a x-b} \left (d+\frac {(b c-a d) x}{a x-b}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}-\frac {4 a \int \frac {\sqrt {x}}{(a x-b)^{3/4}}d\sqrt [4]{x}}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-4 \left (b-\frac {a d}{c}\right ) \left (\frac {\int \frac {1}{\sqrt {d}-\frac {\sqrt {a d-b c} \sqrt {x}}{\sqrt {a x-b}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a d-b c}}-\frac {\int \frac {1}{\sqrt {d}+\frac {\sqrt {a d-b c} \sqrt {x}}{\sqrt {a x-b}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a d-b c}}\right )-\frac {4 a \int \frac {\sqrt {x}}{(a x-b)^{3/4}}d\sqrt [4]{x}}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-4 \left (b-\frac {a d}{c}\right ) \left (\frac {\int \frac {1}{\sqrt {d}-\frac {\sqrt {a d-b c} \sqrt {x}}{\sqrt {a x-b}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a d-b c}}-\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}\right )-\frac {4 a \int \frac {\sqrt {x}}{(a x-b)^{3/4}}d\sqrt [4]{x}}{c}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-\frac {4 a \int \frac {\sqrt {x}}{(a x-b)^{3/4}}d\sqrt [4]{x}}{c}-4 \left (b-\frac {a d}{c}\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-\frac {4 a \int \frac {\sqrt {x}}{1-a x}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{c}-4 \left (b-\frac {a d}{c}\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-\frac {4 a \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a}}\right )}{c}-4 \left (b-\frac {a d}{c}\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-\frac {4 a \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )}{c}-4 \left (b-\frac {a d}{c}\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-\frac {4 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )}{c}-4 \left (b-\frac {a d}{c}\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a d-b c}}{\sqrt [4]{d} \sqrt [4]{a x-b}}\right )}{2 \sqrt [4]{d} (a d-b c)^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
-(((-(b*x^3) + a*x^4)^(1/4)*((-4*a*(-1/2*ArcTan[(a^(1/4)*x^(1/4))/(-b + a* x)^(1/4)]/a^(3/4) + ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)]/(2*a^(3/4) )))/c - 4*(b - (a*d)/c)*(-1/2*ArcTan[((-(b*c) + a*d)^(1/4)*x^(1/4))/(d^(1/ 4)*(-b + a*x)^(1/4))]/(d^(1/4)*(-(b*c) + a*d)^(3/4)) + ArcTanh[((-(b*c) + a*d)^(1/4)*x^(1/4))/(d^(1/4)*(-b + a*x)^(1/4))]/(2*d^(1/4)*(-(b*c) + a*d)^ (3/4)))))/(x^(3/4)*(-b + a*x)^(1/4)))
3.29.45.3.1 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[(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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]
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 = 0.52 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.71
| method | result | size |
| pseudoelliptic | \(\frac {a^{\frac {1}{4}} \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}\right )+2 a^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )-\ln \left (\frac {\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{-\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}\right ) \left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}-2 \arctan \left (\frac {\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{x \left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}\right ) \left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}{c}\) | \(208\) |
(a^(1/4)*ln((a^(1/4)*x+(x^3*(a*x-b))^(1/4))/(-a^(1/4)*x+(x^3*(a*x-b))^(1/4 )))+2*a^(1/4)*arctan(1/a^(1/4)/x*(x^3*(a*x-b))^(1/4))-ln((((a*d-b*c)/d)^(1 /4)*x+(x^3*(a*x-b))^(1/4))/(-((a*d-b*c)/d)^(1/4)*x+(x^3*(a*x-b))^(1/4)))*( (a*d-b*c)/d)^(1/4)-2*arctan(1/x*(x^3*(a*x-b))^(1/4)/((a*d-b*c)/d)^(1/4))*( (a*d-b*c)/d)^(1/4))/c
Time = 0.28 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=\left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \log \left (\frac {i \, c x \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} \log \left (\frac {-i \, c x \left (-\frac {b c - a d}{c^{4} d}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
(a/c^4)^(1/4)*log((c*x*(a/c^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) - (a/c^4) ^(1/4)*log(-(c*x*(a/c^4)^(1/4) - (a*x^4 - b*x^3)^(1/4))/x) + I*(a/c^4)^(1/ 4)*log((I*c*x*(a/c^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) - I*(a/c^4)^(1/4)* log((-I*c*x*(a/c^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) - (-(b*c - a*d)/(c^4 *d))^(1/4)*log((c*x*(-(b*c - a*d)/(c^4*d))^(1/4) + (a*x^4 - b*x^3)^(1/4))/ x) + (-(b*c - a*d)/(c^4*d))^(1/4)*log(-(c*x*(-(b*c - a*d)/(c^4*d))^(1/4) - (a*x^4 - b*x^3)^(1/4))/x) - I*(-(b*c - a*d)/(c^4*d))^(1/4)*log((I*c*x*(-( b*c - a*d)/(c^4*d))^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) + I*(-(b*c - a*d)/(c ^4*d))^(1/4)*log((-I*c*x*(-(b*c - a*d)/(c^4*d))^(1/4) + (a*x^4 - b*x^3)^(1 /4))/x)
\[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{x \left (c x - d\right )}\, dx \]
\[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=\int { \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{{\left (c x - d\right )} x} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (228) = 456\).
Time = 0.32 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=\frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{c} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{c} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{2 \, c} - \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{2 \, c} - \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}}}\right )}{c d} - \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}}}\right )}{c d} - \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \log \left (\sqrt {2} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + \sqrt {a - \frac {b}{x}} + \sqrt {\frac {b c - a d}{d}}\right )}{2 \, c d} + \frac {\sqrt {2} {\left (b c d^{3} - a d^{4}\right )}^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + \sqrt {a - \frac {b}{x}} + \sqrt {\frac {b c - a d}{d}}\right )}{2 \, c d} \]
sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a - b/x)^(1 /4))/(-a)^(1/4))/c + sqrt(2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^ (1/4) - 2*(a - b/x)^(1/4))/(-a)^(1/4))/c + 1/2*sqrt(2)*(-a)^(1/4)*log(sqrt (2)*(-a)^(1/4)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))/c - 1/2*sqrt(2) *(-a)^(1/4)*log(-sqrt(2)*(-a)^(1/4)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))/c - sqrt(2)*(b*c*d^3 - a*d^4)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*((b* c - a*d)/d)^(1/4) + 2*(a - b/x)^(1/4))/((b*c - a*d)/d)^(1/4))/(c*d) - sqrt (2)*(b*c*d^3 - a*d^4)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*((b*c - a*d)/d)^( 1/4) - 2*(a - b/x)^(1/4))/((b*c - a*d)/d)^(1/4))/(c*d) - 1/2*sqrt(2)*(b*c* d^3 - a*d^4)^(1/4)*log(sqrt(2)*(a - b/x)^(1/4)*((b*c - a*d)/d)^(1/4) + sqr t(a - b/x) + sqrt((b*c - a*d)/d))/(c*d) + 1/2*sqrt(2)*(b*c*d^3 - a*d^4)^(1 /4)*log(-sqrt(2)*(a - b/x)^(1/4)*((b*c - a*d)/d)^(1/4) + sqrt(a - b/x) + s qrt((b*c - a*d)/d))/(c*d)
Timed out. \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{x (-d+c x)} \, dx=\int -\frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x\,\left (d-c\,x\right )} \,d x \]