Integrand size = 31, antiderivative size = 221 \[ \int \frac {a+b x}{x (-d+c x) \sqrt [4]{-x^3+x^4}} \, dx=-\frac {4 a \left (-x^3+x^4\right )^{3/4}}{3 d x^3}+\frac {\sqrt {2} (a c+b d) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c-d} \sqrt [4]{d} x \sqrt [4]{-x^3+x^4}}{\sqrt {c-d} x^2-\sqrt {d} \sqrt {-x^3+x^4}}\right )}{\sqrt [4]{c-d} d^{7/4}}-\frac {\sqrt {2} (a c+b d) \text {arctanh}\left (\frac {\sqrt {c-d} x^2+\sqrt {d} \sqrt {-x^3+x^4}}{\sqrt {2} \sqrt [4]{c-d} \sqrt [4]{d} x \sqrt [4]{-x^3+x^4}}\right )}{\sqrt [4]{c-d} d^{7/4}} \]
-4/3*a*(x^4-x^3)^(3/4)/d/x^3+2^(1/2)*(a*c+b*d)*arctan(2^(1/2)*(c-d)^(1/4)* d^(1/4)*x*(x^4-x^3)^(1/4)/((c-d)^(1/2)*x^2-d^(1/2)*(x^4-x^3)^(1/2)))/(c-d) ^(1/4)/d^(7/4)-2^(1/2)*(a*c+b*d)*arctanh(1/2*((c-d)^(1/2)*x^2+d^(1/2)*(x^4 -x^3)^(1/2))*2^(1/2)/(c-d)^(1/4)/d^(1/4)/x/(x^4-x^3)^(1/4))/(c-d)^(1/4)/d^ (7/4)
Time = 6.07 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.04 \[ \int \frac {a+b x}{x (-d+c x) \sqrt [4]{-x^3+x^4}} \, dx=\frac {-4 a \sqrt [4]{c-d} d^{3/4} (-1+x)+3 \sqrt {2} (a c+b d) \sqrt [4]{-1+x} x^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c-d} \sqrt [4]{d} \sqrt [4]{-1+x} \sqrt [4]{x}}{-\sqrt {d} \sqrt {-1+x}+\sqrt {c-d} \sqrt {x}}\right )-3 \sqrt {2} (a c+b d) \sqrt [4]{-1+x} x^{3/4} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {-1+x}+\sqrt {c-d} \sqrt {x}}{\sqrt {2} \sqrt [4]{c-d} \sqrt [4]{d} \sqrt [4]{-1+x} \sqrt [4]{x}}\right )}{3 \sqrt [4]{c-d} d^{7/4} \sqrt [4]{(-1+x) x^3}} \]
(-4*a*(c - d)^(1/4)*d^(3/4)*(-1 + x) + 3*Sqrt[2]*(a*c + b*d)*(-1 + x)^(1/4 )*x^(3/4)*ArcTan[(Sqrt[2]*(c - d)^(1/4)*d^(1/4)*(-1 + x)^(1/4)*x^(1/4))/(- (Sqrt[d]*Sqrt[-1 + x]) + Sqrt[c - d]*Sqrt[x])] - 3*Sqrt[2]*(a*c + b*d)*(-1 + x)^(1/4)*x^(3/4)*ArcTanh[(Sqrt[d]*Sqrt[-1 + x] + Sqrt[c - d]*Sqrt[x])/( Sqrt[2]*(c - d)^(1/4)*d^(1/4)*(-1 + x)^(1/4)*x^(1/4))])/(3*(c - d)^(1/4)*d ^(7/4)*((-1 + x)*x^3)^(1/4))
Time = 0.40 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.69, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2467, 25, 172, 27, 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.
| \(\displaystyle \int \frac {a+b x}{x \sqrt [4]{x^4-x^3} (c x-d)} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x-1} x^{3/4} \int -\frac {a+b x}{\sqrt [4]{x-1} x^{7/4} (d-c x)}dx}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x-1} x^{3/4} \int \frac {a+b x}{\sqrt [4]{x-1} x^{7/4} (d-c x)}dx}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle -\frac {\sqrt [4]{x-1} x^{3/4} \left (\frac {4 \int \frac {3 (a c+b d)}{4 \sqrt [4]{x-1} x^{3/4} (d-c x)}dx}{3 d}+\frac {4 a (x-1)^{3/4}}{3 d x^{3/4}}\right )}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [4]{x-1} x^{3/4} \left (\frac {(a c+b d) \int \frac {1}{\sqrt [4]{x-1} x^{3/4} (d-c x)}dx}{d}+\frac {4 a (x-1)^{3/4}}{3 d x^{3/4}}\right )}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {\sqrt [4]{x-1} x^{3/4} \left (\frac {4 (a c+b d) \int \frac {1}{d+\frac {(c-d) x}{x-1}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{d}+\frac {4 a (x-1)^{3/4}}{3 d x^{3/4}}\right )}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {\sqrt [4]{x-1} x^{3/4} \left (\frac {4 (a c+b d) \left (\frac {\int \frac {1}{\sqrt {d}-\frac {\sqrt {d-c} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {d}}+\frac {\int \frac {1}{\sqrt {d}+\frac {\sqrt {d-c} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {d}}\right )}{d}+\frac {4 a (x-1)^{3/4}}{3 d x^{3/4}}\right )}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\sqrt [4]{x-1} x^{3/4} \left (\frac {4 (a c+b d) \left (\frac {\int \frac {1}{\sqrt {d}-\frac {\sqrt {d-c} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{d-c}}{\sqrt [4]{d} \sqrt [4]{x-1}}\right )}{2 d^{3/4} \sqrt [4]{d-c}}\right )}{d}+\frac {4 a (x-1)^{3/4}}{3 d x^{3/4}}\right )}{\sqrt [4]{x^4-x^3}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\sqrt [4]{x-1} x^{3/4} \left (\frac {4 (a c+b d) \left (\frac {\arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{d-c}}{\sqrt [4]{d} \sqrt [4]{x-1}}\right )}{2 d^{3/4} \sqrt [4]{d-c}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{d-c}}{\sqrt [4]{d} \sqrt [4]{x-1}}\right )}{2 d^{3/4} \sqrt [4]{d-c}}\right )}{d}+\frac {4 a (x-1)^{3/4}}{3 d x^{3/4}}\right )}{\sqrt [4]{x^4-x^3}}\) |
-(((-1 + x)^(1/4)*x^(3/4)*((4*a*(-1 + x)^(3/4))/(3*d*x^(3/4)) + (4*(a*c + b*d)*(ArcTan[((-c + d)^(1/4)*x^(1/4))/(d^(1/4)*(-1 + x)^(1/4))]/(2*d^(3/4) *(-c + d)^(1/4)) + ArcTanh[((-c + d)^(1/4)*x^(1/4))/(d^(1/4)*(-1 + x)^(1/4 ))]/(2*d^(3/4)*(-c + d)^(1/4))))/d))/(-x^3 + x^4)^(1/4))
3.26.74.3.1 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_))/((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_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[ (b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1) *(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f )*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1 ])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
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 = 1.29 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(-\frac {4 \left (a \left (x^{3} \left (-1+x \right )\right )^{\frac {3}{4}} d \left (\frac {c -d}{d}\right )^{\frac {1}{4}}-\frac {3 \left (a c +b d \right ) x^{3} \left (\ln \left (\frac {-\left (\frac {c -d}{d}\right )^{\frac {1}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c -d}{d}}\, x^{2}+\sqrt {x^{3} \left (-1+x \right )}}{\left (\frac {c -d}{d}\right )^{\frac {1}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c -d}{d}}\, x^{2}+\sqrt {x^{3} \left (-1+x \right )}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-\left (\frac {c -d}{d}\right )^{\frac {1}{4}} x}{\left (\frac {c -d}{d}\right )^{\frac {1}{4}} x}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+\left (\frac {c -d}{d}\right )^{\frac {1}{4}} x}{\left (\frac {c -d}{d}\right )^{\frac {1}{4}} x}\right )\right ) \sqrt {2}}{8}\right )}{3 \left (\frac {c -d}{d}\right )^{\frac {1}{4}} d^{2} x^{3}}\) | \(256\) |
-4/3/((c-d)/d)^(1/4)*(a*(x^3*(-1+x))^(3/4)*d*((c-d)/d)^(1/4)-3/8*(a*c+b*d) *x^3*(ln((-((c-d)/d)^(1/4)*(x^3*(-1+x))^(1/4)*2^(1/2)*x+((c-d)/d)^(1/2)*x^ 2+(x^3*(-1+x))^(1/2))/(((c-d)/d)^(1/4)*(x^3*(-1+x))^(1/4)*2^(1/2)*x+((c-d) /d)^(1/2)*x^2+(x^3*(-1+x))^(1/2)))+2*arctan((2^(1/2)*(x^3*(-1+x))^(1/4)-(( c-d)/d)^(1/4)*x)/((c-d)/d)^(1/4)/x)+2*arctan((2^(1/2)*(x^3*(-1+x))^(1/4)+( (c-d)/d)^(1/4)*x)/((c-d)/d)^(1/4)/x))*2^(1/2))/d^2/x^3
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 843, normalized size of antiderivative = 3.81 \[ \int \frac {a+b x}{x (-d+c x) \sqrt [4]{-x^3+x^4}} \, dx=\frac {3 \, d x^{3} \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (c d^{5} - d^{6}\right )} x \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {3}{4}} + {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} + b^{3} d^{3}\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 3 \, d x^{3} \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (c d^{5} - d^{6}\right )} x \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {3}{4}} - {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} + b^{3} d^{3}\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 3 i \, d x^{3} \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {1}{4}} \log \left (\frac {i \, {\left (c d^{5} - d^{6}\right )} x \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {3}{4}} + {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} + b^{3} d^{3}\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 3 i \, d x^{3} \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, {\left (c d^{5} - d^{6}\right )} x \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {3}{4}} + {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} + b^{3} d^{3}\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {3}{4}} a}{3 \, d x^{3}} \]
1/3*(3*d*x^3*(-(a^4*c^4 + 4*a^3*b*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a*b^3*c*d^ 3 + b^4*d^4)/(c*d^7 - d^8))^(1/4)*log(((c*d^5 - d^6)*x*(-(a^4*c^4 + 4*a^3* b*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a*b^3*c*d^3 + b^4*d^4)/(c*d^7 - d^8))^(3/4 ) + (a^3*c^3 + 3*a^2*b*c^2*d + 3*a*b^2*c*d^2 + b^3*d^3)*(x^4 - x^3)^(1/4)) /x) - 3*d*x^3*(-(a^4*c^4 + 4*a^3*b*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a*b^3*c*d ^3 + b^4*d^4)/(c*d^7 - d^8))^(1/4)*log(-((c*d^5 - d^6)*x*(-(a^4*c^4 + 4*a^ 3*b*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a*b^3*c*d^3 + b^4*d^4)/(c*d^7 - d^8))^(3 /4) - (a^3*c^3 + 3*a^2*b*c^2*d + 3*a*b^2*c*d^2 + b^3*d^3)*(x^4 - x^3)^(1/4 ))/x) - 3*I*d*x^3*(-(a^4*c^4 + 4*a^3*b*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a*b^3 *c*d^3 + b^4*d^4)/(c*d^7 - d^8))^(1/4)*log((I*(c*d^5 - d^6)*x*(-(a^4*c^4 + 4*a^3*b*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a*b^3*c*d^3 + b^4*d^4)/(c*d^7 - d^8 ))^(3/4) + (a^3*c^3 + 3*a^2*b*c^2*d + 3*a*b^2*c*d^2 + b^3*d^3)*(x^4 - x^3) ^(1/4))/x) + 3*I*d*x^3*(-(a^4*c^4 + 4*a^3*b*c^3*d + 6*a^2*b^2*c^2*d^2 + 4* a*b^3*c*d^3 + b^4*d^4)/(c*d^7 - d^8))^(1/4)*log((-I*(c*d^5 - d^6)*x*(-(a^4 *c^4 + 4*a^3*b*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a*b^3*c*d^3 + b^4*d^4)/(c*d^7 - d^8))^(3/4) + (a^3*c^3 + 3*a^2*b*c^2*d + 3*a*b^2*c*d^2 + b^3*d^3)*(x^4 - x^3)^(1/4))/x) - 4*(x^4 - x^3)^(3/4)*a)/(d*x^3)
\[ \int \frac {a+b x}{x (-d+c x) \sqrt [4]{-x^3+x^4}} \, dx=\int \frac {a + b x}{x \sqrt [4]{x^{3} \left (x - 1\right )} \left (c x - d\right )}\, dx \]
\[ \int \frac {a+b x}{x (-d+c x) \sqrt [4]{-x^3+x^4}} \, dx=\int { \frac {b x + a}{{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (c x - d\right )} x} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.53 \[ \int \frac {a+b x}{x (-d+c x) \sqrt [4]{-x^3+x^4}} \, dx=-\frac {\sqrt {2} {\left (a c + b d\right )} \log \left (\sqrt {2} \left (\frac {c - d}{d}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {c - d}{d}} + \sqrt {-\frac {1}{x} + 1}\right )}{2 \, {\left (c d^{3} - d^{4}\right )}^{\frac {1}{4}} d} + \frac {{\left ({\left (c d^{3} - d^{4}\right )}^{\frac {3}{4}} a c + {\left (c d^{3} - d^{4}\right )}^{\frac {3}{4}} b d\right )} \log \left (-\sqrt {2} \left (\frac {c - d}{d}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {c - d}{d}} + \sqrt {-\frac {1}{x} + 1}\right )}{\sqrt {2} c d^{4} - \sqrt {2} d^{5}} - \frac {4 \, a {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{4}}}{3 \, d} + \frac {{\left (\sqrt {2} a c + \sqrt {2} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c - d}{d}\right )^{\frac {1}{4}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c - d}{d}\right )^{\frac {1}{4}}}\right )}{{\left (c d^{3} - d^{4}\right )}^{\frac {1}{4}} d} + \frac {{\left (\sqrt {2} a c + \sqrt {2} b d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c - d}{d}\right )^{\frac {1}{4}} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c - d}{d}\right )^{\frac {1}{4}}}\right )}{{\left (c d^{3} - d^{4}\right )}^{\frac {1}{4}} d} \]
-1/2*sqrt(2)*(a*c + b*d)*log(sqrt(2)*((c - d)/d)^(1/4)*(-1/x + 1)^(1/4) + sqrt((c - d)/d) + sqrt(-1/x + 1))/((c*d^3 - d^4)^(1/4)*d) + ((c*d^3 - d^4) ^(3/4)*a*c + (c*d^3 - d^4)^(3/4)*b*d)*log(-sqrt(2)*((c - d)/d)^(1/4)*(-1/x + 1)^(1/4) + sqrt((c - d)/d) + sqrt(-1/x + 1))/(sqrt(2)*c*d^4 - sqrt(2)*d ^5) - 4/3*a*(-1/x + 1)^(3/4)/d + (sqrt(2)*a*c + sqrt(2)*b*d)*arctan(1/2*sq rt(2)*(sqrt(2)*((c - d)/d)^(1/4) + 2*(-1/x + 1)^(1/4))/((c - d)/d)^(1/4))/ ((c*d^3 - d^4)^(1/4)*d) + (sqrt(2)*a*c + sqrt(2)*b*d)*arctan(-1/2*sqrt(2)* (sqrt(2)*((c - d)/d)^(1/4) - 2*(-1/x + 1)^(1/4))/((c - d)/d)^(1/4))/((c*d^ 3 - d^4)^(1/4)*d)
Timed out. \[ \int \frac {a+b x}{x (-d+c x) \sqrt [4]{-x^3+x^4}} \, dx=\int -\frac {a+b\,x}{x\,{\left (x^4-x^3\right )}^{1/4}\,\left (d-c\,x\right )} \,d x \]