Integrand size = 25, antiderivative size = 264 \[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx=\frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac {i \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a}-\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a} \]
4*I*(a-I*a*x)^(1/4)/a/(a+I*a*x)^(1/4)+1/2*I*ln(1-(a-I*a*x)^(1/4)*2^(1/2)/( a+I*a*x)^(1/4)+(a-I*a*x)^(1/2)/(a+I*a*x)^(1/2))/a*2^(1/2)-1/2*I*ln(1+(a-I* a*x)^(1/4)*2^(1/2)/(a+I*a*x)^(1/4)+(a-I*a*x)^(1/2)/(a+I*a*x)^(1/2))/a*2^(1 /2)+I*arctan(1-(a-I*a*x)^(1/4)*2^(1/2)/(a+I*a*x)^(1/4))*2^(1/2)/a-I*arctan (1+(a-I*a*x)^(1/4)*2^(1/2)/(a+I*a*x)^(1/4))*2^(1/2)/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.69 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx=\frac {i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{5/4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2}-\frac {i x}{2}\right )}{5 a^2 \sqrt [4]{a+i a x}} \]
((I/5)*2^(3/4)*(1 + I*x)^(1/4)*(a - I*a*x)^(5/4)*Hypergeometric2F1[5/4, 5/ 4, 9/4, 1/2 - (I/2)*x])/(a^2*(a + I*a*x)^(1/4))
Time = 0.34 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.91, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {57, 73, 770, 755, 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.
| \(\displaystyle \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\int \frac {1}{(a-i a x)^{3/4} \sqrt [4]{i x a+a}}dx\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {4 i \int \frac {1}{\sqrt [4]{i x a+a}}d\sqrt [4]{a-i a x}}{a}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {4 i \int \frac {1}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{a}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {4 i \left (\frac {1}{2} \int \frac {1-\sqrt {a-i a x}}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+\frac {1}{2} \int \frac {\sqrt {a-i a x}+1}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}\right )}{a}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {4 i \left (\frac {1}{2} \int \frac {1-\sqrt {a-i a x}}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {a-i a x}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+\frac {1}{2} \int \frac {1}{\sqrt {a-i a x}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}\right )\right )}{a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {4 i \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {a-i a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {a-i a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {a-i a x}}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}\right )}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {4 i \left (\frac {1}{2} \int \frac {1-\sqrt {a-i a x}}{-i x a+a+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {4 i \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{\sqrt {a-i a x}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1\right )}{\sqrt {a-i a x}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {4 i \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{\sqrt {a-i a x}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1\right )}{\sqrt {a-i a x}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {4 i \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{\sqrt {a-i a x}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}{\sqrt {a-i a x}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}+1}d\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{i x a+a}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\right )\right )}{a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac {4 i \left (\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {a-i a x}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {a-i a x}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}\) |
((4*I)*(a - I*a*x)^(1/4))/(a*(a + I*a*x)^(1/4)) - ((4*I)*((-(ArcTan[1 - (S qrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2 ]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)]/Sqrt[2])/2 + (-1/2*Log[1 + Sqrt[a - I*a*x] - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[a - I*a*x] + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)]/(2*Sqrt [2]))/2))/a
3.13.13.3.1 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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, 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_)^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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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_)^(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[((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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.04 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.81
| method | result | size |
| risch | \(-\frac {4 \left (x +i\right ) \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}{a \left (i x -1\right ) \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}+\frac {\left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+i\right ) \ln \left (\frac {-\left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+i\right ) x^{2}-x^{3}+i \operatorname {RootOf}\left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {3}{4}}-2 i \operatorname {RootOf}\left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x +i \sqrt {-x^{4}-2 i x^{3}-2 i x +1}\, x -2 i x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}}-\sqrt {-x^{4}-2 i x^{3}-2 i x +1}+x}{\left (i x -1\right )^{2}}\right )-i \operatorname {RootOf}\left (\textit {\_Z}^{2}+i\right ) \ln \left (-\frac {-i \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+i\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}} x +x^{3}+i \sqrt {-x^{4}-2 i x^{3}-2 i x +1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {3}{4}}+i \operatorname {RootOf}\left (\textit {\_Z}^{2}+i\right ) \left (-x^{4}-2 i x^{3}-2 i x +1\right )^{\frac {1}{4}}+2 i x^{2}-\sqrt {-x^{4}-2 i x^{3}-2 i x +1}-x}{\left (i x -1\right )^{2}}\right )\right ) \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (-\left (i x -1\right )^{3} \left (i x +1\right )\right )^{\frac {1}{4}}}{a \left (i x -1\right ) \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(479\) |
-4*(x+I)/a*(-a*(I*x-1))^(1/4)/(I*x-1)/(a*(I*x+1))^(1/4)+(RootOf(_Z^2+I)*ln ((-(1-x^4-2*I*x^3-2*I*x)^(1/4)*RootOf(_Z^2+I)*x^2-x^3+I*RootOf(_Z^2+I)*(1- x^4-2*I*x^3-2*I*x)^(3/4)-2*I*RootOf(_Z^2+I)*(1-x^4-2*I*x^3-2*I*x)^(1/4)*x+ I*(1-x^4-2*I*x^3-2*I*x)^(1/2)*x-2*I*x^2+RootOf(_Z^2+I)*(1-x^4-2*I*x^3-2*I* x)^(1/4)-(1-x^4-2*I*x^3-2*I*x)^(1/2)+x)/(I*x-1)^2)-I*RootOf(_Z^2+I)*ln(-(- I*(1-x^4-2*I*x^3-2*I*x)^(1/4)*RootOf(_Z^2+I)*x^2+2*RootOf(_Z^2+I)*(1-x^4-2 *I*x^3-2*I*x)^(1/4)*x+x^3+I*(1-x^4-2*I*x^3-2*I*x)^(1/2)*x+RootOf(_Z^2+I)*( 1-x^4-2*I*x^3-2*I*x)^(3/4)+I*RootOf(_Z^2+I)*(1-x^4-2*I*x^3-2*I*x)^(1/4)+2* I*x^2-(1-x^4-2*I*x^3-2*I*x)^(1/2)-x)/(I*x-1)^2))/a*(-a*(I*x-1))^(1/4)/(I*x -1)*(-(I*x-1)^3*(I*x+1))^(1/4)/(a*(I*x+1))^(1/4)
Time = 0.24 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx=-\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, {\left (x - i\right )}}\right ) - {\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, {\left (x - i\right )}}\right ) + {\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, {\left (x - i\right )}}\right ) - {\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, {\left (x - i\right )}}\right ) - 8 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{2 \, {\left (a^{2} x - i \, a^{2}\right )}} \]
-1/2*((a^2*x - I*a^2)*sqrt(4*I/a^2)*log(1/2*((a^2*x - I*a^2)*sqrt(4*I/a^2) + 2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/(x - I)) - (a^2*x - I*a^2)*sqrt (4*I/a^2)*log(-1/2*((a^2*x - I*a^2)*sqrt(4*I/a^2) - 2*(I*a*x + a)^(3/4)*(- I*a*x + a)^(1/4))/(x - I)) + (a^2*x - I*a^2)*sqrt(-4*I/a^2)*log(1/2*((a^2* x - I*a^2)*sqrt(-4*I/a^2) + 2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/(x - I )) - (a^2*x - I*a^2)*sqrt(-4*I/a^2)*log(-1/2*((a^2*x - I*a^2)*sqrt(-4*I/a^ 2) - 2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/(x - I)) - 8*(I*a*x + a)^(3/4 )*(-I*a*x + a)^(1/4))/(a^2*x - I*a^2)
\[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx=\int \frac {\sqrt [4]{- i a \left (x + i\right )}}{\left (i a \left (x - i\right )\right )^{\frac {5}{4}}}\, dx \]
\[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {5}{4}}} \,d x } \]
\[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {5}{4}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{1/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{5/4}} \,d x \]