\(\int \frac {(a-i a x)^{3/4}}{(a+i a x)^{7/4}} \, dx\) [1195]

Optimal result

Integrand size = 25, antiderivative size = 266 \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{7/4}} \, dx=\frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}+\frac {i \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a}+\frac {i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a} \]

[Out]

4/3*I*(a-I*a*x)^(3/4)/a/(a+I*a*x)^(3/4)-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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {49, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{7/4}} \, dx=\frac {i \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt {2} a}+\frac {i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt {2} a} \]

[In]

Int[(a - I*a*x)^(3/4)/(a + I*a*x)^(7/4),x]

[Out]

(((4*I)/3)*(a - I*a*x)^(3/4))/(a*(a + I*a*x)^(3/4)) + (I*Sqrt[2]*ArcTan[1 - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I
*a*x)^(1/4)])/a - (I*Sqrt[2]*ArcTan[1 + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/a - (I*Log[1 + Sqrt[a
- I*a*x]/Sqrt[a + I*a*x] - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/(Sqrt[2]*a) + (I*Log[1 + Sqrt[a - I
*a*x]/Sqrt[a + I*a*x] + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/(Sqrt[2]*a)

Rule 49

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] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && 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]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

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])

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = \frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{3/4}} \, dx \\ & = \frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {(4 i) \text {Subst}\left (\int \frac {x^2}{\left (2 a-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{a-i a x}\right )}{a} \\ & = \frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {(4 i) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \\ & = \frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}+\frac {(2 i) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {(2 i) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \\ & = \frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac {i \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a}-\frac {i \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2} a} \\ & = \frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\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}-\frac {\left (i \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac {\left (i \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \\ & = \frac {4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}+\frac {i \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac {i \sqrt {2} \tan ^{-1}\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} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.97 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.26 \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{7/4}} \, dx=\frac {i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{7/4} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},\frac {7}{4},\frac {11}{4},\frac {1}{2}-\frac {i x}{2}\right )}{7 a^2 (a+i a x)^{3/4}} \]

[In]

Integrate[(a - I*a*x)^(3/4)/(a + I*a*x)^(7/4),x]

[Out]

((I/7)*2^(1/4)*(1 + I*x)^(3/4)*(a - I*a*x)^(7/4)*Hypergeometric2F1[7/4, 7/4, 11/4, 1/2 - (I/2)*x])/(a^2*(a + I
*a*x)^(3/4))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.97 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.73

method	result	size
risch	\(\frac {\frac {4 x}{3}+\frac {4 i}{3}}{\left (a \left (i x +1\right )\right )^{\frac {3}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}-\frac {\left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \ln \left (\frac {-\left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) x^{2}-x^{3}-i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {3}{4}}+2 i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} x -i \sqrt {-x^{4}+2 i x^{3}+2 i x +1}\, x +2 i x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}}-\sqrt {-x^{4}+2 i x^{3}+2 i x +1}+x}{\left (i x +1\right )^{2}}\right )-i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \ln \left (-\frac {-i \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} x +x^{3}-i \sqrt {-x^{4}+2 i x^{3}+2 i x +1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {3}{4}}+i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}}-2 i x^{2}-\sqrt {-x^{4}+2 i x^{3}+2 i x +1}-x}{\left (i x +1\right )^{2}}\right )\right ) \left (-\left (i x -1\right ) \left (i x +1\right )^{3}\right )^{\frac {1}{4}}}{\left (a \left (i x +1\right )\right )^{\frac {3}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}\)	\(459\)

[In]

int((a-I*a*x)^(3/4)/(a+I*a*x)^(7/4),x,method=_RETURNVERBOSE)

[Out]

4/3*(x+I)/(a*(I*x+1))^(3/4)/(-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*(I*x+1))^(3/4)*(-(I*x-1)*
(I*x+1)^3)^(1/4)/(-a*(I*x-1))^(1/4)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.12 \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{7/4}} \, dx=-\frac {3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, {\left (x + i\right )}}\right ) - 3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, {\left (x + i\right )}}\right ) + 3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} \log \left (\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} + 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, {\left (x + i\right )}}\right ) - 3 \, {\left (a^{2} x - i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} \log \left (-\frac {{\left (a^{2} x + i \, a^{2}\right )} \sqrt {-\frac {4 i}{a^{2}}} - 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, {\left (x + i\right )}}\right ) - 8 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{6 \, {\left (a^{2} x - i \, a^{2}\right )}} \]

[In]

integrate((a-I*a*x)^(3/4)/(a+I*a*x)^(7/4),x, algorithm="fricas")

[Out]

-1/6*(3*(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)^(1/4)*(-I*a*x + a
)^(3/4))/(x + I)) - 3*(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)^(1
/4)*(-I*a*x + a)^(3/4))/(x + I)) + 3*(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)^(1/4)*(-I*a*x + a)^(3/4))/(x + I)) - 3*(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)^(1/4)*(-I*a*x + a)^(3/4))/(x + I)) - 8*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4))/(a
^2*x - I*a^2)

Sympy [F]

\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{7/4}} \, dx=\int \frac {\left (- i a \left (x + i\right )\right )^{\frac {3}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {7}{4}}}\, dx \]

[In]

integrate((a-I*a*x)**(3/4)/(a+I*a*x)**(7/4),x)

[Out]

Integral((-I*a*(x + I))**(3/4)/(I*a*(x - I))**(7/4), x)

Maxima [F]

\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{7/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {7}{4}}} \,d x } \]

[In]

integrate((a-I*a*x)^(3/4)/(a+I*a*x)^(7/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(3/4)/(I*a*x + a)^(7/4), x)

Giac [F]

\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{7/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {7}{4}}} \,d x } \]

[In]

integrate((a-I*a*x)^(3/4)/(a+I*a*x)^(7/4),x, algorithm="giac")

[Out]

integrate((-I*a*x + a)^(3/4)/(I*a*x + a)^(7/4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{7/4}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \]

[In]

int((a - a*x*1i)^(3/4)/(a + a*x*1i)^(7/4),x)

[Out]

int((a - a*x*1i)^(3/4)/(a + a*x*1i)^(7/4), x)