\(\int \frac {(a-i a x)^{5/4}}{(a+i a x)^{3/4}} \, dx\) [1189]

Optimal result

Integrand size = 25, antiderivative size = 112 \[ \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{3/4}} \, dx=-\frac {10}{3} i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}-\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{3 a}+\frac {10 a^2 \left (1+x^2\right )^{3/4} \operatorname {EllipticF}\left (\frac {\arctan (x)}{2},2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

[Out]

-10/3*I*(a-I*a*x)^(1/4)*(a+I*a*x)^(1/4)-2/3*I*(a-I*a*x)^(5/4)*(a+I*a*x)^(1/4)/a+10/3*a^2*(x^2+1)^(3/4)*(cos(1/
2*arctan(x))^2)^(1/2)/cos(1/2*arctan(x))*EllipticF(sin(1/2*arctan(x)),2^(1/2))/(a-I*a*x)^(3/4)/(a+I*a*x)^(3/4)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {52, 42, 239, 237} \[ \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{3/4}} \, dx=\frac {10 a^2 \left (x^2+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {\arctan (x)}{2},2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {10}{3} i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}-\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{3 a} \]

[In]

Int[(a - I*a*x)^(5/4)/(a + I*a*x)^(3/4),x]

[Out]

((-10*I)/3)*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4) - (((2*I)/3)*(a - I*a*x)^(5/4)*(a + I*a*x)^(1/4))/a + (10*a^2*
(1 + x^2)^(3/4)*EllipticF[ArcTan[x]/2, 2])/(3*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))

Rule 42

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[(a + b*x)^FracPart[m]*((c + d*x)^Frac
Part[m]/(a*c + b*d*x^2)^FracPart[m]), Int[(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c +
a*d, 0] &&  !IntegerQ[2*m]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 237

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]))*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 239

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[(1 + b*(x^2/a))^(3/4)/(a + b*x^2)^(3/4), Int[1/(1 + b*(x^2
/a))^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{3 a}+\frac {1}{3} (5 a) \int \frac {\sqrt [4]{a-i a x}}{(a+i a x)^{3/4}} \, dx \\ & = -\frac {10}{3} i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}-\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{3 a}+\frac {1}{3} \left (5 a^2\right ) \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx \\ & = -\frac {10}{3} i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}-\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{3 a}+\frac {\left (5 a^2 \left (a^2+a^2 x^2\right )^{3/4}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{3/4}} \, dx}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \\ & = -\frac {10}{3} i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}-\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{3 a}+\frac {\left (5 a^2 \left (1+x^2\right )^{3/4}\right ) \int \frac {1}{\left (1+x^2\right )^{3/4}} \, dx}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \\ & = -\frac {10}{3} i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}-\frac {2 i (a-i a x)^{5/4} \sqrt [4]{a+i a x}}{3 a}+\frac {10 a^2 \left (1+x^2\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.62 \[ \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{3/4}} \, dx=\frac {2 i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{9/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {9}{4},\frac {13}{4},\frac {1}{2}-\frac {i x}{2}\right )}{9 a (a+i a x)^{3/4}} \]

[In]

Integrate[(a - I*a*x)^(5/4)/(a + I*a*x)^(3/4),x]

[Out]

(((2*I)/9)*2^(1/4)*(1 + I*x)^(3/4)*(a - I*a*x)^(9/4)*Hypergeometric2F1[3/4, 9/4, 13/4, 1/2 - (I/2)*x])/(a*(a +
I*a*x)^(3/4))

Maple [F]

\[\int \frac {\left (-i a x +a \right )^{\frac {5}{4}}}{\left (i a x +a \right )^{\frac {3}{4}}}d x\]

[In]

int((a-I*a*x)^(5/4)/(a+I*a*x)^(3/4),x)

[Out]

int((a-I*a*x)^(5/4)/(a+I*a*x)^(3/4),x)

Fricas [F]

\[ \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{3/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {5}{4}}}{{\left (i \, a x + a\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate((a-I*a*x)^(5/4)/(a+I*a*x)^(3/4),x, algorithm="fricas")

[Out]

-2/3*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4)*(x + 6*I) + integral(5/3*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4)/(x^2 +
1), x)

Sympy [F]

\[ \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{3/4}} \, dx=\int \frac {\left (- i a \left (x + i\right )\right )^{\frac {5}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {3}{4}}}\, dx \]

[In]

integrate((a-I*a*x)**(5/4)/(a+I*a*x)**(3/4),x)

[Out]

Integral((-I*a*(x + I))**(5/4)/(I*a*(x - I))**(3/4), x)

Maxima [F]

\[ \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{3/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {5}{4}}}{{\left (i \, a x + a\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate((a-I*a*x)^(5/4)/(a+I*a*x)^(3/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(5/4)/(I*a*x + a)^(3/4), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{3/4}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((a-I*a*x)^(5/4)/(a+I*a*x)^(3/4),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:The choice was do
ne assuming 0=[0,0]ext_reduce Error: Bad Argument Typeintegrate(1/4*16*i*((sageVARa+(-i)*sageVARa*sageVARx)^(1
/4))^8/((-((sa

Mupad [F(-1)]

Timed out. \[ \int \frac {(a-i a x)^{5/4}}{(a+i a x)^{3/4}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{5/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{3/4}} \,d x \]

[In]

int((a - a*x*1i)^(5/4)/(a + a*x*1i)^(3/4),x)

[Out]

int((a - a*x*1i)^(5/4)/(a + a*x*1i)^(3/4), x)