\(\int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx\) [1175]

Optimal result

Integrand size = 25, antiderivative size = 82 \[ \int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx=-\frac {4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {2 \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

[Out]

-4/5*I/a/(a-I*a*x)^(5/4)/(a+I*a*x)^(1/4)+2/5*(x^2+1)^(1/4)*(cos(1/2*arctan(x))^2)^(1/2)/cos(1/2*arctan(x))*Ell
ipticE(sin(1/2*arctan(x)),2^(1/2))/a^2/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {48, 42, 203, 202} \[ \int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx=\frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]

[In]

Int[1/((a - I*a*x)^(9/4)*(a + I*a*x)^(1/4)),x]

[Out]

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

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 48

Int[1/(((a_) + (b_.)*(x_))^(9/4)*((c_) + (d_.)*(x_))^(1/4)), x_Symbol] :> Simp[-4/(5*b*(a + b*x)^(5/4)*(c + d*
x)^(1/4)), x] - Dist[d/(5*b), Int[1/((a + b*x)^(5/4)*(c + d*x)^(5/4)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ
[b*c + a*d, 0] && NegQ[a^2*b^2]

Rule 202

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]))*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Dist[(1 + b*(x^2/a))^(1/4)/(a*(a + b*x^2)^(1/4)), Int[1/(1 + b*
(x^2/a))^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a] && PosQ[b/a]

Rubi steps \begin{align*} \text {integral}& = -\frac {4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {1}{5} \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx \\ & = -\frac {4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {\sqrt [4]{a^2+a^2 x^2} \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = -\frac {4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {\sqrt [4]{1+x^2} \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = -\frac {4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {2 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ \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 = 0.85 \[ \int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx=-\frac {2 i 2^{3/4} \sqrt [4]{1+i x} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{4},-\frac {1}{4},\frac {1}{2}-\frac {i x}{2}\right )}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]

[In]

Integrate[1/((a - I*a*x)^(9/4)*(a + I*a*x)^(1/4)),x]

[Out]

(((-2*I)/5)*2^(3/4)*(1 + I*x)^(1/4)*Hypergeometric2F1[-5/4, 1/4, -1/4, 1/2 - (I/2)*x])/(a*(a - I*a*x)^(5/4)*(a
+ I*a*x)^(1/4))

Maple [C] (verified)

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

Time = 0.38 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.28

method	result	size
risch	\(\frac {\frac {2}{5} x^{2}+\frac {2}{5} i x +\frac {4}{5}}{\left (x +i\right ) a^{2} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}-\frac {x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-x^{2}\right ) \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{5 \left (a^{2}\right )^{\frac {1}{4}} a^{2} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\)	\(105\)

[In]

int(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {9}{4}}} \,d x } \]

[In]

integrate(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(1/4),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx=\int \frac {1}{\sqrt [4]{i a \left (x - i\right )} \left (- i a \left (x + i\right )\right )^{\frac {9}{4}}}\, dx \]

[In]

integrate(1/(a-I*a*x)**(9/4)/(a+I*a*x)**(1/4),x)

[Out]

Integral(1/((I*a*(x - I))**(1/4)*(-I*a*(x + I))**(9/4)), x)

Maxima [F]

\[ \int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {9}{4}}} \,d x } \]

[In]

integrate(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(9/4)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(1/4),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:The choice was done assuming 0=[0,0]ext_reduce Error: Bad Argument TypeDone

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{9/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \]

[In]

int(1/((a - a*x*1i)^(9/4)*(a + a*x*1i)^(1/4)),x)

[Out]

int(1/((a - a*x*1i)^(9/4)*(a + a*x*1i)^(1/4)), x)