\(\int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx\) [1196]

Optimal result

Integrand size = 25, antiderivative size = 33 \[ \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx=\frac {2 i (a-i a x)^{3/4}}{3 a^2 (a+i a x)^{3/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {37} \[ \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx=\frac {2 i (a-i a x)^{3/4}}{3 a^2 (a+i a x)^{3/4}} \]

[In]

Int[1/((a - I*a*x)^(1/4)*(a + I*a*x)^(7/4)),x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 i (a-i a x)^{3/4}}{3 a^2 (a+i a x)^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.71 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx=\frac {2 i (a-i a x)^{3/4}}{3 a^2 (a+i a x)^{3/4}} \]

[In]

Integrate[1/((a - I*a*x)^(1/4)*(a + I*a*x)^(7/4)),x]

[Out]

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94

method	result	size
risch	\(\frac {\frac {2 x}{3}+\frac {2 i}{3}}{a \left (a \left (i x +1\right )\right )^{\frac {3}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}\)	\(31\)
gosper	\(-\frac {2 i \left (x +i\right ) \left (-x +i\right )}{3 \left (-i a x +a \right )^{\frac {1}{4}} \left (i a x +a \right )^{\frac {7}{4}}}\)	\(32\)

[In]

int(1/(a-I*a*x)^(1/4)/(a+I*a*x)^(7/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx=\frac {2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{3 \, {\left (a^{3} x - i \, a^{3}\right )}} \]

[In]

integrate(1/(a-I*a*x)^(1/4)/(a+I*a*x)^(7/4),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx=\int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {7}{4}} \sqrt [4]{- i a \left (x + i\right )}}\, dx \]

[In]

integrate(1/(a-I*a*x)**(1/4)/(a+I*a*x)**(7/4),x)

[Out]

Integral(1/((I*a*(x - I))**(7/4)*(-I*a*(x + I))**(1/4)), x)

Maxima [F]

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

[In]

integrate(1/(a-I*a*x)^(1/4)/(a+I*a*x)^(7/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(1/4)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a-I*a*x)^(1/4)/(a+I*a*x)^(7/4),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. \[ \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{1/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \]

[In]

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

[Out]

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