3.12.86 \(\int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{3/4}} \, dx\) [1186]

Optimal. Leaf size=31 \[ -\frac {2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {37} \begin {gather*} -\frac {2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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*} \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{3/4}} \, dx &=-\frac {2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 31, normalized size = 1.00 \begin {gather*} -\frac {2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(3/4)),x]')

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4371 deep

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Maple [A]
time = 0.16, size = 31, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(3/4),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(3/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.30, size = 31, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{a^{3} x + i \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {3}{4}} \left (- i a \left (x + i\right )\right )^{\frac {5}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)**(5/4)/(a+I*a*x)**(3/4),x)

[Out]

Integral(1/((I*a*(x - I))**(3/4)*(-I*a*(x + I))**(5/4)), x)

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{5/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{3/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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