3.13.14 \(\int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx\) [1214]

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)^(3/4)*(a + I*a*x)^(5/4)),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)^{3/4} (a+i a x)^{5/4}} \, 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 = 0.05, 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)^(3/4)*(a + I*a*x)^(5/4)),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)^(3/4)*(a + I*a*x)^(5/4)),x]')

[Out]

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

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Maple [A]
time = 0.18, 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)^(3/4)/(a+I*a*x)^(5/4),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)^(3/4)/(a+I*a*x)^(5/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(5/4)*(-I*a*x + a)^(3/4)), 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 {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{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)^(3/4)/(a+I*a*x)^(5/4),x, algorithm="fricas")

[Out]

2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)/(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 {5}{4}} \left (- i a \left (x + i\right )\right )^{\frac {3}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Could not integrate

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Mupad [B]
time = 1.16, size = 27, normalized size = 0.87 \begin {gather*} \frac {{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,2{}\mathrm {i}}{a^2\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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