3.1186 \(\int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{3/4}} \, dx\)

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 = 0.0031736, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {37} \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]

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*}

Mathematica [A]  time = 0.0126921, size = 31, normalized size = 1. \[ -\frac{2 i \sqrt [4]{a+i a x}}{a^2 \sqrt [4]{a-i a x}} \]

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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Maple [A]  time = 0.029, size = 31, normalized size = 1. \begin{align*} 2\,{\frac{x-i}{a \left ( a \left ( 1+ix \right ) \right ) ^{3/4}\sqrt [4]{-a \left ( -1+ix \right ) }}} \end{align*}

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)

[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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}

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 = 1.54243, size = 76, normalized size = 2.45 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \left (i x + 1\right )\right )^{\frac{3}{4}} \left (- a \left (i x - 1\right )\right )^{\frac{5}{4}}}\, dx \end{align*}

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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="giac")

[Out]

Exception raised: TypeError