3.1225 \(\int \frac{\sqrt [4]{a-i a x}}{(a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=33 \[ \frac{2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \]

[Out]

(((2*I)/5)*(a - I*a*x)^(5/4))/(a^2*(a + I*a*x)^(5/4))

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Rubi [A]  time = 0.0032964, antiderivative size = 33, 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} \[ \frac{2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((2*I)/5)*(a - I*a*x)^(5/4))/(a^2*(a + I*a*x)^(5/4))

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{\sqrt [4]{a-i a x}}{(a+i a x)^{9/4}} \, dx &=\frac{2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}}\\ \end{align*}

Mathematica [A]  time = 0.011016, size = 33, normalized size = 1. \[ \frac{2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((2*I)/5)*(a - I*a*x)^(5/4))/(a^2*(a + I*a*x)^(5/4))

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Maple [B]  time = 0.034, size = 50, normalized size = 1.5 \begin{align*}{\frac{4\,ix+2\,{x}^{2}-2}{5\,{a}^{2} \left ( -1+ix \right ) \left ( x-i \right ) }\sqrt [4]{-a \left ( -1+ix \right ) }{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{9}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.985, size = 113, normalized size = 3.42 \begin{align*} -\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (2 \, x + 2 i\right )}}{5 \, a^{3} x^{2} - 10 i \, a^{3} x - 5 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*(2*x + 2*I)/(5*a^3*x^2 - 10*I*a^3*x - 5*a^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [4]{- a \left (i x - 1\right )}}{\left (a \left (i x + 1\right )\right )^{\frac{9}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11823, size = 46, normalized size = 1.39 \begin{align*} -\frac{{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (-\frac{4 i \, a}{i \, a x + a} + 2 i\right )}}{5 \,{\left (i \, a x + a\right )}^{\frac{1}{4}} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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