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

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

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Rubi [A]  time = 0.00, antiderivative size = 33, 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 (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.11, size = 33, normalized size = 1.00 \begin {gather*} \frac {2 i (a-i a x)^{5/4}}{5 a^2 (a+i a x)^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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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fricas [B]  time = 1.43, size = 45, normalized size = 1.36 \begin {gather*} -\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 {gather*}

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

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:

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maple [B]  time = 0.04, size = 50, normalized size = 1.52 \begin {gather*} \frac {2 \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (x^{2}+2 i x -1\right )}{5 \left (i x -1\right ) \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} \left (x -i\right ) a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-i \, a x + a\right )}^{\frac {1}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}}\,{d x} \end {gather*}

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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mupad [B]  time = 0.55, size = 38, normalized size = 1.15 \begin {gather*} -\frac {2\,\left (-1+x\,1{}\mathrm {i}\right )\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}}{5\,a^2\,\left (x-\mathrm {i}\right )\,{\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((a - a*x*1i)^(1/4)/(a + a*x*1i)^(9/4),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{- i a \left (x + i\right )}}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}}}\, dx \end {gather*}

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

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