3.12.25 \(\int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=67 \[ \frac {4 i \sqrt [4]{a-i a x}}{5 a^3 \sqrt [4]{a+i a x}}+\frac {2 i \sqrt [4]{a-i a x}}{5 a^2 (a+i a x)^{5/4}} \]

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Rubi [A]  time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {45, 37} \begin {gather*} \frac {4 i \sqrt [4]{a-i a x}}{5 a^3 \sqrt [4]{a+i a x}}+\frac {2 i \sqrt [4]{a-i a x}}{5 a^2 (a+i a x)^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(3/4)*(a + I*a*x)^(9/4)),x]

[Out]

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

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{9/4}} \, dx &=\frac {2 i \sqrt [4]{a-i a x}}{5 a^2 (a+i a x)^{5/4}}+\frac {2 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{5 a}\\ &=\frac {2 i \sqrt [4]{a-i a x}}{5 a^2 (a+i a x)^{5/4}}+\frac {4 i \sqrt [4]{a-i a x}}{5 a^3 \sqrt [4]{a+i a x}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 45, normalized size = 0.67 \begin {gather*} \frac {2 (3+2 i x) \sqrt [4]{a-i a x}}{5 a^3 (x-i) \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)^(9/4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.14, size = 54, normalized size = 0.81 \begin {gather*} \frac {i \sqrt [4]{a-i a x} \left (5+\frac {a-i a x}{a+i a x}\right )}{5 a^3 \sqrt [4]{a+i a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a - I*a*x)^(3/4)*(a + I*a*x)^(9/4)),x]

[Out]

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

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fricas [A]  time = 1.36, size = 44, normalized size = 0.66 \begin {gather*} \frac {{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (4 \, x - 6 i\right )}}{5 \, {\left (a^{4} x^{2} - 2 i \, a^{4} x - a^{4}\right )}} \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)^(9/4),x, algorithm="fricas")

[Out]

1/5*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*(4*x - 6*I)/(a^4*x^2 - 2*I*a^4*x - a^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}\,{d x} \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)^(9/4),x, algorithm="giac")

[Out]

integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(3/4)), x)

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maple [A]  time = 0.04, size = 44, normalized size = 0.66 \begin {gather*} \frac {\frac {4}{5} x^{2}-\frac {2}{5} i x +\frac {6}{5}}{\left (-\left (i x -1\right ) a \right )^{\frac {3}{4}} \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(1/(-I*a*x+a)^(3/4)/(I*a*x+a)^(9/4),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}\,{d x} \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)^(9/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(3/4)), x)

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mupad [B]  time = 0.63, size = 38, normalized size = 0.57 \begin {gather*} \frac {2\,\left (3+x\,2{}\mathrm {i}\right )\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}}{5\,a^3\,\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(1/((a - a*x*1i)^(3/4)*(a + a*x*1i)^(9/4)),x)

[Out]

(2*(x*2i + 3)*(-a*(x*1i - 1))^(1/4))/(5*a^3*(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 {1}{\left (i a \left (x - i\right )\right )^{\frac {9}{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)**(9/4),x)

[Out]

Integral(1/((I*a*(x - I))**(9/4)*(-I*a*(x + I))**(3/4)), x)

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