3.12.16 \(\int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{7/4}} \, dx\)

Optimal. Leaf size=65 \[ \frac {4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}-\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \]

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Rubi [A]  time = 0.01, antiderivative size = 65, 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 (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}-\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(7/4)),x]

[Out]

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

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)^{5/4} (a+i a x)^{7/4}} \, dx &=-\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}}+\frac {2 \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{7/4}} \, dx}{a}\\ &=-\frac {2 i}{a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}}+\frac {4 i (a-i a x)^{3/4}}{3 a^3 (a+i a x)^{3/4}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.58 \begin {gather*} \frac {4 x-2 i}{3 a^2 \sqrt [4]{a-i a x} (a+i a x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(7/4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.10, size = 55, normalized size = 0.85 \begin {gather*} -\frac {i (a-i a x)^{3/4} \left (-1+\frac {3 (a+i a x)}{a-i a x}\right )}{3 a^3 (a+i a x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a - I*a*x)^(5/4)*(a + I*a*x)^(7/4)),x]

[Out]

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

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fricas [A]  time = 1.10, size = 36, normalized size = 0.55 \begin {gather*} \frac {{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (4 \, x - 2 i\right )}}{3 \, {\left (a^{4} x^{2} + a^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(7/4),x, algorithm="fricas")

[Out]

1/3*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4)*(4*x - 2*I)/(a^4*x^2 + 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 {7}{4}} {\left (-i \, a x + a\right )}^{\frac {5}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(7/4),x, algorithm="giac")

[Out]

integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(5/4)), x)

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maple [A]  time = 0.05, size = 33, normalized size = 0.51 \begin {gather*} \frac {\frac {4 x}{3}-\frac {2 i}{3}}{\left (\left (i x +1\right ) a \right )^{\frac {3}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-I*a*x+a)^(5/4)/(I*a*x+a)^(7/4),x)

[Out]

2/3/a^2/((I*x+1)*a)^(3/4)/(-(I*x-1)*a)^(1/4)*(2*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 {7}{4}} {\left (-i \, a x + a\right )}^{\frac {5}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(5/4)/(a+I*a*x)^(7/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(5/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{5/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a - a*x*1i)^(5/4)*(a + a*x*1i)^(7/4)),x)

[Out]

int(1/((a - a*x*1i)^(5/4)*(a + a*x*1i)^(7/4)), x)

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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 {7}{4}} \left (- i a \left (x + i\right )\right )^{\frac {5}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)**(5/4)/(a+I*a*x)**(7/4),x)

[Out]

Integral(1/((I*a*(x - I))**(7/4)*(-I*a*(x + I))**(5/4)), x)

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