3.1188 \(\int \frac{1}{(a-i a x)^{13/4} (a+i a x)^{3/4}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{16 i \sqrt [4]{a+i a x}}{45 a^4 \sqrt [4]{a-i a x}}-\frac{8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac{2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}} \]

[Out]

(((-2*I)/9)*(a + I*a*x)^(1/4))/(a^2*(a - I*a*x)^(9/4)) - (((8*I)/45)*(a + I*a*x)^(1/4))/(a^3*(a - I*a*x)^(5/4)
) - (((16*I)/45)*(a + I*a*x)^(1/4))/(a^4*(a - I*a*x)^(1/4))

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Rubi [A]  time = 0.0179842, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {45, 37} \[ -\frac{16 i \sqrt [4]{a+i a x}}{45 a^4 \sqrt [4]{a-i a x}}-\frac{8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac{2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(3/4)),x]

[Out]

(((-2*I)/9)*(a + I*a*x)^(1/4))/(a^2*(a - I*a*x)^(9/4)) - (((8*I)/45)*(a + I*a*x)^(1/4))/(a^3*(a - I*a*x)^(5/4)
) - (((16*I)/45)*(a + I*a*x)^(1/4))/(a^4*(a - I*a*x)^(1/4))

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])

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)^{13/4} (a+i a x)^{3/4}} \, dx &=-\frac{2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}}+\frac{4 \int \frac{1}{(a-i a x)^{9/4} (a+i a x)^{3/4}} \, dx}{9 a}\\ &=-\frac{2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}}-\frac{8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}+\frac{8 \int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{3/4}} \, dx}{45 a^2}\\ &=-\frac{2 i \sqrt [4]{a+i a x}}{9 a^2 (a-i a x)^{9/4}}-\frac{8 i \sqrt [4]{a+i a x}}{45 a^3 (a-i a x)^{5/4}}-\frac{16 i \sqrt [4]{a+i a x}}{45 a^4 \sqrt [4]{a-i a x}}\\ \end{align*}

Mathematica [A]  time = 0.0235545, size = 52, normalized size = 0.52 \[ \frac{2 \left (-8 i x^2+20 x+17 i\right ) \sqrt [4]{a+i a x}}{45 a^4 (x+i)^2 \sqrt [4]{a-i a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(3/4)),x]

[Out]

(2*(a + I*a*x)^(1/4)*(17*I + 20*x - (8*I)*x^2))/(45*a^4*(I + x)^2*(a - I*a*x)^(1/4))

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Maple [A]  time = 0.038, size = 50, normalized size = 0.5 \begin{align*}{\frac{24\,i{x}^{2}+16\,{x}^{3}+6\,x+34\,i}{45\,{a}^{3} \left ( x+i \right ) ^{2}} \left ( a \left ( 1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\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)^(13/4)/(a+I*a*x)^(3/4),x)

[Out]

2/45/a^3/(a*(1+I*x))^(3/4)/(-a*(-1+I*x))^(1/4)*(12*I*x^2+8*x^3+3*x+17*I)/(x+I)^2

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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{13}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(13/4)), x)

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Fricas [A]  time = 1.45051, size = 154, normalized size = 1.54 \begin{align*} \frac{2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (8 \, x^{2} + 20 i \, x - 17\right )}}{45 \, a^{5} x^{3} + 135 i \, a^{5} x^{2} - 135 \, a^{5} x - 45 i \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(3/4),x, algorithm="fricas")

[Out]

2*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4)*(8*x^2 + 20*I*x - 17)/(45*a^5*x^3 + 135*I*a^5*x^2 - 135*a^5*x - 45*I*a^
5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)**(13/4)/(a+I*a*x)**(3/4),x)

[Out]

Timed out

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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)^(13/4)/(a+I*a*x)^(3/4),x, algorithm="giac")

[Out]

Exception raised: TypeError