3.1216 \(\int \frac{1}{(a-i a x)^{11/4} (a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=100 \[ \frac{16 i \sqrt [4]{a-i a x}}{21 a^4 \sqrt [4]{a+i a x}}-\frac{8 i}{21 a^3 \sqrt [4]{a+i a x} (a-i a x)^{3/4}}-\frac{2 i}{7 a^2 \sqrt [4]{a+i a x} (a-i a x)^{7/4}} \]

[Out]

((-2*I)/7)/(a^2*(a - I*a*x)^(7/4)*(a + I*a*x)^(1/4)) - ((8*I)/21)/(a^3*(a - I*a*x)^(3/4)*(a + I*a*x)^(1/4)) +
(((16*I)/21)*(a - I*a*x)^(1/4))/(a^4*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.0181517, 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} \[ \frac{16 i \sqrt [4]{a-i a x}}{21 a^4 \sqrt [4]{a+i a x}}-\frac{8 i}{21 a^3 \sqrt [4]{a+i a x} (a-i a x)^{3/4}}-\frac{2 i}{7 a^2 \sqrt [4]{a+i a x} (a-i a x)^{7/4}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(5/4)),x]

[Out]

((-2*I)/7)/(a^2*(a - I*a*x)^(7/4)*(a + I*a*x)^(1/4)) - ((8*I)/21)/(a^3*(a - I*a*x)^(3/4)*(a + I*a*x)^(1/4)) +
(((16*I)/21)*(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)^{11/4} (a+i a x)^{5/4}} \, dx &=-\frac{2 i}{7 a^2 (a-i a x)^{7/4} \sqrt [4]{a+i a x}}+\frac{4 \int \frac{1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx}{7 a}\\ &=-\frac{2 i}{7 a^2 (a-i a x)^{7/4} \sqrt [4]{a+i a x}}-\frac{8 i}{21 a^3 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac{8 \int \frac{1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{21 a^2}\\ &=-\frac{2 i}{7 a^2 (a-i a x)^{7/4} \sqrt [4]{a+i a x}}-\frac{8 i}{21 a^3 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac{16 i \sqrt [4]{a-i a x}}{21 a^4 \sqrt [4]{a+i a x}}\\ \end{align*}

Mathematica [A]  time = 0.020365, size = 50, normalized size = 0.5 \[ \frac{16 x^2+24 i x-2}{21 a^3 (x+i) (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(5/4)),x]

[Out]

(-2 + (24*I)*x + 16*x^2)/(21*a^3*(I + x)*(a - I*a*x)^(3/4)*(a + I*a*x)^(1/4))

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Maple [A]  time = 0.041, size = 44, normalized size = 0.4 \begin{align*}{\frac{16\,{x}^{2}+24\,ix-2}{21\,{a}^{3} \left ( x+i \right ) } \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)^(11/4)/(a+I*a*x)^(5/4),x)

[Out]

2/21/a^3/(-a*(-1+I*x))^(3/4)/(a*(1+I*x))^(1/4)*(8*x^2+12*I*x-1)/(x+I)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(5/4),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.91745, size = 150, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (8 \, x^{2} + 12 i \, x - 1\right )}}{21 \, a^{5} x^{3} + 21 i \, a^{5} x^{2} + 21 \, a^{5} x + 21 i \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(5/4),x, algorithm="fricas")

[Out]

2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*(8*x^2 + 12*I*x - 1)/(21*a^5*x^3 + 21*I*a^5*x^2 + 21*a^5*x + 21*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)**(11/4)/(a+I*a*x)**(5/4),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)^(11/4)/(a+I*a*x)^(5/4),x, algorithm="giac")

[Out]

Exception raised: TypeError