3.1227 \(\int \frac{1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=100 \[ \frac{16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}+\frac{8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}-\frac{2 i}{3 a^2 (a+i a x)^{5/4} (a-i a x)^{3/4}} \]

[Out]

((-2*I)/3)/(a^2*(a - I*a*x)^(3/4)*(a + I*a*x)^(5/4)) + (((8*I)/15)*(a - I*a*x)^(1/4))/(a^3*(a + I*a*x)^(5/4))
+ (((16*I)/15)*(a - I*a*x)^(1/4))/(a^4*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.018242, 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} \[ \frac{16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}+\frac{8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}-\frac{2 i}{3 a^2 (a+i a x)^{5/4} (a-i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(9/4)),x]

[Out]

((-2*I)/3)/(a^2*(a - I*a*x)^(3/4)*(a + I*a*x)^(5/4)) + (((8*I)/15)*(a - I*a*x)^(1/4))/(a^3*(a + I*a*x)^(5/4))
+ (((16*I)/15)*(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)^{7/4} (a+i a x)^{9/4}} \, dx &=-\frac{2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac{4 \int \frac{1}{(a-i a x)^{3/4} (a+i a x)^{9/4}} \, dx}{3 a}\\ &=-\frac{2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac{8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac{8 \int \frac{1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{15 a^2}\\ &=-\frac{2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac{8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac{16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}\\ \end{align*}

Mathematica [A]  time = 0.0213835, size = 50, normalized size = 0.5 \[ \frac{2 \left (8 x^2-4 i x+7\right )}{15 a^3 (x-i) (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(9/4)),x]

[Out]

(2*(7 - (4*I)*x + 8*x^2))/(15*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}-8\,ix+14}{15\,{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)^(7/4)/(a+I*a*x)^(9/4),x)

[Out]

2/15/a^3/(-a*(-1+I*x))^(3/4)/(a*(1+I*x))^(1/4)*(8*x^2-4*I*x+7)/(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)^(7/4)/(a+I*a*x)^(9/4),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.21088, size = 149, normalized size = 1.49 \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} - 4 i \, x + 7\right )}}{15 \, a^{5} x^{3} - 15 i \, a^{5} x^{2} + 15 \, a^{5} x - 15 i \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(9/4),x, algorithm="fricas")

[Out]

2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*(8*x^2 - 4*I*x + 7)/(15*a^5*x^3 - 15*I*a^5*x^2 + 15*a^5*x - 15*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)**(7/4)/(a+I*a*x)**(9/4),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)^(7/4)/(a+I*a*x)^(9/4),x, algorithm="giac")

[Out]

Exception raised: TypeError