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 = 0.0809503, 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 \[ \frac{16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}+\frac{8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}-\frac{2 i}{3 a^2 (a+i a x)^{5/4} (a-i a x)^{3/4}} \]

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))

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Rubi in Sympy [A]  time = 16.575, size = 85, normalized size = 0.85 \[ \frac{2 i}{5 a^{2} \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{5}{4}}} + \frac{8 i}{5 a^{3} \left (- i a x + a\right )^{\frac{3}{4}} \sqrt [4]{i a x + a}} - \frac{16 i \left (i a x + a\right )^{\frac{3}{4}}}{15 a^{4} \left (- i a x + a\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0519934, size = 52, normalized size = 0.52 \[ -\frac{2 i \left (8 x^2-4 i x+7\right ) (a+i a x)^{3/4}}{15 a^4 (x-i)^2 (a-i a x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(((-2*I)/15)*(a + I*a*x)^(3/4)*(7 - (4*I)*x + 8*x^2))/(a^4*(-I + x)^2*(a - I*a*x
)^(3/4))

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Maple [A]  time = 0.072, size = 44, normalized size = 0.4 \[{\frac{16\,{x}^{2}-8\,ix+14}{15\,{a}^{3} \left ( x-i \right ) } \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(7/4)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.244274, size = 55, normalized size = 0.55 \[ \frac{2 \,{\left (8 \, x^{2} - 4 i \, x + 7\right )}}{15 \,{\left (a^{3} x - i \, a^{3}\right )}{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(8*x^2 - 4*I*x + 7)/((a^3*x - I*a^3)*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4))

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

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/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(7/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError