3.1215 \(\int \frac{1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=67 \[ \frac{4 i \sqrt [4]{a-i a x}}{3 a^3 \sqrt [4]{a+i a x}}-\frac{2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]

[Out]

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

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Rubi [A]  time = 0.0523563, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{4 i \sqrt [4]{a-i a x}}{3 a^3 \sqrt [4]{a+i a x}}-\frac{2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.8001, size = 54, normalized size = 0.81 \[ \frac{2 i}{a^{2} \left (- i a x + a\right )^{\frac{3}{4}} \sqrt [4]{i a x + a}} - \frac{4 i \left (i a x + a\right )^{\frac{3}{4}}}{3 a^{3} \left (- i a x + a\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0441605, size = 45, normalized size = 0.67 \[ \frac{2 (1-2 i x) (a+i a x)^{3/4}}{3 a^3 (x-i) (a-i a x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.059, size = 33, normalized size = 0.5 \[{\frac{4\,x+2\,i}{3\,{a}^{2}} \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{5}{4}}{\left (-i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225401, size = 35, normalized size = 0.52 \[ \frac{4 \, x + 2 i}{3 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: TypeError