3.1228 \(\int \frac{1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=133 \[ \frac{32 i \sqrt [4]{a-i a x}}{35 a^5 \sqrt [4]{a+i a x}}+\frac{16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}-\frac{4 i}{7 a^3 (a+i a x)^{5/4} (a-i a x)^{3/4}}-\frac{2 i}{7 a^2 (a+i a x)^{5/4} (a-i a x)^{7/4}} \]

[Out]

((-2*I)/7)/(a^2*(a - I*a*x)^(7/4)*(a + I*a*x)^(5/4)) - ((4*I)/7)/(a^3*(a - I*a*x
)^(3/4)*(a + I*a*x)^(5/4)) + (((16*I)/35)*(a - I*a*x)^(1/4))/(a^4*(a + I*a*x)^(5
/4)) + (((32*I)/35)*(a - I*a*x)^(1/4))/(a^5*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.114492, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{32 i \sqrt [4]{a-i a x}}{35 a^5 \sqrt [4]{a+i a x}}+\frac{16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}-\frac{4 i}{7 a^3 (a+i a x)^{5/4} (a-i a x)^{3/4}}-\frac{2 i}{7 a^2 (a+i a x)^{5/4} (a-i a x)^{7/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(9/4)),x]

[Out]

((-2*I)/7)/(a^2*(a - I*a*x)^(7/4)*(a + I*a*x)^(5/4)) - ((4*I)/7)/(a^3*(a - I*a*x
)^(3/4)*(a + I*a*x)^(5/4)) + (((16*I)/35)*(a - I*a*x)^(1/4))/(a^4*(a + I*a*x)^(5
/4)) + (((32*I)/35)*(a - I*a*x)^(1/4))/(a^5*(a + I*a*x)^(1/4))

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Rubi in Sympy [A]  time = 23.8811, size = 114, normalized size = 0.86 \[ \frac{2 i}{5 a^{2} \left (- i a x + a\right )^{\frac{7}{4}} \left (i a x + a\right )^{\frac{5}{4}}} + \frac{12 i}{5 a^{3} \left (- i a x + a\right )^{\frac{7}{4}} \sqrt [4]{i a x + a}} - \frac{48 i \left (i a x + a\right )^{\frac{3}{4}}}{35 a^{4} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{32 i \left (i a x + a\right )^{\frac{3}{4}}}{35 a^{5} \left (- i a x + a\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(11/4)/(a+I*a*x)**(9/4),x)

[Out]

2*I/(5*a**2*(-I*a*x + a)**(7/4)*(I*a*x + a)**(5/4)) + 12*I/(5*a**3*(-I*a*x + a)*
*(7/4)*(I*a*x + a)**(1/4)) - 48*I*(I*a*x + a)**(3/4)/(35*a**4*(-I*a*x + a)**(7/4
)) - 32*I*(I*a*x + a)**(3/4)/(35*a**5*(-I*a*x + a)**(3/4))

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Mathematica [A]  time = 0.062245, size = 64, normalized size = 0.48 \[ \frac{2 \left (-16 i x^3+8 x^2-22 i x+9\right ) (a+i a x)^{3/4}}{35 a^5 (x-i)^2 (x+i) (a-i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(9/4)),x]

[Out]

(2*(a + I*a*x)^(3/4)*(9 - (22*I)*x + 8*x^2 - (16*I)*x^3))/(35*a^5*(-I + x)^2*(I
+ x)*(a - I*a*x)^(3/4))

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Maple [A]  time = 0.08, size = 56, normalized size = 0.4 \[{\frac{32\,{x}^{3}+16\,i{x}^{2}+44\,x+18\,i}{35\,{a}^{4} \left ( x-i \right ) \left ( x+i \right ) } \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(9/4),x)

[Out]

2/35/a^4/(-a*(-1+I*x))^(3/4)/(a*(1+I*x))^(1/4)*(16*x^3+8*I*x^2+22*x+9*I)/(x-I)/(
x+I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{9}{4}}{\left (-i \, a x + a\right )}^{\frac{11}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(11/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(11/4)), x)

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Fricas [A]  time = 0.243909, size = 62, normalized size = 0.47 \[ \frac{32 \, x^{3} + 16 i \, x^{2} + 44 \, x + 18 i}{35 \,{\left (a^{4} x^{2} + a^{4}\right )}{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(11/4)),x, algorithm="fricas")

[Out]

1/35*(32*x^3 + 16*I*x^2 + 44*x + 18*I)/((a^4*x^2 + a^4)*(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)**(11/4)/(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)^(11/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError