3.1181 \(\int \frac{1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{4 i (a+i a x)^{3/4}}{21 a^3 (a-i a x)^{3/4}}-\frac{2 i (a+i a x)^{3/4}}{7 a^2 (a-i a x)^{7/4}} \]

[Out]

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

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Rubi [A]  time = 0.0506923, 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 \[ -\frac{4 i (a+i a x)^{3/4}}{21 a^3 (a-i a x)^{3/4}}-\frac{2 i (a+i a x)^{3/4}}{7 a^2 (a-i a x)^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.0996, size = 58, normalized size = 0.87 \[ - \frac{2 i \left (i a x + a\right )^{\frac{3}{4}}}{7 a^{2} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{4 i \left (i a x + a\right )^{\frac{3}{4}}}{21 a^{3} \left (- i a x + a\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0441887, size = 45, normalized size = 0.67 \[ \frac{2 (5-2 i x) (a+i a x)^{3/4}}{21 a^3 (x+i) (a-i a x)^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(5 - (2*I)*x)*(a + I*a*x)^(3/4))/(21*a^3*(I + x)*(a - I*a*x)^(3/4))

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Maple [A]  time = 0.069, size = 44, normalized size = 0.7 \[{\frac{4\,{x}^{2}+10+6\,ix}{21\,{a}^{2} \left ( x+i \right ) } \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21/a^2/(-a*(-1+I*x))^(3/4)/(a*(1+I*x))^(1/4)*(2*x^2+5+3*I*x)/(x+I)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206083, size = 55, normalized size = 0.82 \[ \frac{2 \,{\left (2 \, x^{2} + 3 i \, x + 5\right )}}{21 \,{\left (a^{2} x + i \, a^{2}\right )}{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError