3.1182 \(\int \frac{1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}}-\frac{8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac{2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}} \]

[Out]

(((-2*I)/11)*(a + I*a*x)^(3/4))/(a^2*(a - I*a*x)^(11/4)) - (((8*I)/77)*(a + I*a*
x)^(3/4))/(a^3*(a - I*a*x)^(7/4)) - (((16*I)/231)*(a + I*a*x)^(3/4))/(a^4*(a - I
*a*x)^(3/4))

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Rubi [A]  time = 0.0814795, 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 \[ -\frac{16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}}-\frac{8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac{2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(15/4)*(a + I*a*x)^(1/4)),x]

[Out]

(((-2*I)/11)*(a + I*a*x)^(3/4))/(a^2*(a - I*a*x)^(11/4)) - (((8*I)/77)*(a + I*a*
x)^(3/4))/(a^3*(a - I*a*x)^(7/4)) - (((16*I)/231)*(a + I*a*x)^(3/4))/(a^4*(a - I
*a*x)^(3/4))

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Rubi in Sympy [A]  time = 18.3467, size = 87, normalized size = 0.87 \[ - \frac{2 i \left (i a x + a\right )^{\frac{3}{4}}}{11 a^{2} \left (- i a x + a\right )^{\frac{11}{4}}} - \frac{8 i \left (i a x + a\right )^{\frac{3}{4}}}{77 a^{3} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{16 i \left (i a x + a\right )^{\frac{3}{4}}}{231 a^{4} \left (- i a x + a\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(15/4)/(a+I*a*x)**(1/4),x)

[Out]

-2*I*(I*a*x + a)**(3/4)/(11*a**2*(-I*a*x + a)**(11/4)) - 8*I*(I*a*x + a)**(3/4)/
(77*a**3*(-I*a*x + a)**(7/4)) - 16*I*(I*a*x + a)**(3/4)/(231*a**4*(-I*a*x + a)**
(3/4))

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Mathematica [A]  time = 0.0524535, size = 52, normalized size = 0.52 \[ \frac{2 \left (-8 i x^2+28 x+41 i\right ) (a+i a x)^{3/4}}{231 a^4 (x+i)^2 (a-i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(15/4)*(a + I*a*x)^(1/4)),x]

[Out]

(2*(a + I*a*x)^(3/4)*(41*I + 28*x - (8*I)*x^2))/(231*a^4*(I + x)^2*(a - I*a*x)^(
3/4))

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Maple [A]  time = 0.068, size = 50, normalized size = 0.5 \[{\frac{40\,i{x}^{2}+16\,{x}^{3}-26\,x+82\,i}{231\,{a}^{3} \left ( x+i \right ) ^{2}} \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(15/4)/(a+I*a*x)^(1/4),x)

[Out]

2/231/a^3/(-a*(-1+I*x))^(3/4)/(a*(1+I*x))^(1/4)*(20*I*x^2+8*x^3-13*x+41*I)/(x+I)
^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{15}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(15/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(15/4)), x)

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Fricas [A]  time = 0.205212, size = 73, normalized size = 0.73 \[ \frac{16 \, x^{3} + 40 i \, x^{2} - 26 \, x + 82 i}{{\left (231 \, a^{3} x^{2} + 462 i \, a^{3} x - 231 \, a^{3}\right )}{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(15/4)),x, algorithm="fricas")

[Out]

(16*x^3 + 40*I*x^2 - 26*x + 82*I)/((231*a^3*x^2 + 462*I*a^3*x - 231*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)**(15/4)/(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)^(15/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError