3.1183 \(\int \frac{1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=133 \[ -\frac{32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}}-\frac{16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac{4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac{2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}} \]

[Out]

(((-2*I)/15)*(a + I*a*x)^(3/4))/(a^2*(a - I*a*x)^(15/4)) - (((4*I)/55)*(a + I*a*
x)^(3/4))/(a^3*(a - I*a*x)^(11/4)) - (((16*I)/385)*(a + I*a*x)^(3/4))/(a^4*(a -
I*a*x)^(7/4)) - (((32*I)/1155)*(a + I*a*x)^(3/4))/(a^5*(a - I*a*x)^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.116078, 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 \[ -\frac{32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}}-\frac{16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac{4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac{2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(19/4)*(a + I*a*x)^(1/4)),x]

[Out]

(((-2*I)/15)*(a + I*a*x)^(3/4))/(a^2*(a - I*a*x)^(15/4)) - (((4*I)/55)*(a + I*a*
x)^(3/4))/(a^3*(a - I*a*x)^(11/4)) - (((16*I)/385)*(a + I*a*x)^(3/4))/(a^4*(a -
I*a*x)^(7/4)) - (((32*I)/1155)*(a + I*a*x)^(3/4))/(a^5*(a - I*a*x)^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.8857, size = 116, normalized size = 0.87 \[ - \frac{2 i \left (i a x + a\right )^{\frac{3}{4}}}{15 a^{2} \left (- i a x + a\right )^{\frac{15}{4}}} - \frac{4 i \left (i a x + a\right )^{\frac{3}{4}}}{55 a^{3} \left (- i a x + a\right )^{\frac{11}{4}}} - \frac{16 i \left (i a x + a\right )^{\frac{3}{4}}}{385 a^{4} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{32 i \left (i a x + a\right )^{\frac{3}{4}}}{1155 a^{5} \left (- i a x + a\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(19/4)/(a+I*a*x)**(1/4),x)

[Out]

-2*I*(I*a*x + a)**(3/4)/(15*a**2*(-I*a*x + a)**(15/4)) - 4*I*(I*a*x + a)**(3/4)/
(55*a**3*(-I*a*x + a)**(11/4)) - 16*I*(I*a*x + a)**(3/4)/(385*a**4*(-I*a*x + a)*
*(7/4)) - 32*I*(I*a*x + a)**(3/4)/(1155*a**5*(-I*a*x + a)**(3/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0611999, size = 57, normalized size = 0.43 \[ \frac{2 \left (-16 i x^3+72 x^2+138 i x-159\right ) (a+i a x)^{3/4}}{1155 a^5 (x+i)^3 (a-i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(19/4)*(a + I*a*x)^(1/4)),x]

[Out]

(2*(a + I*a*x)^(3/4)*(-159 + (138*I)*x + 72*x^2 - (16*I)*x^3))/(1155*a^5*(I + x)
^3*(a - I*a*x)^(3/4))

_______________________________________________________________________________________

Maple [A]  time = 0.072, size = 55, normalized size = 0.4 \[{\frac{112\,i{x}^{3}+32\,{x}^{4}-42\,ix-318-132\,{x}^{2}}{1155\,{a}^{4} \left ( x+i \right ) ^{3}} \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(19/4)/(a+I*a*x)^(1/4),x)

[Out]

2/1155/a^4/(-a*(-1+I*x))^(3/4)/(a*(1+I*x))^(1/4)*(56*I*x^3+16*x^4-21*I*x-159-66*
x^2)/(x+I)^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{19}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(19/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(19/4)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.21272, size = 90, normalized size = 0.68 \[ \frac{32 \, x^{4} + 112 i \, x^{3} - 132 \, x^{2} - 42 i \, x - 318}{{\left (1155 \, a^{4} x^{3} + 3465 i \, a^{4} x^{2} - 3465 \, a^{4} x - 1155 i \, a^{4}\right )}{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(19/4)),x, algorithm="fricas")

[Out]

(32*x^4 + 112*I*x^3 - 132*x^2 - 42*I*x - 318)/((1155*a^4*x^3 + 3465*I*a^4*x^2 -
3465*a^4*x - 1155*I*a^4)*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4))

_______________________________________________________________________________________

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)**(19/4)/(a+I*a*x)**(1/4),x)

[Out]

Timed out

_______________________________________________________________________________________

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)^(19/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError