3.1223 \(\int \frac{1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=154 \[ \frac{14 x}{65 a^6 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{42 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}-\frac{2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}} \]

[Out]

((-2*I)/13)/(a^2*(a - I*a*x)^(13/4)*(a + I*a*x)^(5/4)) - ((2*I)/13)/(a^3*(a - I*
a*x)^(9/4)*(a + I*a*x)^(5/4)) + (14*x)/(65*a^6*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/
4)*(1 + x^2)) + (42*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(65*a^6*(a - I*a*
x)^(1/4)*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.126479, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{14 x}{65 a^6 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{42 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}-\frac{2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(17/4)*(a + I*a*x)^(9/4)),x]

[Out]

((-2*I)/13)/(a^2*(a - I*a*x)^(13/4)*(a + I*a*x)^(5/4)) - ((2*I)/13)/(a^3*(a - I*
a*x)^(9/4)*(a + I*a*x)^(5/4)) + (14*x)/(65*a^6*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/
4)*(1 + x^2)) + (42*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(65*a^6*(a - I*a*
x)^(1/4)*(a + I*a*x)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 i}{5 a^{2} \left (- i a x + a\right )^{\frac{13}{4}} \left (i a x + a\right )^{\frac{5}{4}}} + \frac{18 i}{5 a^{3} \left (- i a x + a\right )^{\frac{13}{4}} \sqrt [4]{i a x + a}} - \frac{126 i \left (i a x + a\right )^{\frac{3}{4}}}{65 a^{4} \left (- i a x + a\right )^{\frac{13}{4}}} - \frac{84 i}{65 a^{5} \left (- i a x + a\right )^{\frac{5}{4}} \sqrt [4]{i a x + a}} - \frac{14 i \left (i a x + a\right )^{\frac{3}{4}}}{13 a^{5} \left (- i a x + a\right )^{\frac{9}{4}}} - \frac{21 \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\sqrt [4]{a^{2} x^{2} + a^{2}}}\, dx}{65 a^{6} \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{42 x \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{65 a^{8} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(17/4)/(a+I*a*x)**(9/4),x)

[Out]

2*I/(5*a**2*(-I*a*x + a)**(13/4)*(I*a*x + a)**(5/4)) + 18*I/(5*a**3*(-I*a*x + a)
**(13/4)*(I*a*x + a)**(1/4)) - 126*I*(I*a*x + a)**(3/4)/(65*a**4*(-I*a*x + a)**(
13/4)) - 84*I/(65*a**5*(-I*a*x + a)**(5/4)*(I*a*x + a)**(1/4)) - 14*I*(I*a*x + a
)**(3/4)/(13*a**5*(-I*a*x + a)**(9/4)) - 21*(-I*a*x + a)**(3/4)*(I*a*x + a)**(3/
4)*Integral((a**2*x**2 + a**2)**(-1/4), x)/(65*a**6*(a**2*x**2 + a**2)**(3/4)) +
42*x*(-I*a*x + a)**(3/4)*(I*a*x + a)**(3/4)/(65*a**8*(x**2 + 1))

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Mathematica [C]  time = 0.187928, size = 127, normalized size = 0.82 \[ \frac{2 \left (-7\ 2^{3/4} \sqrt [4]{1+i x} (x-i) (x+i)^4 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )+21 x^5+42 i x^4+7 x^3+56 i x^2-23 x+10 i\right )}{65 a^6 (x-i) (x+i)^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(17/4)*(a + I*a*x)^(9/4)),x]

[Out]

(2*(10*I - 23*x + (56*I)*x^2 + 7*x^3 + (42*I)*x^4 + 21*x^5 - 7*2^(3/4)*(1 + I*x)
^(1/4)*(-I + x)*(I + x)^4*Hypergeometric2F1[1/4, 3/4, 7/4, 1/2 - (I/2)*x]))/(65*
a^6*(-I + x)*(I + x)^3*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

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Maple [C]  time = 0.128, size = 130, normalized size = 0.8 \[{\frac{84\,i{x}^{4}+42\,{x}^{5}+112\,i{x}^{2}-46\,x+14\,{x}^{3}+20\,i}{ \left ( 65\,x-65\,i \right ) \left ( x+i \right ) ^{3}{a}^{6}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{21\,x}{65\,{a}^{6}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(17/4)/(a+I*a*x)^(9/4),x)

[Out]

2/65*(42*I*x^4+21*x^5+56*I*x^2-23*x+7*x^3+10*I)/(x-I)/(x+I)^3/a^6/(-a*(-1+I*x))^
(1/4)/(a*(1+I*x))^(1/4)-21/65/(a^2)^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x^2)/a^6*
(-a^2*(-1+I*x)*(1+I*x))^(1/4)/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(17/4)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (42 \, x^{5} + 84 i \, x^{4} + 14 \, x^{3} + 112 i \, x^{2} - 46 \, x + 20 i\right )}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} +{\left (65 \, a^{8} x^{6} + 130 i \, a^{8} x^{5} + 65 \, a^{8} x^{4} + 260 i \, a^{8} x^{3} - 65 \, a^{8} x^{2} + 130 i \, a^{8} x - 65 \, a^{8}\right )}{\rm integral}\left (-\frac{21 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{65 \,{\left (a^{8} x^{2} + a^{8}\right )}}, x\right )}{65 \, a^{8} x^{6} + 130 i \, a^{8} x^{5} + 65 \, a^{8} x^{4} + 260 i \, a^{8} x^{3} - 65 \, a^{8} x^{2} + 130 i \, a^{8} x - 65 \, a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(17/4)),x, algorithm="fricas")

[Out]

((42*x^5 + 84*I*x^4 + 14*x^3 + 112*I*x^2 - 46*x + 20*I)*(I*a*x + a)^(3/4)*(-I*a*
x + a)^(3/4) + (65*a^8*x^6 + 130*I*a^8*x^5 + 65*a^8*x^4 + 260*I*a^8*x^3 - 65*a^8
*x^2 + 130*I*a^8*x - 65*a^8)*integral(-21/65*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4
)/(a^8*x^2 + a^8), x))/(65*a^8*x^6 + 130*I*a^8*x^5 + 65*a^8*x^4 + 260*I*a^8*x^3
- 65*a^8*x^2 + 130*I*a^8*x - 65*a^8)

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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)**(17/4)/(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)^(17/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError