3.1212 \(\int \frac{(a-i a x)^{5/4}}{(a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=287 \[ \frac{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i (a+i a x)^{3/4} \sqrt [4]{a-i a x}}{a}+\frac{5 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}-\frac{5 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{5 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{5 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]

[Out]

((4*I)*(a - I*a*x)^(5/4))/(a*(a + I*a*x)^(1/4)) + ((5*I)*(a - I*a*x)^(1/4)*(a +
I*a*x)^(3/4))/a + ((5*I)*ArcTan[1 - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4
)])/Sqrt[2] - ((5*I)*ArcTan[1 + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/
Sqrt[2] + (((5*I)/2)*Log[1 + Sqrt[a - I*a*x]/Sqrt[a + I*a*x] - (Sqrt[2]*(a - I*a
*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2] - (((5*I)/2)*Log[1 + Sqrt[a - I*a*x]/Sqrt
[a + I*a*x] + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2]

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Rubi [A]  time = 0.302966, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ \frac{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i (a+i a x)^{3/4} \sqrt [4]{a-i a x}}{a}+\frac{5 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}-\frac{5 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{5 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{5 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(5/4)/(a + I*a*x)^(5/4),x]

[Out]

((4*I)*(a - I*a*x)^(5/4))/(a*(a + I*a*x)^(1/4)) + ((5*I)*(a - I*a*x)^(1/4)*(a +
I*a*x)^(3/4))/a + ((5*I)*ArcTan[1 - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4
)])/Sqrt[2] - ((5*I)*ArcTan[1 + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/
Sqrt[2] + (((5*I)/2)*Log[1 + Sqrt[a - I*a*x]/Sqrt[a + I*a*x] - (Sqrt[2]*(a - I*a
*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2] - (((5*I)/2)*Log[1 + Sqrt[a - I*a*x]/Sqrt
[a + I*a*x] + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2]

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Rubi in Sympy [A]  time = 45.5538, size = 246, normalized size = 0.86 \[ \frac{5 \sqrt{2} i \log{\left (- \frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + \frac{\sqrt{- i a x + a}}{\sqrt{i a x + a}} + 1 \right )}}{4} - \frac{5 \sqrt{2} i \log{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + \frac{\sqrt{- i a x + a}}{\sqrt{i a x + a}} + 1 \right )}}{4} - \frac{5 \sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} - 1 \right )}}{2} - \frac{5 \sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + 1 \right )}}{2} + \frac{4 i \left (- i a x + a\right )^{\frac{5}{4}}}{a \sqrt [4]{i a x + a}} + \frac{5 i \sqrt [4]{- i a x + a} \left (i a x + a\right )^{\frac{3}{4}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(5/4)/(a+I*a*x)**(5/4),x)

[Out]

5*sqrt(2)*I*log(-sqrt(2)*(-I*a*x + a)**(1/4)/(I*a*x + a)**(1/4) + sqrt(-I*a*x +
a)/sqrt(I*a*x + a) + 1)/4 - 5*sqrt(2)*I*log(sqrt(2)*(-I*a*x + a)**(1/4)/(I*a*x +
a)**(1/4) + sqrt(-I*a*x + a)/sqrt(I*a*x + a) + 1)/4 - 5*sqrt(2)*I*atan(sqrt(2)*
(-I*a*x + a)**(1/4)/(I*a*x + a)**(1/4) - 1)/2 - 5*sqrt(2)*I*atan(sqrt(2)*(-I*a*x
+ a)**(1/4)/(I*a*x + a)**(1/4) + 1)/2 + 4*I*(-I*a*x + a)**(5/4)/(a*(I*a*x + a)*
*(1/4)) + 5*I*(-I*a*x + a)**(1/4)*(I*a*x + a)**(3/4)/a

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Mathematica [C]  time = 0.0612595, size = 72, normalized size = 0.25 \[ -\frac{\sqrt [4]{a-i a x} \left (5 i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )+x-9 i\right )}{\sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(5/4)/(a + I*a*x)^(5/4),x]

[Out]

-(((a - I*a*x)^(1/4)*(-9*I + x + (5*I)*2^(3/4)*(1 + I*x)^(1/4)*Hypergeometric2F1
[1/4, 1/4, 5/4, 1/2 - (I/2)*x]))/(a + I*a*x)^(1/4))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{{\frac{5}{4}}} \left ( a+iax \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(5/4)/(a+I*a*x)^(5/4),x)

[Out]

int((a-I*a*x)^(5/4)/(a+I*a*x)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{5}{4}}}{{\left (i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(5/4)/(I*a*x + a)^(5/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(5/4)/(I*a*x + a)^(5/4), x)

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Fricas [A]  time = 0.239538, size = 323, normalized size = 1.13 \[ -\frac{\sqrt{25 i}{\left (a x - i \, a\right )} \log \left (\frac{\sqrt{25 i}{\left (a x - i \, a\right )} + 5 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{5 \, x - 5 i}\right ) - \sqrt{25 i}{\left (a x - i \, a\right )} \log \left (-\frac{\sqrt{25 i}{\left (a x - i \, a\right )} - 5 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{5 \, x - 5 i}\right ) + \sqrt{-25 i}{\left (a x - i \, a\right )} \log \left (\frac{\sqrt{-25 i}{\left (a x - i \, a\right )} + 5 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{5 \, x - 5 i}\right ) - \sqrt{-25 i}{\left (a x - i \, a\right )} \log \left (-\frac{\sqrt{-25 i}{\left (a x - i \, a\right )} - 5 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{5 \, x - 5 i}\right ) + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, x - 9\right )}}{2 \,{\left (a x - i \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(5/4)/(I*a*x + a)^(5/4),x, algorithm="fricas")

[Out]

-1/2*(sqrt(25*I)*(a*x - I*a)*log((sqrt(25*I)*(a*x - I*a) + 5*(I*a*x + a)^(3/4)*(
-I*a*x + a)^(1/4))/(5*x - 5*I)) - sqrt(25*I)*(a*x - I*a)*log(-(sqrt(25*I)*(a*x -
I*a) - 5*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/(5*x - 5*I)) + sqrt(-25*I)*(a*x
- I*a)*log((sqrt(-25*I)*(a*x - I*a) + 5*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/(5
*x - 5*I)) - sqrt(-25*I)*(a*x - I*a)*log(-(sqrt(-25*I)*(a*x - I*a) - 5*(I*a*x +
a)^(3/4)*(-I*a*x + a)^(1/4))/(5*x - 5*I)) + 2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/
4)*(-I*x - 9))/(a*x - I*a)

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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((a-I*a*x)**(5/4)/(a+I*a*x)**(5/4),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((-I*a*x + a)^(5/4)/(I*a*x + a)^(5/4),x, algorithm="giac")

[Out]

Exception raised: TypeError