3.1213 \(\int \frac{\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=264 \[ \frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}-\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]

[Out]

((4*I)*(a - I*a*x)^(1/4))/(a*(a + I*a*x)^(1/4)) + (I*Sqrt[2]*ArcTan[1 - (Sqrt[2]
*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/a - (I*Sqrt[2]*ArcTan[1 + (Sqrt[2]*(a -
I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/a + (I*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]*a) - (I*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)])/(S
qrt[2]*a)

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Rubi [A]  time = 0.223806, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ \frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}-\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(1/4)/(a + I*a*x)^(5/4),x]

[Out]

((4*I)*(a - I*a*x)^(1/4))/(a*(a + I*a*x)^(1/4)) + (I*Sqrt[2]*ArcTan[1 - (Sqrt[2]
*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/a - (I*Sqrt[2]*ArcTan[1 + (Sqrt[2]*(a -
I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/a + (I*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]*a) - (I*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)])/(S
qrt[2]*a)

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Rubi in Sympy [A]  time = 35.4123, size = 218, normalized size = 0.83 \[ \frac{4 i \sqrt [4]{- i a x + a}}{a \sqrt [4]{i a x + a}} + \frac{\sqrt{2} i \log{\left (1 + \frac{\sqrt{i a x + a}}{\sqrt{- i a x + a}} - \frac{\sqrt{2} \sqrt [4]{i a x + a}}{\sqrt [4]{- i a x + a}} \right )}}{2 a} - \frac{\sqrt{2} i \log{\left (1 + \frac{\sqrt{i a x + a}}{\sqrt{- i a x + a}} + \frac{\sqrt{2} \sqrt [4]{i a x + a}}{\sqrt [4]{- i a x + a}} \right )}}{2 a} - \frac{\sqrt{2} i \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{i a x + a}}{\sqrt [4]{- i a x + a}} \right )}}{a} + \frac{\sqrt{2} i \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{i a x + a}}{\sqrt [4]{- i a x + a}} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(1/4)/(a+I*a*x)**(5/4),x)

[Out]

4*I*(-I*a*x + a)**(1/4)/(a*(I*a*x + a)**(1/4)) + sqrt(2)*I*log(1 + sqrt(I*a*x +
a)/sqrt(-I*a*x + a) - sqrt(2)*(I*a*x + a)**(1/4)/(-I*a*x + a)**(1/4))/(2*a) - sq
rt(2)*I*log(1 + sqrt(I*a*x + a)/sqrt(-I*a*x + a) + sqrt(2)*(I*a*x + a)**(1/4)/(-
I*a*x + a)**(1/4))/(2*a) - sqrt(2)*I*atan(1 - sqrt(2)*(I*a*x + a)**(1/4)/(-I*a*x
+ a)**(1/4))/a + sqrt(2)*I*atan(1 + sqrt(2)*(I*a*x + a)**(1/4)/(-I*a*x + a)**(1
/4))/a

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Mathematica [C]  time = 0.0523719, size = 71, normalized size = 0.27 \[ -\frac{2 i \sqrt [4]{a-i a x} \left (-2+2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )\right )}{a \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(1/4)/(a + I*a*x)^(5/4),x]

[Out]

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

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{a-iax} \left ( a+iax \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(1/4)/(a+I*a*x)^(5/4),x)

[Out]

int((a-I*a*x)^(1/4)/(a+I*a*x)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(5/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(5/4), x)

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Fricas [A]  time = 0.242823, size = 408, normalized size = 1.55 \[ -\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} \log \left (\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) -{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} \log \left (-\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} - 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) +{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} \log \left (\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) -{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} \log \left (-\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} - 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) - 8 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \,{\left (a^{2} x - i \, a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(5/4),x, algorithm="fricas")

[Out]

-1/2*((a^2*x - I*a^2)*sqrt(4*I/a^2)*log(((a^2*x - I*a^2)*sqrt(4*I/a^2) + 2*(I*a*
x + a)^(3/4)*(-I*a*x + a)^(1/4))/(2*x - 2*I)) - (a^2*x - I*a^2)*sqrt(4*I/a^2)*lo
g(-((a^2*x - I*a^2)*sqrt(4*I/a^2) - 2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/(2*x
- 2*I)) + (a^2*x - I*a^2)*sqrt(-4*I/a^2)*log(((a^2*x - I*a^2)*sqrt(-4*I/a^2) +
2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/(2*x - 2*I)) - (a^2*x - I*a^2)*sqrt(-4*I
/a^2)*log(-((a^2*x - I*a^2)*sqrt(-4*I/a^2) - 2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1
/4))/(2*x - 2*I)) - 8*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))/(a^2*x - I*a^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{- a \left (i x - 1\right )}}{\left (a \left (i x + 1\right )\right )^{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a-I*a*x)**(1/4)/(a+I*a*x)**(5/4),x)

[Out]

Integral((-a*(I*x - 1))**(1/4)/(a*(I*x + 1))**(5/4), x)

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GIAC/XCAS [A]  time = 0.247305, size = 244, normalized size = 0.92 \[ -\frac{2 i \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + \frac{2 \,{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}}\right )}\right ) + 2 i \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - \frac{2 \,{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}}\right )}\right ) + i \, \sqrt{2}{\rm ln}\left (\frac{\sqrt{2}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}} + \frac{\sqrt{-i \, a x + a}}{\sqrt{i \, a x + a}} + 1\right ) - i \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}} + \frac{\sqrt{-i \, a x + a}}{\sqrt{i \, a x + a}} + 1\right ) - \frac{8 i \,{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(5/4),x, algorithm="giac")

[Out]

-1/2*(2*I*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(-I*a*x + a)^(1/4)/(I*a*x + a)
^(1/4))) + 2*I*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(-I*a*x + a)^(1/4)/(I*a*
x + a)^(1/4))) + I*sqrt(2)*ln(sqrt(2)*(-I*a*x + a)^(1/4)/(I*a*x + a)^(1/4) + sqr
t(-I*a*x + a)/sqrt(I*a*x + a) + 1) - I*sqrt(2)*ln(-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) - 8*I*(-I*a*x + a)^(1/4)
/(I*a*x + a)^(1/4))/a