3.1224 \(\int \frac{(a-i a x)^{5/4}}{(a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=297 \[ \frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}-\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.244222, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ \frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}-\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(5/4)/(a + I*a*x)^(9/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 44.0861, size = 245, normalized size = 0.82 \[ \frac{4 i \left (- i a x + a\right )^{\frac{5}{4}}}{5 a \left (i a x + a\right )^{\frac{5}{4}}} - \frac{4 i \sqrt [4]{- i a x + a}}{a \sqrt [4]{i a x + a}} - \frac{\sqrt{2} i \log{\left (1 + \frac{\sqrt{i a x + a}}{\sqrt{- i a x + a}} - \frac{\sqrt{2} \sqrt [4]{i a x + a}}{\sqrt [4]{- i a x + a}} \right )}}{2 a} + \frac{\sqrt{2} i \log{\left (1 + \frac{\sqrt{i a x + a}}{\sqrt{- i a x + a}} + \frac{\sqrt{2} \sqrt [4]{i a x + a}}{\sqrt [4]{- i a x + a}} \right )}}{2 a} + \frac{\sqrt{2} i \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{i a x + a}}{\sqrt [4]{- i a x + a}} \right )}}{a} - \frac{\sqrt{2} i \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{i a x + a}}{\sqrt [4]{- i a x + a}} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(5/4)/(a+I*a*x)**(9/4),x)

[Out]

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

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Mathematica [C]  time = 0.0894426, size = 84, normalized size = 0.28 \[ \frac{2 \sqrt [4]{a-i a x} \left (5\ 2^{3/4} (1+i x)^{5/4} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )-12 i x-8\right )}{5 a (x-i) \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(5/4)/(a + I*a*x)^(9/4),x]

[Out]

(2*(a - I*a*x)^(1/4)*(-8 - (12*I)*x + 5*2^(3/4)*(1 + I*x)^(5/4)*Hypergeometric2F
1[1/4, 1/4, 5/4, 1/2 - (I/2)*x]))/(5*a*(-I + x)*(a + I*a*x)^(1/4))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{{\frac{5}{4}}} \left ( a+iax \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(5/4)/(a+I*a*x)^(9/4),x)

[Out]

int((a-I*a*x)^(5/4)/(a+I*a*x)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{5}{4}}}{{\left (i \, a x + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(5/4)/(I*a*x + a)^(9/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(5/4)/(I*a*x + a)^(9/4), x)

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Fricas [A]  time = 0.254097, size = 474, normalized size = 1.6 \[ \frac{{\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} \log \left (\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) -{\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} \log \left (-\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} - 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) +{\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} \log \left (\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) -{\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} \log \left (-\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} - 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) -{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (48 \, x - 32 i\right )}}{10 \, a^{2} x^{2} - 20 i \, a^{2} x - 10 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(5/4)/(I*a*x + a)^(9/4),x, algorithm="fricas")

[Out]

((5*a^2*x^2 - 10*I*a^2*x - 5*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)) - (5*a^2*x^2 - 10*I*a^
2*x - 5*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)) + (5*a^2*x^2 - 10*I*a^2*x - 5*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)) - (5*a^2*x^2 - 10*I*a^2*x - 5*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))
- (I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*(48*x - 32*I))/(10*a^2*x^2 - 20*I*a^2*x
- 10*a^2)

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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)**(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((-I*a*x + a)^(5/4)/(I*a*x + a)^(9/4),x, algorithm="giac")

[Out]

Exception raised: TypeError