3.1199 \(\int \frac{(a-i a x)^{9/4}}{(a+i a x)^{7/4}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{10 a^2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{(a+i a x)^{3/4} (a-i a x)^{3/4}}+\frac{4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+\frac{2 i \sqrt [4]{a+i a x} (a-i a x)^{5/4}}{a}+10 i \sqrt [4]{a+i a x} \sqrt [4]{a-i a x} \]

[Out]

(((4*I)/3)*(a - I*a*x)^(9/4))/(a*(a + I*a*x)^(3/4)) + (10*I)*(a - I*a*x)^(1/4)*(
a + I*a*x)^(1/4) + ((2*I)*(a - I*a*x)^(5/4)*(a + I*a*x)^(1/4))/a - (10*a^2*(1 +
x^2)^(3/4)*EllipticF[ArcTan[x]/2, 2])/((a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))

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Rubi [A]  time = 0.120654, antiderivative size = 139, 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 \[ -\frac{10 a^2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{(a+i a x)^{3/4} (a-i a x)^{3/4}}+\frac{4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+\frac{2 i \sqrt [4]{a+i a x} (a-i a x)^{5/4}}{a}+10 i \sqrt [4]{a+i a x} \sqrt [4]{a-i a x} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(9/4)/(a + I*a*x)^(7/4),x]

[Out]

(((4*I)/3)*(a - I*a*x)^(9/4))/(a*(a + I*a*x)^(3/4)) + (10*I)*(a - I*a*x)^(1/4)*(
a + I*a*x)^(1/4) + ((2*I)*(a - I*a*x)^(5/4)*(a + I*a*x)^(1/4))/a - (10*a^2*(1 +
x^2)^(3/4)*EllipticF[ArcTan[x]/2, 2])/((a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))

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Rubi in Sympy [A]  time = 28.3696, size = 112, normalized size = 0.81 \[ 10 i \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} - \frac{10 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{\sqrt [4]{x^{2} + 1}} + \frac{4 i \left (- i a x + a\right )^{\frac{9}{4}}}{3 a \left (i a x + a\right )^{\frac{3}{4}}} + \frac{2 i \left (- i a x + a\right )^{\frac{5}{4}} \sqrt [4]{i a x + a}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(9/4)/(a+I*a*x)**(7/4),x)

[Out]

10*I*(-I*a*x + a)**(1/4)*(I*a*x + a)**(1/4) - 10*(-I*a*x + a)**(1/4)*(I*a*x + a)
**(1/4)*elliptic_f(atan(x)/2, 2)/(x**2 + 1)**(1/4) + 4*I*(-I*a*x + a)**(9/4)/(3*
a*(I*a*x + a)**(3/4)) + 2*I*(-I*a*x + a)**(5/4)*(I*a*x + a)**(1/4)/a

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Mathematica [C]  time = 0.0710493, size = 80, normalized size = 0.58 \[ \frac{2 i a \sqrt [4]{a-i a x} \left (-15 \sqrt [4]{2} (1+i x)^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )+x^2+11 i x+20\right )}{3 (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(9/4)/(a + I*a*x)^(7/4),x]

[Out]

(((2*I)/3)*a*(a - I*a*x)^(1/4)*(20 + (11*I)*x + x^2 - 15*2^(1/4)*(1 + I*x)^(3/4)
*Hypergeometric2F1[1/4, 3/4, 5/4, 1/2 - (I/2)*x]))/(a + I*a*x)^(3/4)

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{{\frac{9}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(9/4)/(a+I*a*x)^(7/4),x)

[Out]

int((a-I*a*x)^(9/4)/(a+I*a*x)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{9}{4}}}{{\left (i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(9/4)/(I*a*x + a)^(7/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(9/4)/(I*a*x + a)^(7/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (3 \, x - 3 i\right )}{\rm integral}\left (-\frac{5 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{x^{2} + 1}, x\right ) + 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (x^{2} + 11 i \, x + 20\right )}}{3 \, x - 3 i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(9/4)/(I*a*x + a)^(7/4),x, algorithm="fricas")

[Out]

((3*x - 3*I)*integral(-5*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4)/(x^2 + 1), x) + 2*
(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4)*(x^2 + 11*I*x + 20))/(3*x - 3*I)

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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)**(9/4)/(a+I*a*x)**(7/4),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)^(9/4)/(I*a*x + a)^(7/4),x, algorithm="giac")

[Out]

Exception raised: TypeError