3.1204 \(\int \frac{1}{(a-i a x)^{11/4} (a+i a x)^{7/4}} \, dx\)

Optimal. Leaf size=114 \[ \frac{10 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{21 a^3 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{10 x}{21 a^3 (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{3/4}} \]

[Out]

((-2*I)/7)/(a^2*(a - I*a*x)^(7/4)*(a + I*a*x)^(3/4)) + (10*x)/(21*a^3*(a - I*a*x
)^(3/4)*(a + I*a*x)^(3/4)) + (10*(1 + x^2)^(3/4)*EllipticF[ArcTan[x]/2, 2])/(21*
a^3*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))

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Rubi [A]  time = 0.081453, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{10 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{21 a^3 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{10 x}{21 a^3 (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(7/4)),x]

[Out]

((-2*I)/7)/(a^2*(a - I*a*x)^(7/4)*(a + I*a*x)^(3/4)) + (10*x)/(21*a^3*(a - I*a*x
)^(3/4)*(a + I*a*x)^(3/4)) + (10*(1 + x^2)^(3/4)*EllipticF[ArcTan[x]/2, 2])/(21*
a^3*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))

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Rubi in Sympy [A]  time = 26.5213, size = 128, normalized size = 1.12 \[ \frac{2 i}{3 a^{2} \left (- i a x + a\right )^{\frac{7}{4}} \left (i a x + a\right )^{\frac{3}{4}}} - \frac{10 i \sqrt [4]{i a x + a}}{21 a^{3} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{10 i \sqrt [4]{i a x + a}}{21 a^{4} \left (- i a x + a\right )^{\frac{3}{4}}} + \frac{10 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{21 a^{5} \sqrt [4]{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(11/4)/(a+I*a*x)**(7/4),x)

[Out]

2*I/(3*a**2*(-I*a*x + a)**(7/4)*(I*a*x + a)**(3/4)) - 10*I*(I*a*x + a)**(1/4)/(2
1*a**3*(-I*a*x + a)**(7/4)) - 10*I*(I*a*x + a)**(1/4)/(21*a**4*(-I*a*x + a)**(3/
4)) + 10*(-I*a*x + a)**(1/4)*(I*a*x + a)**(1/4)*elliptic_f(atan(x)/2, 2)/(21*a**
5*(x**2 + 1)**(1/4))

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Mathematica [C]  time = 0.116651, size = 96, normalized size = 0.84 \[ \frac{2 \left (5 \sqrt [4]{2} (1+i x)^{3/4} (x+i)^2 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )+5 x^2+5 i x+3\right )}{21 a^3 (x+i) (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(7/4)),x]

[Out]

(2*(3 + (5*I)*x + 5*x^2 + 5*2^(1/4)*(1 + I*x)^(3/4)*(I + x)^2*Hypergeometric2F1[
1/4, 3/4, 5/4, 1/2 - (I/2)*x]))/(21*a^3*(I + x)*(a - I*a*x)^(3/4)*(a + I*a*x)^(3
/4))

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{-{\frac{11}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(7/4),x)

[Out]

int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(7/4),x)

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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)^(7/4)*(-I*a*x + a)^(11/4)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (21 \, a^{5} x^{3} + 21 i \, a^{5} x^{2} + 21 \, a^{5} x + 21 i \, a^{5}\right )}{\rm integral}\left (\frac{5 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{21 \,{\left (a^{5} x^{2} + a^{5}\right )}}, x\right ) + 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (5 \, x^{2} + 5 i \, x + 3\right )}}{21 \, a^{5} x^{3} + 21 i \, a^{5} x^{2} + 21 \, a^{5} x + 21 i \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(11/4)),x, algorithm="fricas")

[Out]

((21*a^5*x^3 + 21*I*a^5*x^2 + 21*a^5*x + 21*I*a^5)*integral(5/21*(I*a*x + a)^(1/
4)*(-I*a*x + a)^(1/4)/(a^5*x^2 + a^5), x) + 2*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/
4)*(5*x^2 + 5*I*x + 3))/(21*a^5*x^3 + 21*I*a^5*x^2 + 21*a^5*x + 21*I*a^5)

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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)**(11/4)/(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(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(11/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError