3.1205 \(\int \frac{1}{(a-i a x)^{15/4} (a+i a x)^{7/4}} \, dx\)

Optimal. Leaf size=147 \[ \frac{10 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{33 a^4 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{10 x}{33 a^4 (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i}{11 a^3 (a-i a x)^{7/4} (a+i a x)^{3/4}}-\frac{2 i}{11 a^2 (a-i a x)^{11/4} (a+i a x)^{3/4}} \]

[Out]

((-2*I)/11)/(a^2*(a - I*a*x)^(11/4)*(a + I*a*x)^(3/4)) - ((2*I)/11)/(a^3*(a - I*
a*x)^(7/4)*(a + I*a*x)^(3/4)) + (10*x)/(33*a^4*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/
4)) + (10*(1 + x^2)^(3/4)*EllipticF[ArcTan[x]/2, 2])/(33*a^4*(a - I*a*x)^(3/4)*(
a + I*a*x)^(3/4))

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Rubi [A]  time = 0.116468, antiderivative size = 147, 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 \[ \frac{10 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{33 a^4 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{10 x}{33 a^4 (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i}{11 a^3 (a-i a x)^{7/4} (a+i a x)^{3/4}}-\frac{2 i}{11 a^2 (a-i a x)^{11/4} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(15/4)*(a + I*a*x)^(7/4)),x]

[Out]

((-2*I)/11)/(a^2*(a - I*a*x)^(11/4)*(a + I*a*x)^(3/4)) - ((2*I)/11)/(a^3*(a - I*
a*x)^(7/4)*(a + I*a*x)^(3/4)) + (10*x)/(33*a^4*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/
4)) + (10*(1 + x^2)^(3/4)*EllipticF[ArcTan[x]/2, 2])/(33*a^4*(a - I*a*x)^(3/4)*(
a + I*a*x)^(3/4))

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Rubi in Sympy [A]  time = 34.5147, size = 156, normalized size = 1.06 \[ \frac{2 i}{3 a^{2} \left (- i a x + a\right )^{\frac{11}{4}} \left (i a x + a\right )^{\frac{3}{4}}} - \frac{14 i \sqrt [4]{i a x + a}}{33 a^{3} \left (- i a x + a\right )^{\frac{11}{4}}} - \frac{10 i \sqrt [4]{i a x + a}}{33 a^{4} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{10 i \sqrt [4]{i a x + a}}{33 a^{5} \left (- i a x + a\right )^{\frac{3}{4}}} + \frac{10 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{33 a^{6} \sqrt [4]{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(15/4)/(a+I*a*x)**(7/4),x)

[Out]

2*I/(3*a**2*(-I*a*x + a)**(11/4)*(I*a*x + a)**(3/4)) - 14*I*(I*a*x + a)**(1/4)/(
33*a**3*(-I*a*x + a)**(11/4)) - 10*I*(I*a*x + a)**(1/4)/(33*a**4*(-I*a*x + a)**(
7/4)) - 10*I*(I*a*x + a)**(1/4)/(33*a**5*(-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)/(33*a**6*(x**2 + 1)**(1/4))

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Mathematica [C]  time = 0.135512, size = 103, normalized size = 0.7 \[ \frac{2 \left (5 \sqrt [4]{2} (1+i x)^{3/4} (x+i)^3 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )+5 x^3+10 i x^2-2 x+6 i\right )}{33 a^4 (x+i)^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(15/4)*(a + I*a*x)^(7/4)),x]

[Out]

(2*(6*I - 2*x + (10*I)*x^2 + 5*x^3 + 5*2^(1/4)*(1 + I*x)^(3/4)*(I + x)^3*Hyperge
ometric2F1[1/4, 3/4, 5/4, 1/2 - (I/2)*x]))/(33*a^4*(I + x)^2*(a - I*a*x)^(3/4)*(
a + I*a*x)^(3/4))

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{-{\frac{15}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(15/4)/(a+I*a*x)^(7/4),x)

[Out]

int(1/(a-I*a*x)^(15/4)/(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)^(15/4)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (33 \, a^{6} x^{4} + 66 i \, a^{6} x^{3} + 66 i \, a^{6} x - 33 \, a^{6}\right )}{\rm integral}\left (\frac{5 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{33 \,{\left (a^{6} x^{2} + a^{6}\right )}}, x\right ) +{\left (10 \, x^{3} + 20 i \, x^{2} - 4 \, x + 12 i\right )}{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{33 \, a^{6} x^{4} + 66 i \, a^{6} x^{3} + 66 i \, a^{6} x - 33 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(15/4)),x, algorithm="fricas")

[Out]

((33*a^6*x^4 + 66*I*a^6*x^3 + 66*I*a^6*x - 33*a^6)*integral(5/33*(I*a*x + a)^(1/
4)*(-I*a*x + a)^(1/4)/(a^6*x^2 + a^6), x) + (10*x^3 + 20*I*x^2 - 4*x + 12*I)*(I*
a*x + a)^(1/4)*(-I*a*x + a)^(1/4))/(33*a^6*x^4 + 66*I*a^6*x^3 + 66*I*a^6*x - 33*
a^6)

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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)**(15/4)/(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)^(15/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError