3.1176 \(\int \frac{1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=115 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{15 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i (a+i a x)^{3/4}}{9 a^2 (a-i a x)^{9/4}}-\frac{4 i}{15 a^2 (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]

[Out]

((-4*I)/15)/(a^2*(a - I*a*x)^(5/4)*(a + I*a*x)^(1/4)) - (((2*I)/9)*(a + I*a*x)^(
3/4))/(a^2*(a - I*a*x)^(9/4)) + (2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(1
5*a^3*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.0961207, antiderivative size = 115, 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 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{15 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i (a+i a x)^{3/4}}{9 a^2 (a-i a x)^{9/4}}-\frac{4 i}{15 a^2 (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(1/4)),x]

[Out]

((-4*I)/15)/(a^2*(a - I*a*x)^(5/4)*(a + I*a*x)^(1/4)) - (((2*I)/9)*(a + I*a*x)^(
3/4))/(a^2*(a - I*a*x)^(9/4)) + (2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(1
5*a^3*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\left (a^{2} x^{2} + a^{2}\right )^{\frac{5}{4}}}\, dx}{15 a \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} - \frac{4 i}{15 a^{2} \left (- i a x + a\right )^{\frac{5}{4}} \sqrt [4]{i a x + a}} - \frac{2 i \left (i a x + a\right )^{\frac{3}{4}}}{9 a^{2} \left (- i a x + a\right )^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(13/4)/(a+I*a*x)**(1/4),x)

[Out]

(-I*a*x + a)**(3/4)*(I*a*x + a)**(3/4)*Integral((a**2*x**2 + a**2)**(-5/4), x)/(
15*a*(a**2*x**2 + a**2)**(3/4)) - 4*I/(15*a**2*(-I*a*x + a)**(5/4)*(I*a*x + a)**
(1/4)) - 2*I*(I*a*x + a)**(3/4)/(9*a**2*(-I*a*x + a)**(9/4))

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Mathematica [C]  time = 0.118992, size = 103, normalized size = 0.9 \[ \frac{-2\ 2^{3/4} \sqrt [4]{1+i x} (x+i)^3 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )+6 x^3+12 i x^2-4 x+22 i}{45 a^3 (x+i)^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(1/4)),x]

[Out]

(22*I - 4*x + (12*I)*x^2 + 6*x^3 - 2*2^(3/4)*(1 + I*x)^(1/4)*(I + x)^3*Hypergeom
etric2F1[1/4, 3/4, 7/4, 1/2 - (I/2)*x])/(45*a^3*(I + x)^2*(a - I*a*x)^(1/4)*(a +
I*a*x)^(1/4))

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Maple [C]  time = 0.089, size = 113, normalized size = 1. \[{\frac{12\,i{x}^{2}+6\,{x}^{3}-4\,x+22\,i}{45\, \left ( x+i \right ) ^{2}{a}^{3}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{x}{15\,{a}^{3}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(1/4),x)

[Out]

2/45*(6*I*x^2+3*x^3-2*x+11*I)/(x+I)^2/a^3/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)-
1/15/(a^2)^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x^2)/a^3*(-a^2*(-1+I*x)*(1+I*x))^(
1/4)/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{13}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(13/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(13/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (3 \, x^{2} + 9 i \, x - 11\right )} +{\left (45 \, a^{5} x^{3} + 135 i \, a^{5} x^{2} - 135 \, a^{5} x - 45 i \, a^{5}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{15 \,{\left (a^{5} x^{2} + a^{5}\right )}}, x\right )}{45 \, a^{5} x^{3} + 135 i \, a^{5} x^{2} - 135 \, a^{5} x - 45 i \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(13/4)),x, algorithm="fricas")

[Out]

(2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(3*x^2 + 9*I*x - 11) + (45*a^5*x^3 + 135
*I*a^5*x^2 - 135*a^5*x - 45*I*a^5)*integral(-1/15*(I*a*x + a)^(3/4)*(-I*a*x + a)
^(3/4)/(a^5*x^2 + a^5), x))/(45*a^5*x^3 + 135*I*a^5*x^2 - 135*a^5*x - 45*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)**(13/4)/(a+I*a*x)**(1/4),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)^(1/4)*(-I*a*x + a)^(13/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError