3.1177 \(\int \frac{1}{(a-i a x)^{17/4} \sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=148 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{39 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}-\frac{4 i}{39 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}} \]

[Out]

((-4*I)/39)/(a^3*(a - I*a*x)^(5/4)*(a + I*a*x)^(1/4)) - (((2*I)/13)*(a + I*a*x)^
(3/4))/(a^2*(a - I*a*x)^(13/4)) - (((10*I)/117)*(a + I*a*x)^(3/4))/(a^3*(a - I*a
*x)^(9/4)) + (2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(39*a^4*(a - I*a*x)^(
1/4)*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.129395, antiderivative size = 148, 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 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{39 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}-\frac{4 i}{39 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - I*a*x)^(17/4)*(a + I*a*x)^(1/4)),x]

[Out]

((-4*I)/39)/(a^3*(a - I*a*x)^(5/4)*(a + I*a*x)^(1/4)) - (((2*I)/13)*(a + I*a*x)^
(3/4))/(a^2*(a - I*a*x)^(13/4)) - (((10*I)/117)*(a + I*a*x)^(3/4))/(a^3*(a - I*a
*x)^(9/4)) + (2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(39*a^4*(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. \[ - \frac{2 i \left (i a x + a\right )^{\frac{3}{4}}}{13 a^{2} \left (- i a x + a\right )^{\frac{13}{4}}} - \frac{4 i}{39 a^{3} \left (- i a x + a\right )^{\frac{5}{4}} \sqrt [4]{i a x + a}} - \frac{10 i \left (i a x + a\right )^{\frac{3}{4}}}{117 a^{3} \left (- i a x + a\right )^{\frac{9}{4}}} - \frac{\left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\sqrt [4]{a^{2} x^{2} + a^{2}}}\, dx}{39 a^{4} \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} + \frac{2 x \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{39 a^{6} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-I*a*x)**(17/4)/(a+I*a*x)**(1/4),x)

[Out]

-2*I*(I*a*x + a)**(3/4)/(13*a**2*(-I*a*x + a)**(13/4)) - 4*I/(39*a**3*(-I*a*x +
a)**(5/4)*(I*a*x + a)**(1/4)) - 10*I*(I*a*x + a)**(3/4)/(117*a**3*(-I*a*x + a)**
(9/4)) - (-I*a*x + a)**(3/4)*(I*a*x + a)**(3/4)*Integral((a**2*x**2 + a**2)**(-1
/4), x)/(39*a**4*(a**2*x**2 + a**2)**(3/4)) + 2*x*(-I*a*x + a)**(3/4)*(I*a*x + a
)**(3/4)/(39*a**6*(x**2 + 1))

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Mathematica [C]  time = 0.138672, size = 102, normalized size = 0.69 \[ -\frac{2 \left (2^{3/4} \sqrt [4]{1+i x} (x+i)^4 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )-3 x^4-9 i x^3+8 x^2+20\right )}{117 a^4 (x+i)^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a - I*a*x)^(17/4)*(a + I*a*x)^(1/4)),x]

[Out]

(-2*(20 + 8*x^2 - (9*I)*x^3 - 3*x^4 + 2^(3/4)*(1 + I*x)^(1/4)*(I + x)^4*Hypergeo
metric2F1[1/4, 3/4, 7/4, 1/2 - (I/2)*x]))/(117*a^4*(I + x)^3*(a - I*a*x)^(1/4)*(
a + I*a*x)^(1/4))

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Maple [C]  time = 0.1, size = 114, normalized size = 0.8 \[{\frac{18\,i{x}^{3}+6\,{x}^{4}-40-16\,{x}^{2}}{117\, \left ( x+i \right ) ^{3}{a}^{4}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{x}{39\,{a}^{4}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-I*a*x)^(17/4)/(a+I*a*x)^(1/4),x)

[Out]

2/117*(9*I*x^3+3*x^4-20-8*x^2)/(x+I)^3/a^4/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)
-1/39/(a^2)^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x^2)/a^4*(-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. \[ \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{17}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(17/4)),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(17/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (6 \, x^{3} + 24 i \, x^{2} - 40 \, x - 40 i\right )}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} +{\left (117 \, a^{6} x^{4} + 468 i \, a^{6} x^{3} - 702 \, a^{6} x^{2} - 468 i \, a^{6} x + 117 \, a^{6}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{39 \,{\left (a^{6} x^{2} + a^{6}\right )}}, x\right )}{117 \, a^{6} x^{4} + 468 i \, a^{6} x^{3} - 702 \, a^{6} x^{2} - 468 i \, a^{6} x + 117 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(17/4)),x, algorithm="fricas")

[Out]

((6*x^3 + 24*I*x^2 - 40*x - 40*I)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4) + (117*a^
6*x^4 + 468*I*a^6*x^3 - 702*a^6*x^2 - 468*I*a^6*x + 117*a^6)*integral(-1/39*(I*a
*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(a^6*x^2 + a^6), x))/(117*a^6*x^4 + 468*I*a^6*x
^3 - 702*a^6*x^2 - 468*I*a^6*x + 117*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)**(17/4)/(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)^(17/4)),x, algorithm="giac")

[Out]

Exception raised: TypeError