3.1171 \(\int \frac{(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{14 a^2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac{2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a} \]

[Out]

(14*a^2*x)/(5*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) - ((14*I)/15)*(a - I*a*x)^(3/
4)*(a + I*a*x)^(3/4) - (((2*I)/5)*(a - I*a*x)^(7/4)*(a + I*a*x)^(3/4))/a - (14*a
^2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5*(a - I*a*x)^(1/4)*(a + I*a*x)^(
1/4))

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Rubi [A]  time = 0.11211, antiderivative size = 144, 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 \[ -\frac{14 a^2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac{2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a} \]

Antiderivative was successfully verified.

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

[Out]

(14*a^2*x)/(5*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) - ((14*I)/15)*(a - I*a*x)^(3/
4)*(a + I*a*x)^(3/4) - (((2*I)/5)*(a - I*a*x)^(7/4)*(a + I*a*x)^(3/4))/a - (14*a
^2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5*(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. \[ \frac{7 a^{2} \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}} \int \frac{1}{\sqrt [4]{a^{2} x^{2} + a^{2}}}\, dx}{5 \left (a^{2} x^{2} + a^{2}\right )^{\frac{3}{4}}} - \frac{14 i \left (- i a x + a\right )^{\frac{3}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{15} - \frac{2 i \left (- i a x + a\right )^{\frac{7}{4}} \left (i a x + a\right )^{\frac{3}{4}}}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*a**2*(-I*a*x + a)**(3/4)*(I*a*x + a)**(3/4)*Integral((a**2*x**2 + a**2)**(-1/4
), x)/(5*(a**2*x**2 + a**2)**(3/4)) - 14*I*(-I*a*x + a)**(3/4)*(I*a*x + a)**(3/4
)/15 - 2*I*(-I*a*x + a)**(7/4)*(I*a*x + a)**(3/4)/(5*a)

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Mathematica [C]  time = 0.0816059, size = 84, normalized size = 0.58 \[ \frac{2 a (a-i a x)^{3/4} \left (7 i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )-3 i x^2+7 x-10 i\right )}{15 \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*(a - I*a*x)^(3/4)*(-10*I + 7*x - (3*I)*x^2 + (7*I)*2^(3/4)*(1 + I*x)^(1/4)*
Hypergeometric2F1[1/4, 3/4, 7/4, 1/2 - (I/2)*x]))/(15*(a + I*a*x)^(1/4))

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Maple [C]  time = 0.266, size = 104, normalized size = 0.7 \[ -{\frac{ \left ( 20\,i+6\,x \right ) \left ( x+i \right ) \left ( x-i \right ){a}^{2}}{15}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}+{\frac{7\,x{a}^{2}}{5}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(10*I+3*x)*(x+I)*(x-I)*a^2/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)+7/5/(a^2)
^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x^2)*a^2*(-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. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{7}{4}}}{{\left (i \, a x + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (3 \, x^{2} + 10 i \, x - 21\right )} - 15 \, x{\rm integral}\left (\frac{14 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (x^{4} + x^{2}\right )}}, x\right )}{15 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(3*x^2 + 10*I*x - 21) - 15*x*integ
ral(14/5*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(x^4 + x^2), x))/x

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

[Out]

Exception raised: TypeError