3.1221 \(\int \frac{1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=88 \[ \frac{2 x}{5 a^4 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{6 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

[Out]

(2*x)/(5*a^4*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)*(1 + x^2)) + (6*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5
*a^4*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.0158891, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {42, 199, 197, 196} \[ \frac{2 x}{5 a^4 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{6 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(9/4)*(a + I*a*x)^(9/4)),x]

[Out]

(2*x)/(5*a^4*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)*(1 + x^2)) + (6*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5
*a^4*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))

Rule 42

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[((a + b*x)^FracPart[m]*(c + d*x)^Frac
Part[m])/(a*c + b*d*x^2)^FracPart[m], Int[(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c +
a*d, 0] &&  !IntegerQ[2*m]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 197

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Dist[(1 + (b*x^2)/a)^(1/4)/(a*(a + b*x^2)^(1/4)), Int[1/(1 + (b
*x^2)/a)^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a] && PosQ[b/a]

Rule 196

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2*EllipticE[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(5/4)*Rt[b
/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx &=\frac{\sqrt [4]{a^2+a^2 x^2} \int \frac{1}{\left (a^2+a^2 x^2\right )^{9/4}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac{2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac{\left (3 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac{1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac{2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac{\left (3 \sqrt [4]{1+x^2}\right ) \int \frac{1}{\left (1+x^2\right )^{5/4}} \, dx}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac{2 x}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac{6 \sqrt [4]{1+x^2} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end{align*}

Mathematica [C]  time = 0.0275565, size = 70, normalized size = 0.8 \[ -\frac{i \sqrt [4]{1+i x} \, _2F_1\left (-\frac{5}{4},\frac{9}{4};-\frac{1}{4};\frac{1}{2}-\frac{i x}{2}\right )}{5 \sqrt [4]{2} a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(9/4)*(a + I*a*x)^(9/4)),x]

[Out]

((-I/5)*(1 + I*x)^(1/4)*Hypergeometric2F1[-5/4, 9/4, -1/4, 1/2 - (I/2)*x])/(2^(1/4)*a^3*(a - I*a*x)^(5/4)*(a +
I*a*x)^(1/4))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a-iax \right ) ^{-{\frac{9}{4}}} \left ( a+iax \right ) ^{-{\frac{9}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(9/4),x)

[Out]

int(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{9}{4}}{\left (-i \, a x + a\right )}^{\frac{9}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(9/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(9/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (3 \, x^{3} + 4 \, x\right )}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} + 5 \,{\left (a^{6} x^{4} + 2 \, a^{6} x^{2} + a^{6}\right )}{\rm integral}\left (-\frac{3 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (a^{6} x^{2} + a^{6}\right )}}, x\right )}{5 \,{\left (a^{6} x^{4} + 2 \, a^{6} x^{2} + a^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(9/4),x, algorithm="fricas")

[Out]

1/5*(2*(3*x^3 + 4*x)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4) + 5*(a^6*x^4 + 2*a^6*x^2 + a^6)*integral(-3/5*(I*a*x
+ a)^(3/4)*(-I*a*x + a)^(3/4)/(a^6*x^2 + a^6), x))/(a^6*x^4 + 2*a^6*x^2 + a^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)**(9/4)/(a+I*a*x)**(9/4),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(9/4)/(a+I*a*x)^(9/4),x, algorithm="giac")

[Out]

Exception raised: TypeError