3.1219 \(\int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=82 \[ \frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i}{5 a \sqrt [4]{a-i a x} (a+i a x)^{5/4}} \]

[Out]

4/5*I/a/(a-I*a*x)^(1/4)/(a+I*a*x)^(5/4)+2/5*(x^2+1)^(1/4)*(cos(1/2*arctan(x))^2)^(1/2)/cos(1/2*arctan(x))*Elli
pticE(sin(1/2*arctan(x)),2^(1/2))/a^2/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4)

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Rubi [A]  time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {46, 42, 197, 196} \[ \frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i}{5 a \sqrt [4]{a-i a x} (a+i a x)^{5/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*I)/5)/(a*(a - I*a*x)^(1/4)*(a + I*a*x)^(5/4)) + (2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5*a^2*(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 46

Int[1/(((a_) + (b_.)*(x_))^(9/4)*((c_) + (d_.)*(x_))^(1/4)), x_Symbol] :> Simp[-4/(5*b*(a + b*x)^(5/4)*(c + d*
x)^(1/4)), x] - Dist[d/(5*b), Int[1/((a + b*x)^(5/4)*(c + d*x)^(5/4)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ
[b*c + a*d, 0] && NegQ[a^2*b^2]

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]

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]

Rubi steps

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

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Mathematica [C]  time = 0.02, size = 70, normalized size = 0.85 \[ \frac {i \sqrt [4]{1+i x} (a-i a x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {9}{4};\frac {7}{4};\frac {1}{2}-\frac {i x}{2}\right )}{3 \sqrt [4]{2} a^3 \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/3)*(1 + I*x)^(1/4)*(a - I*a*x)^(3/4)*Hypergeometric2F1[3/4, 9/4, 7/4, 1/2 - (I/2)*x])/(2^(1/4)*a^3*(a + I*
a*x)^(1/4))

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ \frac {{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (2 \, x - 4 i\right )} + {\left (5 \, a^{4} x^{2} - 10 i \, a^{4} x - 5 \, a^{4}\right )} {\rm integral}\left (-\frac {{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{5 \, {\left (a^{4} x^{2} + a^{4}\right )}}, x\right )}{5 \, a^{4} x^{2} - 10 i \, a^{4} x - 5 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(2*x - 4*I) + (5*a^4*x^2 - 10*I*a^4*x - 5*a^4)*integral(-1/5*(I*a*x + a)
^(3/4)*(-I*a*x + a)^(3/4)/(a^4*x^2 + a^4), x))/(5*a^4*x^2 - 10*I*a^4*x - 5*a^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.04, size = 105, normalized size = 1.28 \[ -\frac {\left (-\left (i x -1\right ) \left (i x +1\right ) a^{2}\right )^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right )}{5 \left (a^{2}\right )^{\frac {1}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a^{2}}+\frac {\frac {2}{5} x^{2}-\frac {2}{5} i x +\frac {4}{5}}{\left (x -i\right ) \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{1/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}} \sqrt [4]{- i a \left (x + i\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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