3.12.76 \(\int \frac {1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx\) [1176]

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

[Out]

-4/15*I/a^2/(a-I*a*x)^(5/4)/(a+I*a*x)^(1/4)-2/9*I*(a+I*a*x)^(3/4)/a^2/(a-I*a*x)^(9/4)+2/15*(x^2+1)^(1/4)*(cos(
1/2*arctan(x))^2)^(1/2)/cos(1/2*arctan(x))*EllipticE(sin(1/2*arctan(x)),2^(1/2))/a^3/(a-I*a*x)^(1/4)/(a+I*a*x)
^(1/4)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {53, 48, 42, 203, 202} \begin {gather*} \frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {\text {ArcTan}(x)}{2}\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}} \end {gather*}

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])/(15*a^3*(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 48

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 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 202

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

Rule 203

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}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx &=-\frac {2 i (a+i a x)^{3/4}}{9 a^2 (a-i a x)^{9/4}}+\frac {\int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx}{3 a}\\ &=-\frac {4 i}{15 a^2 (a-i a x)^{5/4} \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 {\int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx}{15 a}\\ &=-\frac {4 i}{15 a^2 (a-i a x)^{5/4} \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 {\sqrt [4]{a^2+a^2 x^2} \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{15 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {4 i}{15 a^2 (a-i a x)^{5/4} \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 {\sqrt [4]{1+x^2} \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{15 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {4 i}{15 a^2 (a-i a x)^{5/4} \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 {2 \sqrt [4]{1+x^2} 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}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 70, normalized size = 0.61 \begin {gather*} -\frac {2 i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (-\frac {9}{4},\frac {1}{4};-\frac {5}{4};\frac {1}{2}-\frac {i x}{2}\right )}{9 a (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.18, size = 113, normalized size = 0.98

method	result	size
risch	\(\frac {\frac {2}{15} x^{3}+\frac {4}{15} i x^{2}-\frac {4}{45} x +\frac {22}{45} i}{\left (x +i\right )^{2} a^{3} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}-\frac {x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{15 \left (a^{2}\right )^{\frac {1}{4}} a^{3} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\)	\(113\)

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,method=_RETURNVERBOSE)

[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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, algorithm="fricas")

[Out]

1/45*(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 + 3*I*a^5*x^2 - 3*a^5*x - 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))/(a^5*x^3 + 3*I*a^5*x^2 - 3*a^5*x -
I*a^5)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{i a \left (x - i\right )} \left (- i a \left (x + i\right )\right )^{\frac {13}{4}}}\, dx \end {gather*}

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]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:ext_reduce Error: Bad Argument TypeDone

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{13/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a - a*x*1i)^(13/4)*(a + a*x*1i)^(1/4)),x)

[Out]

int(1/((a - a*x*1i)^(13/4)*(a + a*x*1i)^(1/4)), x)

________________________________________________________________________________________