3.12.72 \(\int \frac {(a-i a x)^{3/4}}{\sqrt [4]{a+i a x}} \, dx\) [1172]

Optimal. Leaf size=106 \[ \frac {2 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac {2 a \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

[Out]

2*a*x/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4)-2/3*I*(a-I*a*x)^(3/4)*(a+I*a*x)^(3/4)/a-2*a*(x^2+1)^(1/4)*(cos(1/2*arcta
n(x))^2)^(1/2)/cos(1/2*arctan(x))*EllipticE(sin(1/2*arctan(x)),2^(1/2))/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4)

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Rubi [A]
time = 0.02, antiderivative size = 106, 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 = {52, 42, 235, 233, 202} \begin {gather*} -\frac {2 a \sqrt [4]{x^2+1} E\left (\left .\frac {\text {ArcTan}(x)}{2}\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*x)/((a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) - (((2*I)/3)*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))/a - (2*a*(1 +
x^2)^(1/4)*EllipticE[ArcTan[x]/2, 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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 233

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

Rule 235

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 70, normalized size = 0.66 \begin {gather*} \frac {2 i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{7/4} \, _2F_1\left (\frac {1}{4},\frac {7}{4};\frac {11}{4};\frac {1}{2}-\frac {i x}{2}\right )}{7 a \sqrt [4]{a+i a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((2*I)/7)*2^(3/4)*(1 + I*x)^(1/4)*(a - I*a*x)^(7/4)*Hypergeometric2F1[1/4, 7/4, 11/4, 1/2 - (I/2)*x])/(a*(a +
I*a*x)^(1/4))

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.17, size = 94, normalized size = 0.89

method	result	size
risch	\(-\frac {2 i \left (x -i\right ) \left (x +i\right ) a}{3 \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}+\frac {x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) a \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-I*a*x)^(3/4)/(a+I*a*x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-2/3*I*(x-I)*(x+I)*a/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)+1/(a^2)^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x^2)*a*(
-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 = 0.00, 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((a-I*a*x)^(3/4)/(a+I*a*x)^(1/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(3/4)/(I*a*x + a)^(1/4), x)

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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((a-I*a*x)^(3/4)/(a+I*a*x)^(1/4),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- i a \left (x + i\right )\right )^{\frac {3}{4}}}{\sqrt [4]{i a \left (x - i\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-I*a*x)**(3/4)/(a+I*a*x)**(1/4),x)

[Out]

Integral((-I*a*(x + I))**(3/4)/(I*a*(x - I))**(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-I*a*x)^(3/4)/(a+I*a*x)^(1/4),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:ext_reduce Error:
Bad Argument Typeintegrate(4*i*((sageVARa+(-i)*sageVARa*sageVARx)^(1/4))^6/(-((sageVARa+(-i)*sageVARa*sageVAR
x)^(1/4))^4+2*

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}}{{\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((a - a*x*1i)^(3/4)/(a + a*x*1i)^(1/4),x)

[Out]

int((a - a*x*1i)^(3/4)/(a + a*x*1i)^(1/4), x)

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