\(\int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx\) [1171]

Optimal result

Integrand size = 25, antiderivative size = 144 \[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\frac {14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}-\frac {14 a^2 \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 42, 235, 233, 202} \[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=-\frac {14 a^2 \sqrt [4]{x^2+1} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a} \]

[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))

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*} \text {integral}& = -\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}+\frac {1}{5} (7 a) \int \frac {(a-i a x)^{3/4}}{\sqrt [4]{a+i a x}} \, dx \\ & = -\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}+\frac {1}{5} \left (7 a^2\right ) \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx \\ & = -\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}+\frac {\left (7 a^2 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{a^2+a^2 x^2}} \, dx}{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}+\frac {\left (7 a^2 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{1+x^2}} \, dx}{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}-\frac {\left (7 a^2 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{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}-\frac {14 a^2 \sqrt [4]{1+x^2} 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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.49 \[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\frac {2 i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{11/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{4},\frac {15}{4},\frac {1}{2}-\frac {i x}{2}\right )}{11 a \sqrt [4]{a+i a x}} \]

[In]

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

[Out]

(((2*I)/11)*2^(3/4)*(1 + I*x)^(1/4)*(a - I*a*x)^(11/4)*Hypergeometric2F1[1/4, 11/4, 15/4, 1/2 - (I/2)*x])/(a*(
a + I*a*x)^(1/4))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.72

method	result	size
risch	\(-\frac {2 \left (10 i+3 x \right ) \left (x +i\right ) \left (x -i\right ) a^{2}}{15 \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}+\frac {7 x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-x^{2}\right ) a^{2} \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{5 \left (a^{2}\right )^{\frac {1}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\)	\(104\)

[In]

int((a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-2/15*(10*I+3*x)*(x+I)*(x-I)*a^2/(-a*(I*x-1))^(1/4)/(a*(I*x+1))^(1/4)+7/5/(a^2)^(1/4)*x*hypergeom([1/4,1/2],[3
/2],-x^2)*a^2*(-a^2*(I*x-1)*(I*x+1))^(1/4)/(-a*(I*x-1))^(1/4)/(a*(I*x+1))^(1/4)

Fricas [F]

\[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {7}{4}}}{{\left (i \, a x + a\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((a-I*a*x)^(7/4)/(a+I*a*x)^(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*integral(14/5*(I*a*x + a)^(3/4)*(-I
*a*x + a)^(3/4)/(x^4 + x^2), x))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {7}{4}}}{{\left (i \, a x + a\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:The choice was do
ne assuming 0=[0,0]ext_reduce Error: Bad Argument Typeintegrate(1/4*16*i*((sageVARa+(-i)*sageVARa*sageVARx)^(1
/4))^10/(-((sa

Mupad [F(-1)]

Timed out. \[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{7/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \]

[In]

int((a - a*x*1i)^(7/4)/(a + a*x*1i)^(1/4),x)

[Out]

int((a - a*x*1i)^(7/4)/(a + a*x*1i)^(1/4), x)