3.1217 \(\int \frac{(a-i a x)^{7/4}}{(a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{42 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{42 x}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]

[Out]

(((4*I)/5)*(a - I*a*x)^(7/4))/(a*(a + I*a*x)^(5/4)) + (42*x)/(5*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) - (((28*I
)/5)*(a - I*a*x)^(3/4))/(a*(a + I*a*x)^(1/4)) - (42*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5*(a - I*a*x)^
(1/4)*(a + I*a*x)^(1/4))

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Rubi [A]  time = 0.0303233, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {47, 42, 229, 227, 196} \[ -\frac{42 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{42 x}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]

Antiderivative was successfully verified.

[In]

Int[(a - I*a*x)^(7/4)/(a + I*a*x)^(9/4),x]

[Out]

(((4*I)/5)*(a - I*a*x)^(7/4))/(a*(a + I*a*x)^(5/4)) + (42*x)/(5*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) - (((28*I
)/5)*(a - I*a*x)^(3/4))/(a*(a + I*a*x)^(1/4)) - (42*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(5*(a - I*a*x)^
(1/4)*(a + I*a*x)^(1/4))

Rule 47

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

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 229

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]

Rule 227

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 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{(a-i a x)^{7/4}}{(a+i a x)^{9/4}} \, dx &=\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{7}{5} \int \frac{(a-i a x)^{3/4}}{(a+i a x)^{5/4}} \, dx\\ &=\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{21}{5} \int \frac{1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx\\ &=\frac{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{\left (21 \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{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac{\left (21 \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{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}+\frac{42 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}-\frac{\left (21 \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{4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}+\frac{42 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}-\frac{42 \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]  time = 0.0378105, size = 70, normalized size = 0.5 \[ \frac{i \sqrt [4]{1+i x} (a-i a x)^{11/4} \, _2F_1\left (\frac{9}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2}-\frac{i x}{2}\right )}{11 \sqrt [4]{2} a^3 \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - I*a*x)^(7/4)/(a + I*a*x)^(9/4),x]

[Out]

((I/11)*(1 + I*x)^(1/4)*(a - I*a*x)^(11/4)*Hypergeometric2F1[9/4, 11/4, 15/4, 1/2 - (I/2)*x])/(2^(1/4)*a^3*(a
+ I*a*x)^(1/4))

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Maple [C]  time = 0.05, size = 101, normalized size = 0.7 \begin{align*} -{\frac{32\,{x}^{2}+24+8\,ix}{5\,x-5\,i}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}+{\frac{21\,x}{5}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-I*a*x)^(7/4)/(a+I*a*x)^(9/4),x)

[Out]

-8/5*(4*x^2+3+I*x)/(x-I)/(-a*(-1+I*x))^(1/4)/(a*(1+I*x))^(1/4)+21/5/(a^2)^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x
^2)*(-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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-i \, a x + a\right )}^{\frac{7}{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((a-I*a*x)^(7/4)/(a+I*a*x)^(9/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(7/4)/(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 (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (5 \, x^{2} - 30 i \, x - 21\right )} + 5 \,{\left (a^{2} x^{3} - 2 i \, a^{2} x^{2} - a^{2} x\right )}{\rm integral}\left (\frac{42 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{5 \,{\left (a^{2} x^{4} + a^{2} x^{2}\right )}}, x\right )}{5 \,{\left (a^{2} x^{3} - 2 i \, a^{2} x^{2} - a^{2} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-I*a*x)^(7/4)/(a+I*a*x)^(9/4),x, algorithm="fricas")

[Out]

1/5*(2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(5*x^2 - 30*I*x - 21) + 5*(a^2*x^3 - 2*I*a^2*x^2 - a^2*x)*integral
(42/5*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(a^2*x^4 + a^2*x^2), x))/(a^2*x^3 - 2*I*a^2*x^2 - a^2*x)

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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((a-I*a*x)**(7/4)/(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((a-I*a*x)^(7/4)/(a+I*a*x)^(9/4),x, algorithm="giac")

[Out]

Exception raised: TypeError