3.1206 \(\int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=137 \[ \frac {14 a \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a+i a x)^{3/4} (a-i a x)^{3/4}}{3 a}-\frac {14 a x}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {47, 50, 42, 229, 227, 196} \[ \frac {14 a \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a+i a x)^{3/4} (a-i a x)^{3/4}}{3 a}-\frac {14 a x}{\sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \]

Antiderivative was successfully verified.

[In]

Int[(a - I*a*x)^(7/4)/(a + I*a*x)^(5/4),x]

[Out]

(-14*a*x)/((a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)) + ((4*I)*(a - I*a*x)^(7/4))/(a*(a + I*a*x)^(1/4)) + (((14*I)/3
)*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))/a + (14*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 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 50

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 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 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 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]

Rubi steps

\begin {align*} \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{5/4}} \, dx &=\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}-7 \int \frac {(a-i a x)^{3/4}}{\sqrt [4]{a+i a x}} \, dx\\ &=\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-(7 a) \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}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac {\left (7 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 {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}-\frac {\left (7 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 {14 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac {\left (7 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 {14 a x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i (a-i a x)^{7/4}}{a \sqrt [4]{a+i a x}}+\frac {14 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}+\frac {14 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]  time = 0.03, size = 70, normalized size = 0.51 \[ \frac {i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{11/4} \, _2F_1\left (\frac {5}{4},\frac {11}{4};\frac {15}{4};\frac {1}{2}-\frac {i x}{2}\right )}{11 a^2 \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - I*a*x)^(7/4)/(a + I*a*x)^(5/4),x]

[Out]

((I/11)*2^(3/4)*(1 + I*x)^(1/4)*(a - I*a*x)^(11/4)*Hypergeometric2F1[5/4, 11/4, 15/4, 1/2 - (I/2)*x])/(a^2*(a
+ I*a*x)^(1/4))

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ \frac {{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} {\left (2 i \, x^{2} - 16 \, x + 42 i\right )} + {\left (3 \, a x^{2} - 3 i \, a x\right )} {\rm integral}\left (-\frac {14 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{a x^{4} + a x^{2}}, x\right )}{3 \, a x^{2} - 3 i \, a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-I*a*x)^(7/4)/(a+I*a*x)^(5/4),x, algorithm="fricas")

[Out]

((I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(2*I*x^2 - 16*x + 42*I) + (3*a*x^2 - 3*I*a*x)*integral(-14*(I*a*x + a)^(
3/4)*(-I*a*x + a)^(3/4)/(a*x^4 + a*x^2), x))/(3*a*x^2 - 3*I*a*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-I*a*x)^(7/4)/(a+I*a*x)^(5/4),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, choosing
root of [1,0,0,0,%%%{-2,[1,0]%%%}+%%%{1,[0,4]%%%}] at parameters values [70,22]Warning, choosing root of [1,0
,0,0,%%%{-2,[1,0]%%%}+%%%{1,[0,4]%%%}] at parameters values [42,56]Warning, choosing root of [1,0,0,0,%%%{-2,[
1,0]%%%}+%%%{1,[0,4]%%%}] at parameters values [-9,-13]Warning, choosing root of [1,0,0,0,%%%{-2,[1,0]%%%}+%%%
{1,[0,4]%%%}] at parameters values [46,24]Warning, choosing root of [1,0,0,0,%%%{-2,[1,0]%%%}+%%%{1,[0,4]%%%}]
at parameters values [49,-6]ext_reduce Error: Bad Argument TypeEvaluation time: 0.66integrate(i/4*a/a^2*(16*(
(i*a*x+a)^(1/4))^4*((-((i*a*x+a)^(1/4))^4+2*a)^(1/4))^3-32*a*((-((i*a*x+a)^(1/4))^4+2*a)^(1/4))^3)/((i*a*x+a)^
(1/4))^2/4*i*a*((i*a*x+a)^(1/4))^-3,x)

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maple [C]  time = 0.07, size = 96, normalized size = 0.70 \[ -\frac {7 \left (-\left (i x -1\right ) \left (i x +1\right ) a^{2}\right )^{\frac {1}{4}} a x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right )}{\left (a^{2}\right )^{\frac {1}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}}}+\frac {2 i \left (x^{2}-12 i x +13\right ) a}{3 \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-I*a*x+a)^(7/4)/(I*a*x+a)^(5/4),x)

[Out]

2/3*I*(x^2+13-12*I*x)*a/(-(I*x-1)*a)^(1/4)/((I*x+1)*a)^(1/4)-7/(a^2)^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x^2)*a
*(-(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 \[ \int \frac {{\left (-i \, a x + a\right )}^{\frac {7}{4}}}{{\left (i \, a x + a\right )}^{\frac {5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-I*a*x)^(7/4)/(a+I*a*x)^(5/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(5/4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{7/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{5/4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - a*x*1i)^(7/4)/(a + a*x*1i)^(5/4),x)

[Out]

int((a - a*x*1i)^(7/4)/(a + a*x*1i)^(5/4), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-I*a*x)**(7/4)/(a+I*a*x)**(5/4),x)

[Out]

Integral((-I*a*(x + I))**(7/4)/(I*a*(x - I))**(5/4), x)

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