3.1222 \(\int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=121 \[ \frac {14 x}{45 a^5 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {14 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{15 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}} \]

[Out]

-2/9*I/a^2/(a-I*a*x)^(9/4)/(a+I*a*x)^(5/4)+14/45*x/a^5/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4)/(x^2+1)+14/15*(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^5/(a-I*a*x)^(1/4)
/(a+I*a*x)^(1/4)

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Rubi [A]  time = 0.03, antiderivative size = 121, 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 = {51, 42, 199, 197, 196} \[ \frac {14 x}{45 a^5 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {14 \sqrt [4]{x^2+1} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{15 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(9/4)),x]

[Out]

((-2*I)/9)/(a^2*(a - I*a*x)^(9/4)*(a + I*a*x)^(5/4)) + (14*x)/(45*a^5*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)*(1 +
x^2)) + (14*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(15*a^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 51

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 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 197

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]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rubi steps

\begin {align*} \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{9/4}} \, dx &=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {7 \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx}{9 a}\\ &=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {\left (7 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{9/4}} \, dx}{9 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {14 x}{45 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {\left (7 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{15 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {14 x}{45 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {\left (7 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{15 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac {14 x}{45 a^5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac {14 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{15 a^5 \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 = 0.58 \[ -\frac {i \sqrt [4]{1+i x} \, _2F_1\left (-\frac {9}{4},\frac {9}{4};-\frac {5}{4};\frac {1}{2}-\frac {i x}{2}\right )}{9 \sqrt [4]{2} a^3 (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(9/4)),x]

[Out]

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

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ \frac {{\left (42 \, x^{4} + 42 i \, x^{3} + 56 \, x^{2} + 56 i \, x + 10\right )} {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}} + {\left (45 \, a^{7} x^{5} + 45 i \, a^{7} x^{4} + 90 \, a^{7} x^{3} + 90 i \, a^{7} x^{2} + 45 \, a^{7} x + 45 i \, a^{7}\right )} {\rm integral}\left (-\frac {7 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{15 \, {\left (a^{7} x^{2} + a^{7}\right )}}, x\right )}{45 \, a^{7} x^{5} + 45 i \, a^{7} x^{4} + 90 \, a^{7} x^{3} + 90 i \, a^{7} x^{2} + 45 \, a^{7} x + 45 i \, a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(9/4),x, algorithm="fricas")

[Out]

((42*x^4 + 42*I*x^3 + 56*x^2 + 56*I*x + 10)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4) + (45*a^7*x^5 + 45*I*a^7*x^4
+ 90*a^7*x^3 + 90*I*a^7*x^2 + 45*a^7*x + 45*I*a^7)*integral(-7/15*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(a^7*x^
2 + a^7), x))/(45*a^7*x^5 + 45*I*a^7*x^4 + 90*a^7*x^3 + 90*I*a^7*x^2 + 45*a^7*x + 45*I*a^7)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {13}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(9/4),x, algorithm="giac")

[Out]

integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(13/4)), x)

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maple [C]  time = 0.10, size = 124, normalized size = 1.02 \[ -\frac {7 \left (-\left (i x -1\right ) \left (i x +1\right ) a^{2}\right )^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right )}{15 \left (a^{2}\right )^{\frac {1}{4}} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a^{5}}+\frac {\frac {14}{15} x^{4}+\frac {14}{15} i x^{3}+\frac {56}{45} x^{2}+\frac {56}{45} i x +\frac {2}{9}}{\left (x -i\right ) \left (x +i\right )^{2} \left (-\left (i x -1\right ) a \right )^{\frac {1}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-I*a*x+a)^(13/4)/(I*a*x+a)^(9/4),x)

[Out]

2/45*(21*I*x^3+21*x^4+28*I*x+28*x^2+5)/(x-I)/(x+I)^2/a^5/(-(I*x-1)*a)^(1/4)/((I*x+1)*a)^(1/4)-7/15/(a^2)^(1/4)
*x*hypergeom([1/4,1/2],[3/2],-x^2)/a^5*(-(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(-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(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(9/4),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{13/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a - a*x*1i)^(13/4)*(a + a*x*1i)^(9/4)),x)

[Out]

int(1/((a - a*x*1i)^(13/4)*(a + a*x*1i)^(9/4)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)**(13/4)/(a+I*a*x)**(9/4),x)

[Out]

Timed out

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