3.1228 \(\int \frac{1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx\)

Optimal. Leaf size=133 \[ \frac{32 i \sqrt [4]{a-i a x}}{35 a^5 \sqrt [4]{a+i a x}}+\frac{16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}-\frac{4 i}{7 a^3 (a+i a x)^{5/4} (a-i a x)^{3/4}}-\frac{2 i}{7 a^2 (a+i a x)^{5/4} (a-i a x)^{7/4}} \]

[Out]

((-2*I)/7)/(a^2*(a - I*a*x)^(7/4)*(a + I*a*x)^(5/4)) - ((4*I)/7)/(a^3*(a - I*a*x)^(3/4)*(a + I*a*x)^(5/4)) + (
((16*I)/35)*(a - I*a*x)^(1/4))/(a^4*(a + I*a*x)^(5/4)) + (((32*I)/35)*(a - I*a*x)^(1/4))/(a^5*(a + I*a*x)^(1/4
))

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Rubi [A]  time = 0.0294059, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {45, 37} \[ \frac{32 i \sqrt [4]{a-i a x}}{35 a^5 \sqrt [4]{a+i a x}}+\frac{16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}-\frac{4 i}{7 a^3 (a+i a x)^{5/4} (a-i a x)^{3/4}}-\frac{2 i}{7 a^2 (a+i a x)^{5/4} (a-i a x)^{7/4}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(9/4)),x]

[Out]

((-2*I)/7)/(a^2*(a - I*a*x)^(7/4)*(a + I*a*x)^(5/4)) - ((4*I)/7)/(a^3*(a - I*a*x)^(3/4)*(a + I*a*x)^(5/4)) + (
((16*I)/35)*(a - I*a*x)^(1/4))/(a^4*(a + I*a*x)^(5/4)) + (((32*I)/35)*(a - I*a*x)^(1/4))/(a^5*(a + I*a*x)^(1/4
))

Rule 45

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*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

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] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx &=-\frac{2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{5/4}}+\frac{6 \int \frac{1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx}{7 a}\\ &=-\frac{2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{5/4}}-\frac{4 i}{7 a^3 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac{8 \int \frac{1}{(a-i a x)^{3/4} (a+i a x)^{9/4}} \, dx}{7 a^2}\\ &=-\frac{2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{5/4}}-\frac{4 i}{7 a^3 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac{16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}+\frac{16 \int \frac{1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{35 a^3}\\ &=-\frac{2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{5/4}}-\frac{4 i}{7 a^3 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac{16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}+\frac{32 i \sqrt [4]{a-i a x}}{35 a^5 \sqrt [4]{a+i a x}}\\ \end{align*}

Mathematica [A]  time = 0.0296752, size = 57, normalized size = 0.43 \[ \frac{2 \left (16 x^3+8 i x^2+22 x+9 i\right )}{35 a^4 \left (x^2+1\right ) (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(9/4)),x]

[Out]

(2*(9*I + 22*x + (8*I)*x^2 + 16*x^3))/(35*a^4*(a - I*a*x)^(3/4)*(a + I*a*x)^(1/4)*(1 + x^2))

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Maple [A]  time = 0.048, size = 56, normalized size = 0.4 \begin{align*}{\frac{32\,{x}^{3}+16\,i{x}^{2}+44\,x+18\,i}{35\,{a}^{4} \left ( x-i \right ) \left ( x+i \right ) } \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(9/4),x)

[Out]

2/35/a^4/(-a*(-1+I*x))^(3/4)/(a*(1+I*x))^(1/4)*(16*x^3+8*I*x^2+22*x+9*I)/(x-I)/(x+I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{9}{4}}{\left (-i \, a x + a\right )}^{\frac{11}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(9/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(9/4)*(-I*a*x + a)^(11/4)), x)

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Fricas [A]  time = 2.05048, size = 142, normalized size = 1.07 \begin{align*} \frac{{\left (32 \, x^{3} + 16 i \, x^{2} + 44 \, x + 18 i\right )}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{35 \,{\left (a^{6} x^{4} + 2 \, a^{6} x^{2} + a^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(9/4),x, algorithm="fricas")

[Out]

1/35*(32*x^3 + 16*I*x^2 + 44*x + 18*I)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)/(a^6*x^4 + 2*a^6*x^2 + a^6)

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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(1/(a-I*a*x)**(11/4)/(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(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(9/4),x, algorithm="giac")

[Out]

Exception raised: TypeError