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

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

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Rubi [A]  time = 0.03, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {45, 37} \begin {gather*} \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}} \end {gather*}

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 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]

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])

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*}

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Mathematica [A]  time = 0.03, size = 57, normalized size = 0.43 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.14, size = 99, normalized size = 0.74 \begin {gather*} \frac {i (a+i a x)^{7/4} \left (\frac {7 (a-i a x)^3}{(a+i a x)^3}+\frac {105 (a-i a x)^2}{(a+i a x)^2}-\frac {35 (a-i a x)}{a+i a x}-5\right )}{140 a^5 (a-i a x)^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((I/140)*(a + I*a*x)^(7/4)*(-5 + (7*(a - I*a*x)^3)/(a + I*a*x)^3 + (105*(a - I*a*x)^2)/(a + I*a*x)^2 - (35*(a
- I*a*x))/(a + I*a*x)))/(a^5*(a - I*a*x)^(7/4))

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fricas [A]  time = 1.49, size = 54, normalized size = 0.41 \begin {gather*} \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 {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {11}{4}}}\,{d x} \end {gather*}

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]

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

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maple [A]  time = 0.06, size = 56, normalized size = 0.42 \begin {gather*} \frac {\frac {32}{35} x^{3}+\frac {16}{35} i x^{2}+\frac {44}{35} x +\frac {18}{35} i}{\left (-\left (i x -1\right ) a \right )^{\frac {3}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} \left (x -i\right ) \left (x +i\right ) a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {11}{4}}}\,{d x} \end {gather*}

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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mupad [B]  time = 0.69, size = 56, normalized size = 0.42 \begin {gather*} \frac {2\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (x^4\,16{}\mathrm {i}+8\,x^3+x^2\,30{}\mathrm {i}+13\,x+9{}\mathrm {i}\right )}{35\,a^5\,{\left (x^2+1\right )}^2\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(-a*(x*1i - 1))^(1/4)*(13*x + x^2*30i + 8*x^3 + x^4*16i + 9i))/(35*a^5*(x^2 + 1)^2*(a*(x*1i + 1))^(1/4))

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

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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