3.1215 \(\int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx\)

Optimal. Leaf size=67 \[ \frac {4 i \sqrt [4]{a-i a x}}{3 a^3 \sqrt [4]{a+i a x}}-\frac {2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]

[Out]

-2/3*I/a^2/(a-I*a*x)^(3/4)/(a+I*a*x)^(1/4)+4/3*I*(a-I*a*x)^(1/4)/a^3/(a+I*a*x)^(1/4)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {45, 37} \[ \frac {4 i \sqrt [4]{a-i a x}}{3 a^3 \sqrt [4]{a+i a x}}-\frac {2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(5/4)),x]

[Out]

((-2*I)/3)/(a^2*(a - I*a*x)^(3/4)*(a + I*a*x)^(1/4)) + (((4*I)/3)*(a - I*a*x)^(1/4))/(a^3*(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)^{7/4} (a+i a x)^{5/4}} \, dx &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac {2 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{3 a}\\ &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac {4 i \sqrt [4]{a-i a x}}{3 a^3 \sqrt [4]{a+i a x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 38, normalized size = 0.57 \[ \frac {4 x+2 i}{3 a^2 (a-i a x)^{3/4} \sqrt [4]{a+i a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(5/4)),x]

[Out]

(2*I + 4*x)/(3*a^2*(a - I*a*x)^(3/4)*(a + I*a*x)^(1/4))

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 36, normalized size = 0.54 \[ \frac {{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (4 \, x + 2 i\right )}}{3 \, {\left (a^{4} x^{2} + a^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(5/4),x, algorithm="fricas")

[Out]

1/3*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*(4*x + 2*I)/(a^4*x^2 + a^4)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {5}{4}} {\left (-i \, a x + a\right )}^{\frac {7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((I*a*x + a)^(5/4)*(-I*a*x + a)^(7/4)), x)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 33, normalized size = 0.49 \[ \frac {\frac {4 x}{3}+\frac {2 i}{3}}{\left (-\left (i x -1\right ) a \right )^{\frac {3}{4}} \left (\left (i x +1\right ) a \right )^{\frac {1}{4}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/a^2/(-(I*x-1)*a)^(3/4)/((I*x+1)*a)^(1/4)*(2*x+I)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {5}{4}} {\left (-i \, a x + a\right )}^{\frac {7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(5/4),x, algorithm="maxima")

[Out]

integrate(1/((I*a*x + a)^(5/4)*(-I*a*x + a)^(7/4)), x)

________________________________________________________________________________________

mupad [B]  time = 0.60, size = 40, normalized size = 0.60 \[ -\frac {2\,\left (2\,x+1{}\mathrm {i}\right )\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}}{3\,a^3\,\left (-1+x\,1{}\mathrm {i}\right )\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(2*x + 1i)*(-a*(x*1i - 1))^(1/4))/(3*a^3*(x*1i - 1)*(a*(x*1i + 1))^(1/4))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {5}{4}} \left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-I*a*x)**(7/4)/(a+I*a*x)**(5/4),x)

[Out]

Integral(1/((I*a*(x - I))**(5/4)*(-I*a*(x + I))**(7/4)), x)

________________________________________________________________________________________