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*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))

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Rubi [A]  time = 0.0094655, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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 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)^{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 = 0.0169206, 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))

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Maple [A]  time = 0.033, size = 33, normalized size = 0.5 \begin{align*}{\frac{2\,i+4\,x}{3\,{a}^{2}} \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)^(7/4)/(a+I*a*x)^(5/4),x)

[Out]

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

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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{5}{4}}{\left (-i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \end{align*}

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)

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Fricas [A]  time = 1.73902, size = 95, normalized size = 1.42 \begin{align*} \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 )}} \end{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \left (i x + 1\right )\right )^{\frac{5}{4}} \left (- a \left (i x - 1\right )\right )^{\frac{7}{4}}}\, dx \end{align*}

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

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

[Out]

Exception raised: TypeError