3.13.15 \(\int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx\) [1215]

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

[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)

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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 = {47, 37} \begin {gather*} \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}} \end {gather*}

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 47

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*}

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Mathematica [A]
time = 0.10, size = 45, normalized size = 0.67 \begin {gather*} \frac {2 (1-2 i x) (a+i a x)^{3/4}}{3 a^3 (-i+x) (a-i a x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.15, size = 33, normalized size = 0.49

method	result	size
risch	\(\frac {\frac {2 i}{3}+\frac {4 x}{3}}{a^{2} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\)	\(33\)

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,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.79, size = 36, normalized size = 0.54 \begin {gather*} \frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (2 \, x + i\right )}}{3 \, {\left (a^{4} x^{2} + a^{4}\right )}} \end {gather*}

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]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:ext_reduce Error: Bad Argument TypeDone

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Mupad [B]
time = 0.60, size = 40, normalized size = 0.60 \begin {gather*} -\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}} \end {gather*}

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))

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