∫ 1________ (a−iax)7∕4(a+iax)9∕4 dx [1227]

Optimal. Leaf size=100 \[ -\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac {16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}} \]

-2/3*I/a^2/(a-I*a*x)^(3/4)/(a+I*a*x)^(5/4)+8/15*I*(a-I*a*x)^(1/4)/a^3/(a+I*a*x)^(5/4)+16/15*I*(a-I*a*x)^(1/4)/

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Rubi [A]
time = 0.01, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}-\frac {2 i}{3 a^2 (a+i a x)^{5/4} (a-i a x)^{3/4}} \end {gather*}

Int[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(9/4)),x]

((-2*I)/3)/(a^2*(a - I*a*x)^(3/4)*(a + I*a*x)^(5/4)) + (((8*I)/15)*(a - I*a*x)^(1/4))/(a^3*(a + I*a*x)^(5/4))

+ (((16*I)/15)*(a - I*a*x)^(1/4))/(a^4*(a + I*a*x)^(1/4))

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

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]))) && (

\begin {align*} \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {4 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{9/4}} \, dx}{3 a}\\ &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac {8 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{15 a^2}\\ &=-\frac {2 i}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 i \sqrt [4]{a-i a x}}{15 a^3 (a+i a x)^{5/4}}+\frac {16 i \sqrt [4]{a-i a x}}{15 a^4 \sqrt [4]{a+i a x}}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 52, normalized size = 0.52 \begin {gather*} -\frac {2 i (a+i a x)^{3/4} \left (7-4 i x+8 x^2\right )}{15 a^4 (-i+x)^2 (a-i a x)^{3/4}} \end {gather*}

Integrate[1/((a - I*a*x)^(7/4)*(a + I*a*x)^(9/4)),x]

(((-2*I)/15)*(a + I*a*x)^(3/4)*(7 - (4*I)*x + 8*x^2))/(a^4*(-I + x)^2*(a - I*a*x)^(3/4))

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method	result	size

risch	\(\frac {\frac {16}{15} x^{2}-\frac {8}{15} i x +\frac {14}{15}}{a^{3} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}} \left (x -i\right )}\)	\(44\)

int(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(9/4),x,method=_RETURNVERBOSE)

2/15/a^3/(-a*(-1+I*x))^(3/4)/(a*(1+I*x))^(1/4)*(8*x^2-4*I*x+7)/(x-I)

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

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(9/4),x, algorithm="maxima")

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [A]
time = 1.29, size = 56, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (8 \, x^{2} - 4 i \, x + 7\right )}}{15 \, {\left (a^{5} x^{3} - i \, a^{5} x^{2} + a^{5} x - i \, a^{5}\right )}} \end {gather*}

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(9/4),x, algorithm="fricas")

2/15*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*(8*x^2 - 4*I*x + 7)/(a^5*x^3 - I*a^5*x^2 + a^5*x - I*a^5)

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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 {9}{4}} \left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}\, dx \end {gather*}

integrate(1/(a-I*a*x)**(7/4)/(a+I*a*x)**(9/4),x)

Integral(1/((I*a*(x - I))**(9/4)*(-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*}

integrate(1/(a-I*a*x)^(7/4)/(a+I*a*x)^(9/4),x, algorithm="giac")

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

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Mupad [B]
time = 0.53, size = 45, normalized size = 0.45 \begin {gather*} \frac {2\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (x^2\,8{}\mathrm {i}+4\,x+7{}\mathrm {i}\right )}{15\,a^4\,\left (x^2+1\right )\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \end {gather*}

int(1/((a - a*x*1i)^(7/4)*(a + a*x*1i)^(9/4)),x)

(2*(-a*(x*1i - 1))^(1/4)*(4*x + x^2*8i + 7i))/(15*a^4*(x^2 + 1)*(a*(x*1i + 1))^(1/4))

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3.13.27 \(\int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx\) [1227]