∫ 1_________ (a−iax)19∕4 4 √ ___

Optimal. Leaf size=133 \[ -\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac {16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac {32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}} \]

-2/15*I*(a+I*a*x)^(3/4)/a^2/(a-I*a*x)^(15/4)-4/55*I*(a+I*a*x)^(3/4)/a^3/(a-I*a*x)^(11/4)-16/385*I*(a+I*a*x)^(3

/4)/a^4/(a-I*a*x)^(7/4)-32/1155*I*(a+I*a*x)^(3/4)/a^5/(a-I*a*x)^(3/4)

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Rubi [A]
time = 0.02, 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 = {47, 37} \begin {gather*} -\frac {32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}}-\frac {16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}} \end {gather*}

(((-2*I)/15)*(a + I*a*x)^(3/4))/(a^2*(a - I*a*x)^(15/4)) - (((4*I)/55)*(a + I*a*x)^(3/4))/(a^3*(a - I*a*x)^(11

/4)) - (((16*I)/385)*(a + I*a*x)^(3/4))/(a^4*(a - I*a*x)^(7/4)) - (((32*I)/1155)*(a + I*a*x)^(3/4))/(a^5*(a -

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)^{19/4} \sqrt [4]{a+i a x}} \, dx &=-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}}+\frac {2 \int \frac {1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx}{5 a}\\ &=-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}+\frac {8 \int \frac {1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx}{55 a^2}\\ &=-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac {16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}+\frac {16 \int \frac {1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx}{385 a^3}\\ &=-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac {16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac {32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 57, normalized size = 0.43 \begin {gather*} \frac {2 (a+i a x)^{3/4} \left (-159+138 i x+72 x^2-16 i x^3\right )}{1155 a^5 (i+x)^3 (a-i a x)^{3/4}} \end {gather*}

(2*(a + I*a*x)^(3/4)*(-159 + (138*I)*x + 72*x^2 - (16*I)*x^3))/(1155*a^5*(I + x)^3*(a - I*a*x)^(3/4))

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method	result	size

risch	\(\frac {\frac {32}{1155} x^{4}+\frac {16}{165} i x^{3}-\frac {4}{35} x^{2}-\frac {2}{55} i x -\frac {106}{385}}{a^{4} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}} \left (x +i\right )^{3}}\)	\(55\)

int(1/(a-I*a*x)^(19/4)/(a+I*a*x)^(1/4),x,method=_RETURNVERBOSE)

2/1155/a^4/(-a*(-1+I*x))^(3/4)/(a*(1+I*x))^(1/4)*(56*I*x^3+16*x^4-21*I*x-159-66*x^2)/(x+I)^3

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

integrate(1/(a-I*a*x)^(19/4)/(a+I*a*x)^(1/4),x, algorithm="maxima")

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Fricas [A]
time = 1.03, size = 68, normalized size = 0.51 \begin {gather*} \frac {2 \, {\left (16 \, x^{3} + 72 i \, x^{2} - 138 \, x - 159 i\right )} {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{1155 \, {\left (a^{6} x^{4} + 4 i \, a^{6} x^{3} - 6 \, a^{6} x^{2} - 4 i \, a^{6} x + a^{6}\right )}} \end {gather*}

integrate(1/(a-I*a*x)^(19/4)/(a+I*a*x)^(1/4),x, algorithm="fricas")

2/1155*(16*x^3 + 72*I*x^2 - 138*x - 159*I)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)/(a^6*x^4 + 4*I*a^6*x^3 - 6*a^6

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

Exception raised: SystemError >> excessive stack use: stack is 7772 deep

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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)^(19/4)/(a+I*a*x)^(1/4),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 = 0.79, size = 57, normalized size = 0.43 \begin {gather*} -\frac {{\left (x-\mathrm {i}\right )}^5\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (-16\,x^3-x^2\,72{}\mathrm {i}+138\,x+159{}\mathrm {i}\right )\,2{}\mathrm {i}}{1155\,a^5\,{\left (x^2+1\right )}^4\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \end {gather*}

-((x - 1i)^5*(-a*(x*1i - 1))^(1/4)*(138*x - x^2*72i - 16*x^3 + 159i)*2i)/(1155*a^5*(x^2 + 1)^4*(a*(x*1i + 1))^

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3.12.83 \(\int \frac {1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx\) [1183]