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

Optimal result

Integrand size = 25, antiderivative size = 133 \[ \int \frac {1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx=-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac {16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac {32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {47, 37} \[ \int \frac {1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx=-\frac {32 i (a+i a x)^{3/4}}{1155 a^5 (a-i a x)^{3/4}}-\frac {16 i (a+i a x)^{3/4}}{385 a^4 (a-i a x)^{7/4}}-\frac {4 i (a+i a x)^{3/4}}{55 a^3 (a-i a x)^{11/4}}-\frac {2 i (a+i a x)^{3/4}}{15 a^2 (a-i a x)^{15/4}} \]

[In]

Int[1/((a - I*a*x)^(19/4)*(a + I*a*x)^(1/4)),x]

[Out]

(((-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 -
I*a*x)^(3/4))

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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 10.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx=\frac {2 (a+i a x)^{3/4} \left (-159+138 i x+72 x^2-16 i x^3\right )}{1155 a^5 (i+x)^3 (a-i a x)^{3/4}} \]

[In]

Integrate[1/((a - I*a*x)^(19/4)*(a + I*a*x)^(1/4)),x]

[Out]

(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))

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.36

method	result	size
gosper	\(-\frac {2 \left (x +i\right ) \left (-x +i\right ) \left (16 x^{3}+72 i x^{2}-138 x -159 i\right )}{1155 \left (-i a x +a \right )^{\frac {19}{4}} \left (i a x +a \right )^{\frac {1}{4}}}\)	\(48\)
risch	\(\frac {\frac {32}{1155} x^{4}+\frac {16}{165} i x^{3}-\frac {4}{35} x^{2}-\frac {2}{55} i x -\frac {106}{385}}{a^{4} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}} \left (x +i\right )^{3}}\)	\(55\)

[In]

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

[Out]

-2/1155*(x+I)*(-x+I)*(72*I*x^2+16*x^3-159*I-138*x)/(a-I*a*x)^(19/4)/(a+I*a*x)^(1/4)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx=\frac {2 \, {\left (16 \, x^{3} + 72 i \, x^{2} - 138 \, x - 159 i\right )} {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{1155 \, {\left (a^{6} x^{4} + 4 i \, a^{6} x^{3} - 6 \, a^{6} x^{2} - 4 i \, a^{6} x + a^{6}\right )}} \]

[In]

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

[Out]

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
*x^2 - 4*I*a^6*x + a^6)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a-I*a*x)**(19/4)/(a+I*a*x)**(1/4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {19}{4}}} \,d x } \]

[In]

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

[Out]

integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(19/4)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:The choice was done assuming 0=[0,0]ext_reduce Error: Bad Argument TypeDone

Mupad [B] (verification not implemented)

Time = 0.89 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(a-i a x)^{19/4} \sqrt [4]{a+i a x}} \, dx=-\frac {{\left (x-\mathrm {i}\right )}^5\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (-16\,x^3-x^2\,72{}\mathrm {i}+138\,x+159{}\mathrm {i}\right )\,2{}\mathrm {i}}{1155\,a^5\,{\left (x^2+1\right )}^4\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \]

[In]

int(1/((a - a*x*1i)^(19/4)*(a + a*x*1i)^(1/4)),x)

[Out]

-((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))^
(1/4))