Integrand size = 25, antiderivative size = 115 \[ \int \frac {1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx=-\frac {4 i}{9 a (a-i a x)^{9/4} \sqrt [4]{a+i a x}}-\frac {2 i}{45 a^2 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {2 \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{15 a^3 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \] Output:
-4/9*I/a/(a-I*a*x)^(9/4)/(a+I*a*x)^(1/4)-2/45*I/a^2/(a-I*a*x)^(5/4)/(a+I*a *x)^(1/4)+2/15*(x^2+1)^(1/4)*EllipticE(sin(1/2*arctan(x)),2^(1/2))/a^3/(a- I*a*x)^(1/4)/(a+I*a*x)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx=-\frac {2 i 2^{3/4} \sqrt [4]{1+i x} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {1}{4},-\frac {5}{4},\frac {1}{2}-\frac {i x}{2}\right )}{9 a (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \] Input:
Integrate[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(1/4)),x]
Output:
(((-2*I)/9)*2^(3/4)*(1 + I*x)^(1/4)*Hypergeometric2F1[-9/4, 1/4, -5/4, 1/2 - (I/2)*x])/(a*(a - I*a*x)^(9/4)*(a + I*a*x)^(1/4))
Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {61, 56, 46, 213, 212}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {\int \frac {1}{(a-i a x)^{9/4} \sqrt [4]{i x a+a}}dx}{3 a}-\frac {2 i (a+i a x)^{3/4}}{9 a^2 (a-i a x)^{9/4}}\) |
\(\Big \downarrow \) 56 |
\(\displaystyle \frac {\frac {1}{5} \int \frac {1}{(a-i a x)^{5/4} (i x a+a)^{5/4}}dx-\frac {4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}}}{3 a}-\frac {2 i (a+i a x)^{3/4}}{9 a^2 (a-i a x)^{9/4}}\) |
\(\Big \downarrow \) 46 |
\(\displaystyle \frac {\frac {\sqrt [4]{a^2 x^2+a^2} \int \frac {1}{\left (x^2 a^2+a^2\right )^{5/4}}dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}}}{3 a}-\frac {2 i (a+i a x)^{3/4}}{9 a^2 (a-i a x)^{9/4}}\) |
\(\Big \downarrow \) 213 |
\(\displaystyle \frac {\frac {\sqrt [4]{x^2+1} \int \frac {1}{\left (x^2+1\right )^{5/4}}dx}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}}}{3 a}-\frac {2 i (a+i a x)^{3/4}}{9 a^2 (a-i a x)^{9/4}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {\frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {4 i}{5 a (a-i a x)^{5/4} \sqrt [4]{a+i a x}}}{3 a}-\frac {2 i (a+i a x)^{3/4}}{9 a^2 (a-i a x)^{9/4}}\) |
Input:
Int[1/((a - I*a*x)^(13/4)*(a + I*a*x)^(1/4)),x]
Output:
(((-2*I)/9)*(a + I*a*x)^(3/4))/(a^2*(a - I*a*x)^(9/4)) + (((-4*I)/5)/(a*(a - I*a*x)^(5/4)*(a + I*a*x)^(1/4)) + (2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x ]/2, 2])/(5*a^2*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4)))/(3*a)
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]) I nt[(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && !IntegerQ[2*m]
Int[1/(((a_) + (b_.)*(x_))^(9/4)*((c_) + (d_.)*(x_))^(1/4)), x_Symbol] :> S imp[-4/(5*b*(a + b*x)^(5/4)*(c + d*x)^(1/4)), x] - Simp[d/(5*b) Int[1/((a + b*x)^(5/4)*(c + d*x)^(5/4)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a^2*b^2]
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] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( a*(a + b*x^2)^(1/4)) Int[1/(1 + b*(x^2/a))^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a] && PosQ[b/a]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.98
method | result | size |
risch | \(\frac {\frac {2}{15} x^{3}+\frac {4}{15} i x^{2}-\frac {4}{45} x +\frac {22}{45} i}{\left (x +i\right )^{2} a^{3} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}-\frac {x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{15 \left (a^{2}\right )^{\frac {1}{4}} a^{3} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(113\) |
Input:
int(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(1/4),x,method=_RETURNVERBOSE)
Output:
2/45*(6*I*x^2+3*x^3-2*x+11*I)/(x+I)^2/a^3/(-a*(I*x-1))^(1/4)/(a*(I*x+1))^( 1/4)-1/15/(a^2)^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x^2)/a^3*(-a^2*(I*x-1)* (I*x+1))^(1/4)/(-a*(I*x-1))^(1/4)/(a*(I*x+1))^(1/4)
\[ \int \frac {1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {13}{4}}} \,d x } \] Input:
integrate(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(1/4),x, algorithm="fricas")
Output:
1/45*(2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(3*x^2 + 9*I*x - 11) + 45*(a^ 5*x^3 + 3*I*a^5*x^2 - 3*a^5*x - I*a^5)*integral(-1/15*(I*a*x + a)^(3/4)*(- I*a*x + a)^(3/4)/(a^5*x^2 + a^5), x))/(a^5*x^3 + 3*I*a^5*x^2 - 3*a^5*x - I *a^5)
\[ \int \frac {1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx=\int \frac {1}{\sqrt [4]{i a \left (x - i\right )} \left (- i a \left (x + i\right )\right )^{\frac {13}{4}}}\, dx \] Input:
integrate(1/(a-I*a*x)**(13/4)/(a+I*a*x)**(1/4),x)
Output:
Integral(1/((I*a*(x - I))**(1/4)*(-I*a*(x + I))**(13/4)), x)
\[ \int \frac {1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {13}{4}}} \,d x } \] Input:
integrate(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((I*a*x + a)^(1/4)*(-I*a*x + a)^(13/4)), x)
Exception generated. \[ \int \frac {1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(1/4),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo r the root of a polynomial with parameters. This might be wrong.The choice was done
Timed out. \[ \int \frac {1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{13/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \] Input:
int(1/((a - a*x*1i)^(13/4)*(a + a*x*1i)^(1/4)),x)
Output:
int(1/((a - a*x*1i)^(13/4)*(a + a*x*1i)^(1/4)), x)
\[ \int \frac {1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx=\frac {\int \frac {1}{\left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}} i \,x^{3}-3 \left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}} i x -3 \left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}} x^{2}+\left (i x +1\right )^{\frac {1}{4}} \left (-i x +1\right )^{\frac {1}{4}}}d x}{\sqrt {a}\, a^{3}} \] Input:
int(1/(a-I*a*x)^(13/4)/(a+I*a*x)^(1/4),x)
Output:
int(1/((i*x + 1)**(1/4)*( - i*x + 1)**(1/4)*i*x**3 - 3*(i*x + 1)**(1/4)*( - i*x + 1)**(1/4)*i*x - 3*(i*x + 1)**(1/4)*( - i*x + 1)**(1/4)*x**2 + (i*x + 1)**(1/4)*( - i*x + 1)**(1/4)),x)/(sqrt(a)*a**3)