Integrand size = 25, antiderivative size = 115 \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{3/4}} \, dx=-\frac {2 i \sqrt [4]{a+i a x}}{7 a^2 (a-i a x)^{7/4}}-\frac {2 i \sqrt [4]{a+i a x}}{7 a^3 (a-i a x)^{3/4}}+\frac {2 \left (1+x^2\right )^{3/4} \operatorname {EllipticF}\left (\frac {\arctan (x)}{2},2\right )}{7 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \] Output:
-2/7*I*(a+I*a*x)^(1/4)/a^2/(a-I*a*x)^(7/4)-2/7*I*(a+I*a*x)^(1/4)/a^3/(a-I* a*x)^(3/4)+2/7*(x^2+1)^(3/4)*InverseJacobiAM(1/2*arctan(x),2^(1/2))/a^2/(a -I*a*x)^(3/4)/(a+I*a*x)^(3/4)
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 \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{3/4}} \, dx=-\frac {2 i \sqrt [4]{2} (1+i x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {3}{4},-\frac {3}{4},\frac {1}{2}-\frac {i x}{2}\right )}{7 a (a-i a x)^{7/4} (a+i a x)^{3/4}} \] Input:
Integrate[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(3/4)),x]
Output:
(((-2*I)/7)*2^(1/4)*(1 + I*x)^(3/4)*Hypergeometric2F1[-7/4, 3/4, -3/4, 1/2 - (I/2)*x])/(a*(a - I*a*x)^(7/4)*(a + I*a*x)^(3/4))
Time = 0.18 (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, 61, 46, 231, 229}
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.
\(\displaystyle \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{3/4}} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {3 \int \frac {1}{(a-i a x)^{7/4} (i x a+a)^{3/4}}dx}{7 a}-\frac {2 i \sqrt [4]{a+i a x}}{7 a^2 (a-i a x)^{7/4}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {1}{(a-i a x)^{3/4} (i x a+a)^{3/4}}dx}{3 a}-\frac {2 i \sqrt [4]{a+i a x}}{3 a^2 (a-i a x)^{3/4}}\right )}{7 a}-\frac {2 i \sqrt [4]{a+i a x}}{7 a^2 (a-i a x)^{7/4}}\) |
\(\Big \downarrow \) 46 |
\(\displaystyle \frac {3 \left (\frac {\left (a^2 x^2+a^2\right )^{3/4} \int \frac {1}{\left (x^2 a^2+a^2\right )^{3/4}}dx}{3 a (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {2 i \sqrt [4]{a+i a x}}{3 a^2 (a-i a x)^{3/4}}\right )}{7 a}-\frac {2 i \sqrt [4]{a+i a x}}{7 a^2 (a-i a x)^{7/4}}\) |
\(\Big \downarrow \) 231 |
\(\displaystyle \frac {3 \left (\frac {\left (x^2+1\right )^{3/4} \int \frac {1}{\left (x^2+1\right )^{3/4}}dx}{3 a (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {2 i \sqrt [4]{a+i a x}}{3 a^2 (a-i a x)^{3/4}}\right )}{7 a}-\frac {2 i \sqrt [4]{a+i a x}}{7 a^2 (a-i a x)^{7/4}}\) |
\(\Big \downarrow \) 229 |
\(\displaystyle \frac {3 \left (\frac {2 \left (x^2+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {\arctan (x)}{2},2\right )}{3 a (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {2 i \sqrt [4]{a+i a x}}{3 a^2 (a-i a x)^{3/4}}\right )}{7 a}-\frac {2 i \sqrt [4]{a+i a x}}{7 a^2 (a-i a x)^{7/4}}\) |
Input:
Int[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(3/4)),x]
Output:
(((-2*I)/7)*(a + I*a*x)^(1/4))/(a^2*(a - I*a*x)^(7/4)) + (3*((((-2*I)/3)*( a + I*a*x)^(1/4))/(a^2*(a - I*a*x)^(3/4)) + (2*(1 + x^2)^(3/4)*EllipticF[A rcTan[x]/2, 2])/(3*a*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))))/(7*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[((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)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(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)^(-3/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(3/4)/( a + b*x^2)^(3/4) Int[1/(1 + b*(x^2/a))^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
\[\int \frac {1}{\left (-i a x +a \right )^{\frac {11}{4}} \left (i a x +a \right )^{\frac {3}{4}}}d x\]
Input:
int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(3/4),x)
Output:
int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(3/4),x)
\[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{3/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {11}{4}}} \,d x } \] Input:
integrate(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(3/4),x, algorithm="fricas")
Output:
1/7*(7*(a^4*x^2 + 2*I*a^4*x - a^4)*integral(1/7*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4)/(a^4*x^2 + a^4), x) + 2*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4)*(x + 2*I))/(a^4*x^2 + 2*I*a^4*x - a^4)
\[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{3/4}} \, dx=\int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {3}{4}} \left (- i a \left (x + i\right )\right )^{\frac {11}{4}}}\, dx \] Input:
integrate(1/(a-I*a*x)**(11/4)/(a+I*a*x)**(3/4),x)
Output:
Integral(1/((I*a*(x - I))**(3/4)*(-I*a*(x + I))**(11/4)), x)
\[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{3/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {11}{4}}} \,d x } \] Input:
integrate(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(3/4),x, algorithm="maxima")
Output:
integrate(1/((I*a*x + a)^(3/4)*(-I*a*x + a)^(11/4)), x)
Exception generated. \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{3/4}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(3/4),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. \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{3/4}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{11/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{3/4}} \,d x \] Input:
int(1/((a - a*x*1i)^(11/4)*(a + a*x*1i)^(3/4)),x)
Output:
int(1/((a - a*x*1i)^(11/4)*(a + a*x*1i)^(3/4)), x)
\[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{3/4}} \, dx=-\frac {\int \frac {1}{2 \left (i x +1\right )^{\frac {3}{4}} \left (-i x +1\right )^{\frac {3}{4}} i x +\left (i x +1\right )^{\frac {3}{4}} \left (-i x +1\right )^{\frac {3}{4}} x^{2}-\left (i x +1\right )^{\frac {3}{4}} \left (-i x +1\right )^{\frac {3}{4}}}d x}{\sqrt {a}\, a^{3}} \] Input:
int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(3/4),x)
Output:
( - int(1/(2*(i*x + 1)**(3/4)*( - i*x + 1)**(3/4)*i*x + (i*x + 1)**(3/4)*( - i*x + 1)**(3/4)*x**2 - (i*x + 1)**(3/4)*( - i*x + 1)**(3/4)),x))/(sqrt( a)*a**3)