Integrand size = 25, antiderivative size = 147 \[ \int \frac {1}{(a-i a x)^{15/4} (a+i a x)^{7/4}} \, dx=-\frac {2 i}{11 a^2 (a-i a x)^{11/4} (a+i a x)^{3/4}}-\frac {2 i}{11 a^3 (a-i a x)^{7/4} (a+i a x)^{3/4}}+\frac {10 x}{33 a^4 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac {10 \left (1+x^2\right )^{3/4} \operatorname {EllipticF}\left (\frac {\arctan (x)}{2},2\right )}{33 a^4 (a-i a x)^{3/4} (a+i a x)^{3/4}} \] Output:
-2/11*I/a^2/(a-I*a*x)^(11/4)/(a+I*a*x)^(3/4)-2/11*I/a^3/(a-I*a*x)^(7/4)/(a +I*a*x)^(3/4)+10/33*x/a^4/(a-I*a*x)^(3/4)/(a+I*a*x)^(3/4)+10/33*(x^2+1)^(3 /4)*InverseJacobiAM(1/2*arctan(x),2^(1/2))/a^4/(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 = 0.48 \[ \int \frac {1}{(a-i a x)^{15/4} (a+i a x)^{7/4}} \, dx=-\frac {i \sqrt [4]{2} (1+i x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},\frac {7}{4},-\frac {7}{4},\frac {1}{2}-\frac {i x}{2}\right )}{11 a^2 (a-i a x)^{11/4} (a+i a x)^{3/4}} \] Input:
Integrate[1/((a - I*a*x)^(15/4)*(a + I*a*x)^(7/4)),x]
Output:
((-1/11*I)*2^(1/4)*(1 + I*x)^(3/4)*Hypergeometric2F1[-11/4, 7/4, -7/4, 1/2 - (I/2)*x])/(a^2*(a - I*a*x)^(11/4)*(a + I*a*x)^(3/4))
Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.25, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {61, 61, 46, 215, 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.
\(\displaystyle \int \frac {1}{(a-i a x)^{15/4} (a+i a x)^{7/4}} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {7 \int \frac {1}{(a-i a x)^{11/4} (i x a+a)^{7/4}}dx}{11 a}-\frac {2 i}{11 a^2 (a-i a x)^{11/4} (a+i a x)^{3/4}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {7 \left (\frac {5 \int \frac {1}{(a-i a x)^{7/4} (i x a+a)^{7/4}}dx}{7 a}-\frac {2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{3/4}}\right )}{11 a}-\frac {2 i}{11 a^2 (a-i a x)^{11/4} (a+i a x)^{3/4}}\) |
\(\Big \downarrow \) 46 |
\(\displaystyle \frac {7 \left (\frac {5 \left (a^2 x^2+a^2\right )^{3/4} \int \frac {1}{\left (x^2 a^2+a^2\right )^{7/4}}dx}{7 a (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{3/4}}\right )}{11 a}-\frac {2 i}{11 a^2 (a-i a x)^{11/4} (a+i a x)^{3/4}}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {7 \left (\frac {5 \left (a^2 x^2+a^2\right )^{3/4} \left (\frac {\int \frac {1}{\left (x^2 a^2+a^2\right )^{3/4}}dx}{3 a^2}+\frac {2 x}{3 a^2 \left (a^2 x^2+a^2\right )^{3/4}}\right )}{7 a (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{3/4}}\right )}{11 a}-\frac {2 i}{11 a^2 (a-i a x)^{11/4} (a+i a x)^{3/4}}\) |
\(\Big \downarrow \) 231 |
\(\displaystyle \frac {7 \left (\frac {5 \left (a^2 x^2+a^2\right )^{3/4} \left (\frac {\left (x^2+1\right )^{3/4} \int \frac {1}{\left (x^2+1\right )^{3/4}}dx}{3 a^2 \left (a^2 x^2+a^2\right )^{3/4}}+\frac {2 x}{3 a^2 \left (a^2 x^2+a^2\right )^{3/4}}\right )}{7 a (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{3/4}}\right )}{11 a}-\frac {2 i}{11 a^2 (a-i a x)^{11/4} (a+i a x)^{3/4}}\) |
\(\Big \downarrow \) 229 |
\(\displaystyle \frac {7 \left (\frac {5 \left (a^2 x^2+a^2\right )^{3/4} \left (\frac {2 \left (x^2+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {\arctan (x)}{2},2\right )}{3 a^2 \left (a^2 x^2+a^2\right )^{3/4}}+\frac {2 x}{3 a^2 \left (a^2 x^2+a^2\right )^{3/4}}\right )}{7 a (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{3/4}}\right )}{11 a}-\frac {2 i}{11 a^2 (a-i a x)^{11/4} (a+i a x)^{3/4}}\) |
Input:
Int[1/((a - I*a*x)^(15/4)*(a + I*a*x)^(7/4)),x]
Output:
((-2*I)/11)/(a^2*(a - I*a*x)^(11/4)*(a + I*a*x)^(3/4)) + (7*(((-2*I)/7)/(a ^2*(a - I*a*x)^(7/4)*(a + I*a*x)^(3/4)) + (5*(a^2 + a^2*x^2)^(3/4)*((2*x)/ (3*a^2*(a^2 + a^2*x^2)^(3/4)) + (2*(1 + x^2)^(3/4)*EllipticF[ArcTan[x]/2, 2])/(3*a^2*(a^2 + a^2*x^2)^(3/4))))/(7*a*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/ 4))))/(11*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)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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]
\[\int \frac {1}{\left (-i a x +a \right )^{\frac {15}{4}} \left (i a x +a \right )^{\frac {7}{4}}}d x\]
Input:
int(1/(a-I*a*x)^(15/4)/(a+I*a*x)^(7/4),x)
Output:
int(1/(a-I*a*x)^(15/4)/(a+I*a*x)^(7/4),x)
\[ \int \frac {1}{(a-i a x)^{15/4} (a+i a x)^{7/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {7}{4}} {\left (-i \, a x + a\right )}^{\frac {15}{4}}} \,d x } \] Input:
integrate(1/(a-I*a*x)^(15/4)/(a+I*a*x)^(7/4),x, algorithm="fricas")
Output:
1/33*(33*(a^6*x^4 + 2*I*a^6*x^3 + 2*I*a^6*x - a^6)*integral(5/33*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4)/(a^6*x^2 + a^6), x) + 2*(5*x^3 + 10*I*x^2 - 2* x + 6*I)*(I*a*x + a)^(1/4)*(-I*a*x + a)^(1/4))/(a^6*x^4 + 2*I*a^6*x^3 + 2* I*a^6*x - a^6)
Timed out. \[ \int \frac {1}{(a-i a x)^{15/4} (a+i a x)^{7/4}} \, dx=\text {Timed out} \] Input:
integrate(1/(a-I*a*x)**(15/4)/(a+I*a*x)**(7/4),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(a-i a x)^{15/4} (a+i a x)^{7/4}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(a-I*a*x)^(15/4)/(a+I*a*x)^(7/4),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {1}{(a-i a x)^{15/4} (a+i a x)^{7/4}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a-I*a*x)^(15/4)/(a+I*a*x)^(7/4),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. \[ \int \frac {1}{(a-i a x)^{15/4} (a+i a x)^{7/4}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{15/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \] Input:
int(1/((a - a*x*1i)^(15/4)*(a + a*x*1i)^(7/4)),x)
Output:
int(1/((a - a*x*1i)^(15/4)*(a + a*x*1i)^(7/4)), x)
\[ \int \frac {1}{(a-i a x)^{15/4} (a+i a x)^{7/4}} \, dx=-\frac {\int \frac {1}{2 \left (i x +1\right )^{\frac {3}{4}} \left (-i x +1\right )^{\frac {3}{4}} i \,x^{3}+2 \left (i x +1\right )^{\frac {3}{4}} \left (-i x +1\right )^{\frac {3}{4}} i x +\left (i x +1\right )^{\frac {3}{4}} \left (-i x +1\right )^{\frac {3}{4}} x^{4}-\left (i x +1\right )^{\frac {3}{4}} \left (-i x +1\right )^{\frac {3}{4}}}d x}{\sqrt {a}\, a^{5}} \] Input:
int(1/(a-I*a*x)^(15/4)/(a+I*a*x)^(7/4),x)
Output:
( - int(1/(2*(i*x + 1)**(3/4)*( - i*x + 1)**(3/4)*i*x**3 + 2*(i*x + 1)**(3 /4)*( - i*x + 1)**(3/4)*i*x + (i*x + 1)**(3/4)*( - i*x + 1)**(3/4)*x**4 - (i*x + 1)**(3/4)*( - i*x + 1)**(3/4)),x))/(sqrt(a)*a**5)