Integrand size = 25, antiderivative size = 115 \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {6 \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
4/5*I*(a-I*a*x)^(3/4)/a/(a+I*a*x)^(5/4)-6/5*I/a/(a-I*a*x)^(1/4)/(a+I*a*x)^ (1/4)-6/5*(x^2+1)^(1/4)*(cos(1/2*arctan(x))^2)^(1/2)/cos(1/2*arctan(x))*El lipticE(sin(1/2*arctan(x)),2^(1/2))/a/(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 \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\frac {i \sqrt [4]{1+i x} (a-i a x)^{7/4} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},\frac {9}{4},\frac {11}{4},\frac {1}{2}-\frac {i x}{2}\right )}{7 \sqrt [4]{2} a^3 \sqrt [4]{a+i a x}} \]
((I/7)*(1 + I*x)^(1/4)*(a - I*a*x)^(7/4)*Hypergeometric2F1[7/4, 9/4, 11/4, 1/2 - (I/2)*x])/(2^(1/4)*a^3*(a + I*a*x)^(1/4))
Time = 0.19 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {57, 58, 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.
\(\displaystyle \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {3}{5} \int \frac {1}{\sqrt [4]{a-i a x} (i x a+a)^{5/4}}dx\) |
\(\Big \downarrow \) 58 |
\(\displaystyle \frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {3}{5} \left (a \int \frac {1}{(a-i a x)^{5/4} (i x a+a)^{5/4}}dx+\frac {2 i}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\right )\) |
\(\Big \downarrow \) 46 |
\(\displaystyle \frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {3}{5} \left (\frac {a \sqrt [4]{a^2 x^2+a^2} \int \frac {1}{\left (x^2 a^2+a^2\right )^{5/4}}dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 i}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\right )\) |
\(\Big \downarrow \) 213 |
\(\displaystyle \frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {3}{5} \left (\frac {\sqrt [4]{x^2+1} \int \frac {1}{\left (x^2+1\right )^{5/4}}dx}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 i}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\right )\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {3}{5} \left (\frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {2 i}{a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\right )\) |
(((4*I)/5)*(a - I*a*x)^(3/4))/(a*(a + I*a*x)^(5/4)) - (3*((2*I)/(a*(a - I* a*x)^(1/4)*(a + I*a*x)^(1/4)) + (2*(1 + x^2)^(1/4)*EllipticE[ArcTan[x]/2, 2])/(a*(a - I*a*x)^(1/4)*(a + I*a*x)^(1/4))))/5
3.13.18.3.1 Defintions of rubi rules used
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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[1/(((a_) + (b_.)*(x_))^(5/4)*((c_) + (d_.)*(x_))^(1/4)), x_Symbol] :> S imp[-2/(b*(a + b*x)^(1/4)*(c + d*x)^(1/4)), x] + Simp[c 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_)^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 = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {2 \left (3 x^{2}+2 i x +1\right )}{5 \left (x -i\right ) a \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}+\frac {3 x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-x^{2}\right ) \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{5 \left (a^{2}\right )^{\frac {1}{4}} a \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(107\) |
-2/5*(3*x^2+1+2*I*x)/(x-I)/a/(-a*(I*x-1))^(1/4)/(a*(I*x+1))^(1/4)+3/5/(a^2 )^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x^2)/a*(-a^2*(I*x-1)*(I*x+1))^(1/4)/( -a*(I*x-1))^(1/4)/(a*(I*x+1))^(1/4)
\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}} \,d x } \]
-1/5*(2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(5*I*x + 3) - 5*(a^3*x^3 - 2* I*a^3*x^2 - a^3*x)*integral(6/5*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(a^3* x^4 + a^3*x^2), x))/(a^3*x^3 - 2*I*a^3*x^2 - a^3*x)
\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\int \frac {\left (- i a \left (x + i\right )\right )^{\frac {3}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}}}\, dx \]
\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}} \,d x } \]
\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}} \,d x } \]
Timed out. \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \]