Integrand size = 25, antiderivative size = 144 \[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\frac {14 a^2 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {14}{15} i (a-i a x)^{3/4} (a+i a x)^{3/4}-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}-\frac {14 a^2 \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \] Output:
14/5*a^2*x/(a-I*a*x)^(1/4)/(a+I*a*x)^(1/4)-14/15*I*(a-I*a*x)^(3/4)*(a+I*a* x)^(3/4)-2/5*I*(a-I*a*x)^(7/4)*(a+I*a*x)^(3/4)/a-14/5*a^2*(x^2+1)^(1/4)*El lipticE(sin(1/2*arctan(x)),2^(1/2))/(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 = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.49 \[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\frac {2 i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{11/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{4},\frac {15}{4},\frac {1}{2}-\frac {i x}{2}\right )}{11 a \sqrt [4]{a+i a x}} \] Input:
Integrate[(a - I*a*x)^(7/4)/(a + I*a*x)^(1/4),x]
Output:
(((2*I)/11)*2^(3/4)*(1 + I*x)^(1/4)*(a - I*a*x)^(11/4)*Hypergeometric2F1[1 /4, 11/4, 15/4, 1/2 - (I/2)*x])/(a*(a + I*a*x)^(1/4))
Time = 0.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {60, 60, 46, 227, 225, 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.
\(\displaystyle \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {7}{5} a \int \frac {(a-i a x)^{3/4}}{\sqrt [4]{i x a+a}}dx-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {7}{5} a \left (a \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{i x a+a}}dx-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}\right )-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}\) |
\(\Big \downarrow \) 46 |
\(\displaystyle \frac {7}{5} a \left (\frac {a \sqrt [4]{a^2 x^2+a^2} \int \frac {1}{\sqrt [4]{x^2 a^2+a^2}}dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}\right )-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}\) |
\(\Big \downarrow \) 227 |
\(\displaystyle \frac {7}{5} a \left (\frac {a \sqrt [4]{x^2+1} \int \frac {1}{\sqrt [4]{x^2+1}}dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}\right )-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}\) |
\(\Big \downarrow \) 225 |
\(\displaystyle \frac {7}{5} a \left (\frac {a \sqrt [4]{x^2+1} \left (\frac {2 x}{\sqrt [4]{x^2+1}}-\int \frac {1}{\left (x^2+1\right )^{5/4}}dx\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}\right )-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {7}{5} a \left (\frac {a \sqrt [4]{x^2+1} \left (\frac {2 x}{\sqrt [4]{x^2+1}}-2 E\left (\left .\frac {\arctan (x)}{2}\right |2\right )\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i (a-i a x)^{3/4} (a+i a x)^{3/4}}{3 a}\right )-\frac {2 i (a-i a x)^{7/4} (a+i a x)^{3/4}}{5 a}\) |
Input:
Int[(a - I*a*x)^(7/4)/(a + I*a*x)^(1/4),x]
Output:
(((-2*I)/5)*(a - I*a*x)^(7/4)*(a + I*a*x)^(3/4))/a + (7*a*((((-2*I)/3)*(a - I*a*x)^(3/4)*(a + I*a*x)^(3/4))/a + (a*(1 + x^2)^(1/4)*((2*x)/(1 + x^2)^ (1/4) - 2*EllipticE[ArcTan[x]/2, 2]))/((a - I*a*x)^(1/4)*(a + I*a*x)^(1/4) )))/5
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 + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) , x] - Simp[a Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ a, 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( a + b*x^2)^(1/4) Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {2 \left (10 i+3 x \right ) \left (x -i\right ) \left (x +i\right ) a^{2}}{15 \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}+\frac {7 x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) a^{2} \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{5 \left (a^{2}\right )^{\frac {1}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(104\) |
Input:
int((a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x,method=_RETURNVERBOSE)
Output:
-2/15*(10*I+3*x)*(x-I)*(x+I)*a^2/(-a*(I*x-1))^(1/4)/(a*(I*x+1))^(1/4)+7/5/ (a^2)^(1/4)*x*hypergeom([1/4,1/2],[3/2],-x^2)*a^2*(-a^2*(I*x-1)*(I*x+1))^( 1/4)/(-a*(I*x-1))^(1/4)/(a*(I*x+1))^(1/4)
\[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {7}{4}}}{{\left (i \, a x + a\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x, algorithm="fricas")
Output:
-1/15*(2*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)*(3*x^2 + 10*I*x - 21) - 15*x *integral(14/5*(I*a*x + a)^(3/4)*(-I*a*x + a)^(3/4)/(x^4 + x^2), x))/x
\[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\int \frac {\left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}{\sqrt [4]{i a \left (x - i\right )}}\, dx \] Input:
integrate((a-I*a*x)**(7/4)/(a+I*a*x)**(1/4),x)
Output:
Integral((-I*a*(x + I))**(7/4)/(I*a*(x - I))**(1/4), x)
\[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {7}{4}}}{{\left (i \, a x + a\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x, algorithm="maxima")
Output:
integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(1/4), x)
Exception generated. \[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Warning, choosing root of [1,0,0,0,%%%{-1,[1,0]%%%}+%%% {i,[0,1]%%%}] at parameters values [99,84]Warning, need to choose a branch for the roo
Timed out. \[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{7/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \] Input:
int((a - a*x*1i)^(7/4)/(a + a*x*1i)^(1/4),x)
Output:
int((a - a*x*1i)^(7/4)/(a + a*x*1i)^(1/4), x)
\[ \int \frac {(a-i a x)^{7/4}}{\sqrt [4]{a+i a x}} \, dx=\sqrt {a}\, a \left (\int \frac {\left (-i x +1\right )^{\frac {3}{4}}}{\left (i x +1\right )^{\frac {1}{4}}}d x -\left (\int \frac {\left (-i x +1\right )^{\frac {3}{4}} x}{\left (i x +1\right )^{\frac {1}{4}}}d x \right ) i \right ) \] Input:
int((a-I*a*x)^(7/4)/(a+I*a*x)^(1/4),x)
Output:
sqrt(a)*a*(int(( - i*x + 1)**(3/4)/(i*x + 1)**(1/4),x) - int((( - i*x + 1) **(3/4)*x)/(i*x + 1)**(1/4),x)*i)