Optimal. Leaf size=79 \[ \frac{4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0580244, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{4 i \sqrt [4]{a-i a x}}{3 a (a+i a x)^{3/4}}-\frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(a - I*a*x)^(1/4)/(a + I*a*x)^(7/4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.3636, size = 68, normalized size = 0.86 \[ \frac{4 i \sqrt [4]{- i a x + a}}{3 a \left (i a x + a\right )^{\frac{3}{4}}} - \frac{2 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{3 a^{2} \sqrt [4]{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a-I*a*x)**(1/4)/(a+I*a*x)**(7/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0508079, size = 73, normalized size = 0.92 \[ -\frac{2 i \sqrt [4]{a-i a x} \left (-2+\sqrt [4]{2} (1+i x)^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )\right )}{3 a (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a - I*a*x)^(1/4)/(a + I*a*x)^(7/4),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.06, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{a-iax} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a-I*a*x)^(1/4)/(a+I*a*x)^(7/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{{\left (i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(7/4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[ \frac{3 \,{\left (a^{2} x - i \, a^{2}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{3 \,{\left (a^{2} x^{2} + a^{2}\right )}}, x\right ) + 4 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{3 \,{\left (a^{2} x - i \, a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(7/4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{- a \left (i x - 1\right )}}{\left (a \left (i x + 1\right )\right )^{\frac{7}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a-I*a*x)**(1/4)/(a+I*a*x)**(7/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-I*a*x + a)^(1/4)/(I*a*x + a)^(7/4),x, algorithm="giac")
[Out]