Optimal. Leaf size=139 \[ -\frac{10 a^2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{(a+i a x)^{3/4} (a-i a x)^{3/4}}+\frac{4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+\frac{2 i \sqrt [4]{a+i a x} (a-i a x)^{5/4}}{a}+10 i \sqrt [4]{a+i a x} \sqrt [4]{a-i a x} \]
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Rubi [A] time = 0.120654, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{10 a^2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{(a+i a x)^{3/4} (a-i a x)^{3/4}}+\frac{4 i (a-i a x)^{9/4}}{3 a (a+i a x)^{3/4}}+\frac{2 i \sqrt [4]{a+i a x} (a-i a x)^{5/4}}{a}+10 i \sqrt [4]{a+i a x} \sqrt [4]{a-i a x} \]
Antiderivative was successfully verified.
[In] Int[(a - I*a*x)^(9/4)/(a + I*a*x)^(7/4),x]
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Rubi in Sympy [A] time = 28.3696, size = 112, normalized size = 0.81 \[ 10 i \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} - \frac{10 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{\sqrt [4]{x^{2} + 1}} + \frac{4 i \left (- i a x + a\right )^{\frac{9}{4}}}{3 a \left (i a x + a\right )^{\frac{3}{4}}} + \frac{2 i \left (- i a x + a\right )^{\frac{5}{4}} \sqrt [4]{i a x + a}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a-I*a*x)**(9/4)/(a+I*a*x)**(7/4),x)
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Mathematica [C] time = 0.0710493, size = 80, normalized size = 0.58 \[ \frac{2 i a \sqrt [4]{a-i a x} \left (-15 \sqrt [4]{2} (1+i x)^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )+x^2+11 i x+20\right )}{3 (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a - I*a*x)^(9/4)/(a + I*a*x)^(7/4),x]
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Maple [F] time = 0.071, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{{\frac{9}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a-I*a*x)^(9/4)/(a+I*a*x)^(7/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{9}{4}}}{{\left (i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-I*a*x + a)^(9/4)/(I*a*x + a)^(7/4),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[ \frac{{\left (3 \, x - 3 i\right )}{\rm integral}\left (-\frac{5 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{x^{2} + 1}, x\right ) + 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (x^{2} + 11 i \, x + 20\right )}}{3 \, x - 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-I*a*x + a)^(9/4)/(I*a*x + a)^(7/4),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a-I*a*x)**(9/4)/(a+I*a*x)**(7/4),x)
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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)^(9/4)/(I*a*x + a)^(7/4),x, algorithm="giac")
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