Optimal. Leaf size=114 \[ \frac{10 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{21 a^3 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{10 x}{21 a^3 (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{3/4}} \]
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Rubi [A] time = 0.081453, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{10 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{21 a^3 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{10 x}{21 a^3 (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac{2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(7/4)),x]
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Rubi in Sympy [A] time = 26.5213, size = 128, normalized size = 1.12 \[ \frac{2 i}{3 a^{2} \left (- i a x + a\right )^{\frac{7}{4}} \left (i a x + a\right )^{\frac{3}{4}}} - \frac{10 i \sqrt [4]{i a x + a}}{21 a^{3} \left (- i a x + a\right )^{\frac{7}{4}}} - \frac{10 i \sqrt [4]{i a x + a}}{21 a^{4} \left (- i a x + a\right )^{\frac{3}{4}}} + \frac{10 \sqrt [4]{- i a x + a} \sqrt [4]{i a x + a} F\left (\frac{\operatorname{atan}{\left (x \right )}}{2}\middle | 2\right )}{21 a^{5} \sqrt [4]{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-I*a*x)**(11/4)/(a+I*a*x)**(7/4),x)
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Mathematica [C] time = 0.116651, size = 96, normalized size = 0.84 \[ \frac{2 \left (5 \sqrt [4]{2} (1+i x)^{3/4} (x+i)^2 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )+5 x^2+5 i x+3\right )}{21 a^3 (x+i) (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - I*a*x)^(11/4)*(a + I*a*x)^(7/4)),x]
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Maple [F] time = 0.092, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{-{\frac{11}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-I*a*x)^(11/4)/(a+I*a*x)^(7/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(11/4)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[ \frac{{\left (21 \, a^{5} x^{3} + 21 i \, a^{5} x^{2} + 21 \, a^{5} x + 21 i \, a^{5}\right )}{\rm integral}\left (\frac{5 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{21 \,{\left (a^{5} x^{2} + a^{5}\right )}}, x\right ) + 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (5 \, x^{2} + 5 i \, x + 3\right )}}{21 \, a^{5} x^{3} + 21 i \, a^{5} x^{2} + 21 \, a^{5} x + 21 i \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(11/4)),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(1/(a-I*a*x)**(11/4)/(a+I*a*x)**(7/4),x)
[Out]
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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(1/((I*a*x + a)^(7/4)*(-I*a*x + a)^(11/4)),x, algorithm="giac")
[Out]