Optimal. Leaf size=109 \[ -\frac{5 a^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a-b x^4\right )^{3/4}}-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b} \]
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Rubi [A] time = 0.139454, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{5 a^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a-b x^4\right )^{3/4}}-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b} \]
Antiderivative was successfully verified.
[In] Int[x^8/(a - b*x^4)^(3/4),x]
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Rubi in Sympy [A] time = 19.2726, size = 95, normalized size = 0.87 \[ - \frac{5 a^{\frac{3}{2}} x^{3} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{12 b^{\frac{3}{2}} \left (a - b x^{4}\right )^{\frac{3}{4}}} - \frac{5 a x \sqrt [4]{a - b x^{4}}}{12 b^{2}} - \frac{x^{5} \sqrt [4]{a - b x^{4}}}{6 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8/(-b*x**4+a)**(3/4),x)
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Mathematica [C] time = 0.0557986, size = 80, normalized size = 0.73 \[ \frac{5 a^2 x \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^4}{a}\right )-5 a^2 x+3 a b x^5+2 b^2 x^9}{12 b^2 \left (a-b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^8/(a - b*x^4)^(3/4),x]
[Out]
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Maple [F] time = 0.028, size = 0, normalized size = 0. \[ \int{{x}^{8} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8/(-b*x^4+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(-b*x^4 + a)^(3/4),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{8}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(-b*x^4 + a)^(3/4),x, algorithm="fricas")
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Sympy [A] time = 3.70027, size = 39, normalized size = 0.36 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8/(-b*x**4+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(-b*x^4 + a)^(3/4),x, algorithm="giac")
[Out]