Optimal. Leaf size=98 \[ \frac{5 a^{3/2} \sqrt{b} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{6 \left (a+b x^4\right )^{3/4}}+\frac{5}{6} b x^2 \sqrt [4]{a+b x^4}-\frac{\left (a+b x^4\right )^{5/4}}{2 x^2} \]
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Rubi [A] time = 0.130865, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5 a^{3/2} \sqrt{b} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{6 \left (a+b x^4\right )^{3/4}}+\frac{5}{6} b x^2 \sqrt [4]{a+b x^4}-\frac{\left (a+b x^4\right )^{5/4}}{2 x^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^(5/4)/x^3,x]
[Out]
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Rubi in Sympy [A] time = 12.0032, size = 87, normalized size = 0.89 \[ \frac{5 a^{\frac{3}{2}} \sqrt{b} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{6 \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{5 b x^{2} \sqrt [4]{a + b x^{4}}}{6} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(5/4)/x**3,x)
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Mathematica [C] time = 0.049086, size = 79, normalized size = 0.81 \[ \frac{-6 a^2+5 a b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )-2 a b x^4+4 b^2 x^8}{12 x^2 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)^(5/4)/x^3,x]
[Out]
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Maple [F] time = 0.042, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(5/4)/x^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(5/4)/x^3,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(5/4)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.30136, size = 32, normalized size = 0.33 \[ - \frac{a^{\frac{5}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**(5/4)/x**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(5/4)/x^3,x, algorithm="giac")
[Out]