Optimal. Leaf size=25 \[ -\frac{f-2 b g x}{2 b \sqrt{a+b x^4}} \]
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Rubi [A] time = 0.0541837, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ -\frac{f-2 b g x}{2 b \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[(a*g + f*x^3 - b*g*x^4)/(a + b*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 14.0076, size = 22, normalized size = 0.88 \[ - \frac{- 2 b g x + f}{2 b \sqrt{a + b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*g*x**4+f*x**3+a*g)/(b*x**4+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0436706, size = 27, normalized size = 1.08 \[ \frac{2 b g x-f}{2 b \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*g + f*x^3 - b*g*x^4)/(a + b*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.005, size = 24, normalized size = 1. \[{\frac{2\,bgx-f}{2\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*g*x^4+f*x^3+a*g)/(b*x^4+a)^(3/2),x)
[Out]
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Maxima [A] time = 1.45987, size = 31, normalized size = 1.24 \[ \frac{2 \, b g x - f}{2 \, \sqrt{b x^{4} + a} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*g*x^4 - f*x^3 - a*g)/(b*x^4 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218192, size = 45, normalized size = 1.8 \[ \frac{\sqrt{b x^{4} + a}{\left (2 \, b g x - f\right )}}{2 \,{\left (b^{2} x^{4} + a b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*g*x^4 - f*x^3 - a*g)/(b*x^4 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 20.0076, size = 109, normalized size = 4.36 \[ f \left (\begin{cases} - \frac{1}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{g x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} - \frac{b g x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*g*x**4+f*x**3+a*g)/(b*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216623, size = 30, normalized size = 1.2 \[ \frac{2 \, g x - \frac{f}{b}}{2 \, \sqrt{b x^{4} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*g*x^4 - f*x^3 - a*g)/(b*x^4 + a)^(3/2),x, algorithm="giac")
[Out]