Optimal. Leaf size=38 \[ -\frac {-2 a b g x+a f-b e x^2}{2 a b \sqrt {a+b x^4}} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {1856} \[ -\frac {-2 a b g x+a f-b e x^2}{2 a b \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 1856
Rubi steps
\begin {align*} \int \frac {a g+e x+f x^3-b g x^4}{\left (a+b x^4\right )^{3/2}} \, dx &=-\frac {a f-2 a b g x-b e x^2}{2 a b \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 38, normalized size = 1.00 \[ \frac {2 a b g x-a f+b e x^2}{2 a b \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 44, normalized size = 1.16 \[ \frac {\sqrt {b x^{4} + a} {\left (2 \, a b g x + b e x^{2} - a f\right )}}{2 \, {\left (a b^{2} x^{4} + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 31, normalized size = 0.82 \[ \frac {{\left (2 \, g + \frac {x e}{a}\right )} x - \frac {f}{b}}{2 \, \sqrt {b x^{4} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 0.92 \[ \frac {2 a b g x +b e \,x^{2}-a f}{2 \sqrt {b \,x^{4}+a}\, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.85, size = 44, normalized size = 1.16 \[ \frac {\sqrt {b x^{4} + a} {\left (2 \, a b g x + b e x^{2} - a f\right )}}{2 \, {\left (a b^{2} x^{4} + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.84, size = 29, normalized size = 0.76 \[ \frac {g\,x-\frac {f}{2\,b}+\frac {e\,x^2}{2\,a}}{\sqrt {b\,x^4+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.51, size = 133, normalized size = 3.50 \[ f \left (\begin {cases} - \frac {1}{2 b \sqrt {a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {g x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} - \frac {b g x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {e x^{2}}{2 a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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