Optimal. Leaf size=57 \[ \frac{f \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{g x}{\sqrt{a+b x^2+c x^4}} \]
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Rubi [A] time = 0.0795246, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {1673, 1588, 12, 1114, 636} \[ \frac{f \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{g x}{\sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1673
Rule 1588
Rule 12
Rule 1114
Rule 636
Rubi steps
\begin{align*} \int \frac{a g+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\int \frac{f x^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx+\int \frac{a g-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\\ &=\frac{g x}{\sqrt{a+b x^2+c x^4}}+f \int \frac{x^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\\ &=\frac{g x}{\sqrt{a+b x^2+c x^4}}+\frac{1}{2} f \operatorname{Subst}\left (\int \frac{x}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{g x}{\sqrt{a+b x^2+c x^4}}+\frac{f \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [F] time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [A] time = 0.004, size = 53, normalized size = 0.9 \begin{align*}{\frac{4\,acgx-{b}^{2}gx-bf{x}^{2}-2\,af}{4\,ac-{b}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13281, size = 66, normalized size = 1.16 \begin{align*} \frac{b f x^{2} + 2 \, a f +{\left (b^{2} g - 4 \, a c g\right )} x}{\sqrt{c x^{4} + b x^{2} + a}{\left (b^{2} - 4 \, a c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30984, size = 171, normalized size = 3. \begin{align*} \frac{\sqrt{c x^{4} + b x^{2} + a}{\left (b f x^{2} +{\left (b^{2} - 4 \, a c\right )} g x + 2 \, a f\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17742, size = 221, normalized size = 3.88 \begin{align*} \frac{{\left (\frac{{\left (b^{3} f - 4 \, a b c f\right )} x}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}} + \frac{b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}}\right )} x + \frac{2 \,{\left (a b^{2} f - 4 \, a^{2} c f\right )}}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}}}{16 \, \sqrt{c x^{4} + b x^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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