Optimal. Leaf size=69 \[ \frac {g x}{\sqrt {a+b x^2+c x^4}}-\frac {-2 a f+x^2 (2 c e-b f)+b e}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
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Rubi [A] time = 0.09, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1673, 1588, 1247, 636} \[ \frac {g x}{\sqrt {a+b x^2+c x^4}}-\frac {-2 a f+x^2 (2 c e-b f)+b e}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 636
Rule 1247
Rule 1588
Rule 1673
Rubi steps
\begin {align*} \int \frac {a g+e x+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\int \frac {x \left (e+f x^2\right )}{\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}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {e+f 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 {b e-2 a f+(2 c e-b f) x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [A] time = 0.75, size = 92, normalized size = 1.33 \[ \frac {\sqrt {c x^{4} + b x^{2} + a} {\left ({\left (b^{2} - 4 \, a c\right )} g x - {\left (2 \, c e - b f\right )} x^{2} - b e + 2 \, a f\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.10, size = 166, normalized size = 2.41 \[ \frac {{\left (\frac {{\left (b^{3} f - 4 \, a b c f - 2 \, b^{2} c e + 8 \, a c^{2} e\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {2 \, a b^{2} f - 8 \, a^{2} c f - b^{3} e + 4 \, a b c e}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}}{\sqrt {c x^{4} + b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 63, normalized size = 0.91 \[ \frac {4 a c g x -b^{2} g x -b f \,x^{2}+2 c e \,x^{2}-2 a f +b e}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 94, normalized size = 1.36 \[ -\frac {\sqrt {c x^{4} + b x^{2} + a} {\left ({\left (2 \, c e - b f\right )} x^{2} + b e - 2 \, a f - {\left (b^{2} g - 4 \, a c g\right )} x\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 62, normalized size = 0.90 \[ -\frac {g\,b^2\,x+f\,b\,x^2-e\,b-2\,c\,e\,x^2-4\,a\,c\,g\,x+2\,a\,f}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^4+b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {a g}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \left (- \frac {e x}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \left (- \frac {f x^{3}}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \frac {c g x^{4}}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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