Optimal. Leaf size=40 \[ \frac{a}{c+d x+e x^2+f x^3}+\frac{b x}{c+d x+e x^2+f x^3} \]
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Rubi [A] time = 0.0968871, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 52, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.058, Rules used = {6, 2102, 1588} \[ \frac{a}{c+d x+e x^2+f x^3}+\frac{b x}{c+d x+e x^2+f x^3} \]
Antiderivative was successfully verified.
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Rule 6
Rule 2102
Rule 1588
Rubi steps
\begin{align*} \int \frac{b c-a d-2 a e x-b e x^2-3 a f x^2-2 b f x^3}{\left (c+d x+e x^2+f x^3\right )^2} \, dx &=\int \frac{b c-a d-2 a e x+(-b e-3 a f) x^2-2 b f x^3}{\left (c+d x+e x^2+f x^3\right )^2} \, dx\\ &=\frac{b x}{c+d x+e x^2+f x^3}-\frac{\int \frac{2 a d f+4 a e f x+6 a f^2 x^2}{\left (c+d x+e x^2+f x^3\right )^2} \, dx}{2 f}\\ &=\frac{a}{c+d x+e x^2+f x^3}+\frac{b x}{c+d x+e x^2+f x^3}\\ \end{align*}
Mathematica [A] time = 0.0619845, size = 23, normalized size = 0.57 \[ \frac{a+b x}{c+d x+e x^2+f x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 28, normalized size = 0.7 \begin{align*} -{\frac{-bx-a}{f{x}^{3}+e{x}^{2}+dx+c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05355, size = 31, normalized size = 0.78 \begin{align*} \frac{b x + a}{f x^{3} + e x^{2} + d x + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24705, size = 50, normalized size = 1.25 \begin{align*} \frac{b x + a}{f x^{3} + e x^{2} + d x + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 30.0342, size = 19, normalized size = 0.48 \begin{align*} \frac{a + b x}{c + d x + e x^{2} + f x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.89861, size = 32, normalized size = 0.8 \begin{align*} \frac{b x + a}{f x^{3} + x^{2} e + d x + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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