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 = 0.10, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 1588
Rule 2102
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*}
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Mathematica [A] time = 0.06, size = 23, normalized size = 0.58 \[ \frac {a+b x}{c+d x+e x^2+f x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 23, normalized size = 0.58 \[ \frac {b x + a}{f x^{3} + e x^{2} + d x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 24, normalized size = 0.60 \[ \frac {b x + a}{f x^{3} + x^{2} e + d x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 0.70 \[ -\frac {-b x -a}{f \,x^{3}+e \,x^{2}+d x +c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 23, normalized size = 0.58 \[ \frac {b x + a}{f x^{3} + e x^{2} + d x + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 23, normalized size = 0.58 \[ \frac {a+b\,x}{f\,x^3+e\,x^2+d\,x+c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 32.56, size = 22, normalized size = 0.55 \[ - \frac {- a - b x}{c + d x + e x^{2} + f x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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