Optimal. Leaf size=28 \[ -\frac {(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \]
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Rubi [A] time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {37} \[ -\frac {(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+b x)^3} \, dx &=-\frac {(c+d x)^2}{2 (b c-a d) (a+b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.93 \[ -\frac {a d+b (c+2 d x)}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 38, normalized size = 1.36 \[ -\frac {2 \, b d x + b c + a d}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 24, normalized size = 0.86 \[ -\frac {2 \, b d x + b c + a d}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 1.25 \[ -\frac {d}{\left (b x +a \right ) b^{2}}-\frac {-a d +b c}{2 \left (b x +a \right )^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 38, normalized size = 1.36 \[ -\frac {2 \, b d x + b c + a d}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 39, normalized size = 1.39 \[ -\frac {\frac {a\,d+b\,c}{2\,b^2}+\frac {d\,x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 39, normalized size = 1.39 \[ \frac {- a d - b c - 2 b d x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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