Optimal. Leaf size=28 \[ -\frac {(d+e x)^2}{2 (a+b x)^2 (b d-a e)} \]
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Rubi [A] time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 37} \[ -\frac {(d+e x)^2}{2 (a+b x)^2 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {d+e x}{(a+b x)^3} \, dx\\ &=-\frac {(d+e x)^2}{2 (b d-a e) (a+b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.93 \[ -\frac {a e+b (d+2 e x)}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 38, normalized size = 1.36 \[ -\frac {2 \, b e x + b d + a e}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 26, normalized size = 0.93 \[ -\frac {2 \, b x e + b d + a e}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 1.25 \[ -\frac {e}{\left (b x +a \right ) b^{2}}-\frac {-a e +b d}{2 \left (b x +a \right )^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 38, normalized size = 1.36 \[ -\frac {2 \, b e x + b d + a e}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.01, size = 39, normalized size = 1.39 \[ -\frac {\frac {a\,e+b\,d}{2\,b^2}+\frac {e\,x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 39, normalized size = 1.39 \[ \frac {- a e - b d - 2 b e x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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