Optimal. Leaf size=38 \[ -\frac{b d-a e}{5 b^2 (a+b x)^5}-\frac{e}{4 b^2 (a+b x)^4} \]
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Rubi [A] time = 0.024086, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 43} \[ -\frac{b d-a e}{5 b^2 (a+b x)^5}-\frac{e}{4 b^2 (a+b x)^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{d+e x}{(a+b x)^6} \, dx\\ &=\int \left (\frac{b d-a e}{b (a+b x)^6}+\frac{e}{b (a+b x)^5}\right ) \, dx\\ &=-\frac{b d-a e}{5 b^2 (a+b x)^5}-\frac{e}{4 b^2 (a+b x)^4}\\ \end{align*}
Mathematica [A] time = 0.0094128, size = 27, normalized size = 0.71 \[ -\frac{a e+4 b d+5 b e x}{20 b^2 (a+b x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 35, normalized size = 0.9 \begin{align*} -{\frac{-ae+bd}{5\,{b}^{2} \left ( bx+a \right ) ^{5}}}-{\frac{e}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14417, size = 97, normalized size = 2.55 \begin{align*} -\frac{5 \, b e x + 4 \, b d + a e}{20 \,{\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50123, size = 153, normalized size = 4.03 \begin{align*} -\frac{5 \, b e x + 4 \, b d + a e}{20 \,{\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.772541, size = 76, normalized size = 2. \begin{align*} - \frac{a e + 4 b d + 5 b e x}{20 a^{5} b^{2} + 100 a^{4} b^{3} x + 200 a^{3} b^{4} x^{2} + 200 a^{2} b^{5} x^{3} + 100 a b^{6} x^{4} + 20 b^{7} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13497, size = 36, normalized size = 0.95 \begin{align*} -\frac{5 \, b x e + 4 \, b d + a e}{20 \,{\left (b x + a\right )}^{5} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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