Optimal. Leaf size=38 \[ -\frac{b d-a e}{4 b^2 (a+b x)^4}-\frac{e}{3 b^2 (a+b x)^3} \]
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Rubi [A] time = 0.0241647, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 43} \[ -\frac{b d-a e}{4 b^2 (a+b x)^4}-\frac{e}{3 b^2 (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) (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)^5} \, dx\\ &=\int \left (\frac{b d-a e}{b (a+b x)^5}+\frac{e}{b (a+b x)^4}\right ) \, dx\\ &=-\frac{b d-a e}{4 b^2 (a+b x)^4}-\frac{e}{3 b^2 (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0089424, size = 27, normalized size = 0.71 \[ -\frac{a e+3 b d+4 b e x}{12 b^2 (a+b x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 35, normalized size = 0.9 \begin{align*} -{\frac{e}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{-ae+bd}{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 [A] time = 0.963735, size = 82, normalized size = 2.16 \begin{align*} -\frac{4 \, b e x + 3 \, b d + a e}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53512, size = 128, normalized size = 3.37 \begin{align*} -\frac{4 \, b e x + 3 \, b d + a e}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.59745, size = 65, normalized size = 1.71 \begin{align*} - \frac{a e + 3 b d + 4 b e x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13264, size = 36, normalized size = 0.95 \begin{align*} -\frac{4 \, b x e + 3 \, b d + a e}{12 \,{\left (b x + a\right )}^{4} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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