Optimal. Leaf size=65 \[ -\frac{2 e (b d-a e)}{3 b^3 (a+b x)^3}-\frac{(b d-a e)^2}{4 b^3 (a+b x)^4}-\frac{e^2}{2 b^3 (a+b x)^2} \]
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Rubi [A] time = 0.04153, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac{2 e (b d-a e)}{3 b^3 (a+b x)^3}-\frac{(b d-a e)^2}{4 b^3 (a+b x)^4}-\frac{e^2}{2 b^3 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^2}{(a+b x)^5} \, dx\\ &=\int \left (\frac{(b d-a e)^2}{b^2 (a+b x)^5}+\frac{2 e (b d-a e)}{b^2 (a+b x)^4}+\frac{e^2}{b^2 (a+b x)^3}\right ) \, dx\\ &=-\frac{(b d-a e)^2}{4 b^3 (a+b x)^4}-\frac{2 e (b d-a e)}{3 b^3 (a+b x)^3}-\frac{e^2}{2 b^3 (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0202566, size = 56, normalized size = 0.86 \[ -\frac{a^2 e^2+2 a b e (d+2 e x)+b^2 \left (3 d^2+8 d e x+6 e^2 x^2\right )}{12 b^3 (a+b x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 71, normalized size = 1.1 \begin{align*}{\frac{2\,e \left ( ae-bd \right ) }{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{{e}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06161, size = 132, normalized size = 2.03 \begin{align*} -\frac{6 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 2 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48163, size = 201, normalized size = 3.09 \begin{align*} -\frac{6 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 2 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.954163, size = 104, normalized size = 1.6 \begin{align*} - \frac{a^{2} e^{2} + 2 a b d e + 3 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (4 a b e^{2} + 8 b^{2} d e\right )}{12 a^{4} b^{3} + 48 a^{3} b^{4} x + 72 a^{2} b^{5} x^{2} + 48 a b^{6} x^{3} + 12 b^{7} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13461, size = 81, normalized size = 1.25 \begin{align*} -\frac{6 \, b^{2} x^{2} e^{2} + 8 \, b^{2} d x e + 3 \, b^{2} d^{2} + 4 \, a b x e^{2} + 2 \, a b d e + a^{2} e^{2}}{12 \,{\left (b x + a\right )}^{4} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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