Optimal. Leaf size=65 \[ \frac{b (b d-a e)}{2 e^3 (d+e x)^4}-\frac{(b d-a e)^2}{5 e^3 (d+e x)^5}-\frac{b^2}{3 e^3 (d+e x)^3} \]
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Rubi [A] time = 0.0364159, antiderivative size = 65, 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} \[ \frac{b (b d-a e)}{2 e^3 (d+e x)^4}-\frac{(b d-a e)^2}{5 e^3 (d+e x)^5}-\frac{b^2}{3 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx &=\int \frac{(a+b x)^2}{(d+e x)^6} \, dx\\ &=\int \left (\frac{(-b d+a e)^2}{e^2 (d+e x)^6}-\frac{2 b (b d-a e)}{e^2 (d+e x)^5}+\frac{b^2}{e^2 (d+e x)^4}\right ) \, dx\\ &=-\frac{(b d-a e)^2}{5 e^3 (d+e x)^5}+\frac{b (b d-a e)}{2 e^3 (d+e x)^4}-\frac{b^2}{3 e^3 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0268726, size = 55, normalized size = 0.85 \[ -\frac{6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )}{30 e^3 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 71, normalized size = 1.1 \begin{align*} -{\frac{{b}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{ \left ( ae-bd \right ) b}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24846, size = 147, normalized size = 2.26 \begin{align*} -\frac{10 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2} + 5 \,{\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{30 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76496, size = 227, normalized size = 3.49 \begin{align*} -\frac{10 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2} + 5 \,{\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{30 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.28312, size = 116, normalized size = 1.78 \begin{align*} - \frac{6 a^{2} e^{2} + 3 a b d e + b^{2} d^{2} + 10 b^{2} e^{2} x^{2} + x \left (15 a b e^{2} + 5 b^{2} d e\right )}{30 d^{5} e^{3} + 150 d^{4} e^{4} x + 300 d^{3} e^{5} x^{2} + 300 d^{2} e^{6} x^{3} + 150 d e^{7} x^{4} + 30 e^{8} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14132, size = 81, normalized size = 1.25 \begin{align*} -\frac{{\left (10 \, b^{2} x^{2} e^{2} + 5 \, b^{2} d x e + b^{2} d^{2} + 15 \, a b x e^{2} + 3 \, a b d e + 6 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{30 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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