Optimal. Leaf size=65 \[ \frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}-\frac{b^2}{4 e^3 (d+e x)^4} \]
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Rubi [A] time = 0.0365427, 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} \[ \frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}-\frac{b^2}{4 e^3 (d+e x)^4} \]
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)^7} \, dx &=\int \frac{(a+b x)^2}{(d+e x)^7} \, dx\\ &=\int \left (\frac{(-b d+a e)^2}{e^2 (d+e x)^7}-\frac{2 b (b d-a e)}{e^2 (d+e x)^6}+\frac{b^2}{e^2 (d+e x)^5}\right ) \, dx\\ &=-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}+\frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{b^2}{4 e^3 (d+e x)^4}\\ \end{align*}
Mathematica [A] time = 0.0211527, size = 55, normalized size = 0.85 \[ -\frac{10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )}{60 e^3 (d+e x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 71, normalized size = 1.1 \begin{align*} -{\frac{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}{6\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{ \left ( 2\,ae-2\,bd \right ) b}{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 [B] time = 1.20014, size = 162, normalized size = 2.49 \begin{align*} -\frac{15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \,{\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77717, size = 251, normalized size = 3.86 \begin{align*} -\frac{15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \,{\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.57847, size = 128, normalized size = 1.97 \begin{align*} - \frac{10 a^{2} e^{2} + 4 a b d e + b^{2} d^{2} + 15 b^{2} e^{2} x^{2} + x \left (24 a b e^{2} + 6 b^{2} d e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1518, size = 81, normalized size = 1.25 \begin{align*} -\frac{{\left (15 \, b^{2} x^{2} e^{2} + 6 \, b^{2} d x e + b^{2} d^{2} + 24 \, a b x e^{2} + 4 \, a b d e + 10 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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