Optimal. Leaf size=28 \[ \frac{(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \]
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Rubi [A] time = 0.0054238, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 37} \[ \frac{(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rubi steps
\begin{align*} \int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx &=\int \frac{(a+b x)^2}{(d+e x)^4} \, dx\\ &=\frac{(a+b x)^3}{3 (b d-a e) (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0248631, size = 53, normalized size = 1.89 \[ -\frac{a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 71, normalized size = 2.5 \begin{align*} -{\frac{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{ \left ( ae-bd \right ) b}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}}{{e}^{3} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.10879, size = 113, normalized size = 4.04 \begin{align*} -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68778, size = 170, normalized size = 6.07 \begin{align*} -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.868145, size = 88, normalized size = 3.14 \begin{align*} - \frac{a^{2} e^{2} + a b d e + b^{2} d^{2} + 3 b^{2} e^{2} x^{2} + x \left (3 a b e^{2} + 3 b^{2} d e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17025, size = 78, normalized size = 2.79 \begin{align*} -\frac{{\left (3 \, b^{2} x^{2} e^{2} + 3 \, b^{2} d x e + b^{2} d^{2} + 3 \, a b x e^{2} + a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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