Optimal. Leaf size=17 \[ -\frac{c^2}{2 e (d+e x)^2} \]
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Rubi [A] time = 0.0051904, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {27, 12, 32} \[ -\frac{c^2}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx &=\int \frac{c^2}{(d+e x)^3} \, dx\\ &=c^2 \int \frac{1}{(d+e x)^3} \, dx\\ &=-\frac{c^2}{2 e (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.001941, size = 17, normalized size = 1. \[ -\frac{c^2}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 16, normalized size = 0.9 \begin{align*} -{\frac{{c}^{2}}{2\,e \left ( ex+d \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21949, size = 36, normalized size = 2.12 \begin{align*} -\frac{c^{2}}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9613, size = 54, normalized size = 3.18 \begin{align*} -\frac{c^{2}}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.690977, size = 27, normalized size = 1.59 \begin{align*} - \frac{c^{2}}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22104, size = 89, normalized size = 5.24 \begin{align*} -\frac{{\left (c^{2} x^{4} e^{8} + 4 \, c^{2} d x^{3} e^{7} + 6 \, c^{2} d^{2} x^{2} e^{6} + 4 \, c^{2} d^{3} x e^{5} + c^{2} d^{4} e^{4}\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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