Optimal. Leaf size=15 \[ -\frac{c}{3 e (d+e x)^3} \]
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Rubi [A] time = 0.0063929, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {24, 21, 32} \[ -\frac{c}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 24
Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^6} \, dx &=\frac{\int \frac{c d e^2+c e^3 x}{(d+e x)^5} \, dx}{e^2}\\ &=c \int \frac{1}{(d+e x)^4} \, dx\\ &=-\frac{c}{3 e (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0036726, size = 15, normalized size = 1. \[ -\frac{c}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 14, normalized size = 0.9 \begin{align*} -{\frac{c}{3\,e \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14637, size = 49, normalized size = 3.27 \begin{align*} -\frac{c}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97777, size = 73, normalized size = 4.87 \begin{align*} -\frac{c}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.488633, size = 37, normalized size = 2.47 \begin{align*} - \frac{c}{3 d^{3} e + 9 d^{2} e^{2} x + 9 d e^{3} x^{2} + 3 e^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2113, size = 46, normalized size = 3.07 \begin{align*} -\frac{{\left (c x^{2} e^{4} + 2 \, c d x e^{3} + c d^{2} e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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