Optimal. Leaf size=15 \[ -\frac {c^2}{e (d+e x)} \]
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Rubi [A] time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \begin {gather*} -\frac {c^2}{e (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
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)^6} \, dx &=\int \frac {c^2}{(d+e x)^2} \, dx\\ &=c^2 \int \frac {1}{(d+e x)^2} \, dx\\ &=-\frac {c^2}{e (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {c^2}{e (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 16, normalized size = 1.07 \begin {gather*} -\frac {c^{2}}{e^{2} x + d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 66, normalized size = 4.40 \begin {gather*} -\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 )}}{{\left (x e + d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 16, normalized size = 1.07 \begin {gather*} -\frac {c^{2}}{\left (e x +d \right ) e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 16, normalized size = 1.07 \begin {gather*} -\frac {c^{2}}{e^{2} x + d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} -\frac {c^2}{e\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 12, normalized size = 0.80 \begin {gather*} - \frac {c^{2}}{d e + e^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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