Optimal. Leaf size=15 \[ -\frac {c}{3 e (d+e x)^3} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {c}{3 e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 24
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*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {c}{3 e (d+e x)^3} \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 {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 36, normalized size = 2.40 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 34, normalized size = 2.27 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 14, normalized size = 0.93 \begin {gather*} -\frac {c}{3 \left (e x +d \right )^{3} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.33, size = 36, normalized size = 2.40 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 35, normalized size = 2.33 \begin {gather*} -\frac {c}{3\,e\,\left (d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.28, size = 37, normalized size = 2.47 \begin {gather*} - \frac {c}{3 d^{3} e + 9 d^{2} e^{2} x + 9 d e^{3} x^{2} + 3 e^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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