Optimal. Leaf size=17 \[ -\frac {1}{3 c e (d+e x)^3} \]
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Rubi [A] time = 0.00, antiderivative size = 17, 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 {1}{3 c e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx &=\int \frac {1}{c (d+e x)^4} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^4} \, dx}{c}\\ &=-\frac {1}{3 c e (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{3 c 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 {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.37, size = 39, normalized size = 2.29 \begin {gather*} -\frac {1}{3 \, {\left (c e^{4} x^{3} + 3 \, c d e^{3} x^{2} + 3 \, c d^{2} e^{2} x + c d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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 = 0.94 \begin {gather*} -\frac {1}{3 \left (e x +d \right )^{3} c e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.35, size = 39, normalized size = 2.29 \begin {gather*} -\frac {1}{3 \, {\left (c e^{4} x^{3} + 3 \, c d e^{3} x^{2} + 3 \, c d^{2} e^{2} x + c d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 41, normalized size = 2.41 \begin {gather*} -\frac {1}{3\,c\,d^3\,e+9\,c\,d^2\,e^2\,x+9\,c\,d\,e^3\,x^2+3\,c\,e^4\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.27, size = 44, normalized size = 2.59 \begin {gather*} - \frac {1}{3 c d^{3} e + 9 c d^{2} e^{2} x + 9 c d e^{3} x^{2} + 3 c e^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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