Optimal. Leaf size=17 \[ -\frac {1}{3 c^3 e (d+e x)^3} \]
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Rubi [A] time = 0.01, 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} \[ -\frac {1}{3 c^3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac {1}{c^3 (d+e x)^4} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^4} \, dx}{c^3}\\ &=-\frac {1}{3 c^3 e (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {1}{3 c^3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 47, normalized size = 2.76 \[ -\frac {1}{3 \, {\left (c^{3} e^{4} x^{3} + 3 \, c^{3} d e^{3} x^{2} + 3 \, c^{3} d^{2} e^{2} x + c^{3} d^{3} e\right )}} \]
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 \[ \text {Exception raised: NotImplementedError} \]
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 \[ -\frac {1}{3 \left (e x +d \right )^{3} c^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 47, normalized size = 2.76 \[ -\frac {1}{3 \, {\left (c^{3} e^{4} x^{3} + 3 \, c^{3} d e^{3} x^{2} + 3 \, c^{3} d^{2} e^{2} x + c^{3} d^{3} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 49, normalized size = 2.88 \[ -\frac {1}{3\,c^3\,d^3\,e+9\,c^3\,d^2\,e^2\,x+9\,c^3\,d\,e^3\,x^2+3\,c^3\,e^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.31, size = 51, normalized size = 3.00 \[ - \frac {1}{3 c^{3} d^{3} e + 9 c^{3} d^{2} e^{2} x + 9 c^{3} d e^{3} x^{2} + 3 c^{3} e^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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