Optimal. Leaf size=17 \[ -\frac {1}{4 c^2 e (d+e x)^4} \]
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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} \[ -\frac {1}{4 c^2 e (d+e x)^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac {1}{c^2 (d+e x)^5} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^5} \, dx}{c^2}\\ &=-\frac {1}{4 c^2 e (d+e x)^4}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {1}{4 c^2 e (d+e x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.08, size = 61, normalized size = 3.59 \[ -\frac {1}{4 \, {\left (c^{2} e^{5} x^{4} + 4 \, c^{2} d e^{4} x^{3} + 6 \, c^{2} d^{2} e^{3} x^{2} + 4 \, c^{2} d^{3} e^{2} x + c^{2} d^{4} e\right )}} \]
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 = 0.21, size = 16, normalized size = 0.94 \[ -\frac {1}{4 \left (e x +d \right )^{4} c^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.37, size = 61, normalized size = 3.59 \[ -\frac {1}{4 \, {\left (c^{2} e^{5} x^{4} + 4 \, c^{2} d e^{4} x^{3} + 6 \, c^{2} d^{2} e^{3} x^{2} + 4 \, c^{2} d^{3} e^{2} x + c^{2} d^{4} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 63, normalized size = 3.71 \[ -\frac {1}{4\,c^2\,d^4\,e+16\,c^2\,d^3\,e^2\,x+24\,c^2\,d^2\,e^3\,x^2+16\,c^2\,d\,e^4\,x^3+4\,c^2\,e^5\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.34, size = 66, normalized size = 3.88 \[ - \frac {1}{4 c^{2} d^{4} e + 16 c^{2} d^{3} e^{2} x + 24 c^{2} d^{2} e^{3} x^{2} + 16 c^{2} d e^{4} x^{3} + 4 c^{2} e^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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