Optimal. Leaf size=17 \[ -\frac {1}{5 c^2 e (d+e x)^5} \]
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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}{5 c^2 e (d+e x)^5} \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 )^2} \, dx &=\int \frac {1}{c^2 (d+e x)^6} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^6} \, dx}{c^2}\\ &=-\frac {1}{5 c^2 e (d+e x)^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{5 c^2 e (d+e x)^5} \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 )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.38, size = 75, normalized size = 4.41 \begin {gather*} -\frac {1}{5 \, {\left (c^{2} e^{6} x^{5} + 5 \, c^{2} d e^{5} x^{4} + 10 \, c^{2} d^{2} e^{4} x^{3} + 10 \, c^{2} d^{3} e^{3} x^{2} + 5 \, c^{2} d^{4} e^{2} x + c^{2} d^{5} 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}{5 \left (e x +d \right )^{5} c^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 75, normalized size = 4.41 \begin {gather*} -\frac {1}{5 \, {\left (c^{2} e^{6} x^{5} + 5 \, c^{2} d e^{5} x^{4} + 10 \, c^{2} d^{2} e^{4} x^{3} + 10 \, c^{2} d^{3} e^{3} x^{2} + 5 \, c^{2} d^{4} e^{2} x + c^{2} d^{5} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 77, normalized size = 4.53 \begin {gather*} -\frac {1}{5\,c^2\,d^5\,e+25\,c^2\,d^4\,e^2\,x+50\,c^2\,d^3\,e^3\,x^2+50\,c^2\,d^2\,e^4\,x^3+25\,c^2\,d\,e^5\,x^4+5\,c^2\,e^6\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.42, size = 82, normalized size = 4.82 \begin {gather*} - \frac {1}{5 c^{2} d^{5} e + 25 c^{2} d^{4} e^{2} x + 50 c^{2} d^{3} e^{3} x^{2} + 50 c^{2} d^{2} e^{4} x^{3} + 25 c^{2} d e^{5} x^{4} + 5 c^{2} e^{6} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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