Optimal. Leaf size=15 \[ -\frac {1}{c^3 e (d+e x)} \]
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Rubi [A] time = 0.00, antiderivative size = 15, 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}{c^3 e (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\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)^2} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^2} \, dx}{c^3}\\ &=-\frac {1}{c^3 e (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{c^3 e (d+e x)} \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 {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 19, normalized size = 1.27 \begin {gather*} -\frac {1}{c^{3} e^{2} x + c^{3} d e} \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 = 1.07 \begin {gather*} -\frac {1}{\left (e x +d \right ) c^{3} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 19, normalized size = 1.27 \begin {gather*} -\frac {1}{c^{3} e^{2} x + c^{3} d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 19, normalized size = 1.27 \begin {gather*} -\frac {1}{x\,c^3\,e^2+d\,c^3\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 17, normalized size = 1.13 \begin {gather*} - \frac {1}{c^{3} d e + c^{3} e^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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