Optimal. Leaf size=48 \[ -\frac {c d^2-a e^2}{c^2 d^2 (a e+c d x)}+\frac {e \log (a e+c d x)}{c^2 d^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45}
\begin {gather*} \frac {e \log (a e+c d x)}{c^2 d^2}-\frac {c d^2-a e^2}{c^2 d^2 (a e+c d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {d+e x}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac {c d^2-a e^2}{c d (a e+c d x)^2}+\frac {e}{c d (a e+c d x)}\right ) \, dx\\ &=-\frac {c d^2-a e^2}{c^2 d^2 (a e+c d x)}+\frac {e \log (a e+c d x)}{c^2 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 0.98 \begin {gather*} \frac {-c d^2+a e^2}{c^2 d^2 (a e+c d x)}+\frac {e \log (a e+c d x)}{c^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 49, normalized size = 1.02
method | result | size |
default | \(-\frac {-e^{2} a +c \,d^{2}}{c^{2} d^{2} \left (c d x +a e \right )}+\frac {e \ln \left (c d x +a e \right )}{d^{2} c^{2}}\) | \(49\) |
risch | \(\frac {e^{2} a}{c^{2} d^{2} \left (c d x +a e \right )}-\frac {1}{c \left (c d x +a e \right )}+\frac {e \ln \left (c d x +a e \right )}{d^{2} c^{2}}\) | \(55\) |
norman | \(\frac {\frac {e^{2} a -c \,d^{2}}{c^{2} d}+\frac {\left (e^{4} a -d^{2} e^{2} c \right ) x}{c^{2} d^{2} e}}{\left (c d x +a e \right ) \left (e x +d \right )}+\frac {e \ln \left (c d x +a e \right )}{d^{2} c^{2}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 54, normalized size = 1.12 \begin {gather*} -\frac {c d^{2} - a e^{2}}{c^{3} d^{3} x + a c^{2} d^{2} e} + \frac {e \log \left (c d x + a e\right )}{c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.77, size = 57, normalized size = 1.19 \begin {gather*} -\frac {c d^{2} - a e^{2} - {\left (c d x e + a e^{2}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3} x + a c^{2} d^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 46, normalized size = 0.96 \begin {gather*} \frac {a e^{2} - c d^{2}}{a c^{2} d^{2} e + c^{3} d^{3} x} + \frac {e \log {\left (a e + c d x \right )}}{c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 51, normalized size = 1.06 \begin {gather*} \frac {e \log \left ({\left | c d x + a e \right |}\right )}{c^{2} d^{2}} - \frac {c d^{2} - a e^{2}}{{\left (c d x + a e\right )} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 47, normalized size = 0.98 \begin {gather*} \frac {a\,e^2-c\,d^2}{c^2\,d^2\,\left (a\,e+c\,d\,x\right )}+\frac {e\,\ln \left (a\,e+c\,d\,x\right )}{c^2\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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