Optimal. Leaf size=38 \[ \frac {e x}{c d}+\frac {\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 38, 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 {\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac {e x}{c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac {d+e x}{a e+c d x} \, dx\\ &=\int \left (\frac {e}{c d}+\frac {c d^2-a e^2}{c d (a e+c d x)}\right ) \, dx\\ &=\frac {e x}{c d}+\frac {\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 0.92 \begin {gather*} \frac {c d e x+\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 39, normalized size = 1.03
method | result | size |
default | \(\frac {e x}{c d}+\frac {\left (-e^{2} a +c \,d^{2}\right ) \ln \left (c d x +a e \right )}{d^{2} c^{2}}\) | \(39\) |
norman | \(\frac {e x}{c d}-\frac {\left (e^{2} a -c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{2} d^{2}}\) | \(40\) |
risch | \(\frac {e x}{c d}-\frac {\ln \left (c d x +a e \right ) e^{2} a}{c^{2} d^{2}}+\frac {\ln \left (c d x +a e \right )}{c}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 39, normalized size = 1.03 \begin {gather*} \frac {x e}{c d} + \frac {{\left (c d^{2} - a e^{2}\right )} \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 = 2.42, size = 36, normalized size = 0.95 \begin {gather*} \frac {c d x e + {\left (c d^{2} - a e^{2}\right )} \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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Sympy [A]
time = 0.07, size = 32, normalized size = 0.84 \begin {gather*} \frac {e x}{c d} - \frac {\left (a e^{2} - c d^{2}\right ) \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 = 0.71, size = 40, normalized size = 1.05 \begin {gather*} \frac {x e}{c d} + \frac {{\left (c d^{2} - a e^{2}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 39, normalized size = 1.03 \begin {gather*} \frac {e\,x}{c\,d}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (a\,e^2-c\,d^2\right )}{c^2\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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