Optimal. Leaf size=33 \[ -\frac {a-\frac {c d^2}{e^2}}{d+e x}+\frac {c d \log (d+e x)}{e^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {24, 45}
\begin {gather*} \frac {c d \log (d+e x)}{e^2}-\frac {a-\frac {c d^2}{e^2}}{d+e x} \end {gather*}
Antiderivative was successfully verified.
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Rule 24
Rule 45
Rubi steps
\begin {align*} \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx &=\frac {\int \frac {a e^3+c d e^2 x}{(d+e x)^2} \, dx}{e^2}\\ &=\frac {\int \left (\frac {-c d^2 e+a e^3}{(d+e x)^2}+\frac {c d e}{d+e x}\right ) \, dx}{e^2}\\ &=-\frac {a-\frac {c d^2}{e^2}}{d+e x}+\frac {c d \log (d+e x)}{e^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 1.09 \begin {gather*} \frac {c d^2-a e^2}{e^2 (d+e x)}+\frac {c d \log (d+e x)}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 38, normalized size = 1.15
method | result | size |
default | \(-\frac {e^{2} a -c \,d^{2}}{e^{2} \left (e x +d \right )}+\frac {c d \ln \left (e x +d \right )}{e^{2}}\) | \(38\) |
risch | \(-\frac {a}{e x +d}+\frac {c \,d^{2}}{e^{2} \left (e x +d \right )}+\frac {c d \ln \left (e x +d \right )}{e^{2}}\) | \(39\) |
norman | \(\frac {\frac {\left (a d \,e^{2}-c \,d^{3}\right ) x^{2}}{d^{2}}+\frac {\left (a d \,e^{2}-c \,d^{3}\right ) x}{d e}}{\left (e x +d \right )^{2}}+\frac {c d \ln \left (e x +d \right )}{e^{2}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 36, normalized size = 1.09 \begin {gather*} c d e^{\left (-2\right )} \log \left (x e + d\right ) + \frac {c d^{2} - a e^{2}}{x e^{3} + d e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.09, size = 43, normalized size = 1.30 \begin {gather*} \frac {c d^{2} - a e^{2} + {\left (c d x e + c d^{2}\right )} \log \left (x e + d\right )}{x e^{3} + d e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 32, normalized size = 0.97 \begin {gather*} \frac {c d \log {\left (d + e x \right )}}{e^{2}} + \frac {- a e^{2} + c d^{2}}{d e^{2} + e^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 36, normalized size = 1.09 \begin {gather*} c d e^{\left (-2\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (c d^{2} - a e^{2}\right )} e^{\left (-2\right )}}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.58, size = 37, normalized size = 1.12 \begin {gather*} \frac {c\,d\,\ln \left (d+e\,x\right )}{e^2}-\frac {a\,e^2-c\,d^2}{e^2\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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