Optimal. Leaf size=33 \[ \frac {c d \log (d+e x)}{e^2}-\frac {a-\frac {c d^2}{e^2}}{d+e x} \]
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Rubi [A] time = 0.03, 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, 43} \[ \frac {c d \log (d+e x)}{e^2}-\frac {a-\frac {c d^2}{e^2}}{d+e x} \]
Antiderivative was successfully verified.
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Rule 24
Rule 43
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 \[ \frac {c d^2-a e^2}{e^2 (d+e x)}+\frac {c d \log (d+e x)}{e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 44, normalized size = 1.33 \[ \frac {c d^{2} - a e^{2} + {\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{e^{3} x + d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 52, normalized size = 1.58 \[ c d e^{\left (-2\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 1.18 \[ -\frac {a}{e x +d}+\frac {c \,d^{2}}{\left (e x +d \right ) e^{2}}+\frac {c d \ln \left (e x +d \right )}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 39, normalized size = 1.18 \[ \frac {c d \log \left (e x + d\right )}{e^{2}} + \frac {c d^{2} - a e^{2}}{e^{3} x + d e^{2}} \]
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 \[ \frac {c\,d\,\ln \left (d+e\,x\right )}{e^2}-\frac {a\,e^2-c\,d^2}{e^2\,\left (d+e\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 32, normalized size = 0.97 \[ \frac {c d \log {\left (d + e x \right )}}{e^{2}} + \frac {- a e^{2} + c d^{2}}{d e^{2} + e^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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