Optimal. Leaf size=38 \[ \frac {\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac {e x}{c d} \]
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Rubi [A] time = 0.03, 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 = {626, 43} \[ \frac {\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac {e x}{c d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
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 \[ \frac {\left (c d^2-a e^2\right ) \log (a e+c d x)+c d e x}{c^2 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 35, normalized size = 0.92 \[ \frac {c d e x + {\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 159, normalized size = 4.18 \[ \frac {x e}{c d} + \frac {{\left (c d^{2} - a e^{2}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{2} d^{2}} + \frac {{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 45, normalized size = 1.18 \[ -\frac {a \,e^{2} \ln \left (c d x +a e \right )}{c^{2} d^{2}}+\frac {e x}{c d}+\frac {\ln \left (c d x +a e \right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 38, normalized size = 1.00 \[ \frac {e x}{c d} + \frac {{\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \]
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 \[ \frac {e\,x}{c\,d}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (a\,e^2-c\,d^2\right )}{c^2\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 32, normalized size = 0.84 \[ \frac {e x}{c d} - \frac {\left (a e^{2} - c d^{2}\right ) \log {\left (a e + c d x \right )}}{c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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