Optimal. Leaf size=45 \[ \frac {d (c d-b e) \log (d+e x)}{e^3}-\frac {x (c d-b e)}{e^2}+\frac {c x^2}{2 e} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {698} \[ -\frac {x (c d-b e)}{e^2}+\frac {d (c d-b e) \log (d+e x)}{e^3}+\frac {c x^2}{2 e} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {b x+c x^2}{d+e x} \, dx &=\int \left (\frac {-c d+b e}{e^2}+\frac {c x}{e}+\frac {d (c d-b e)}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {(c d-b e) x}{e^2}+\frac {c x^2}{2 e}+\frac {d (c d-b e) \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.91 \[ \frac {e x (2 b e-2 c d+c e x)+2 d (c d-b e) \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 47, normalized size = 1.04 \[ \frac {c e^{2} x^{2} - 2 \, {\left (c d e - b e^{2}\right )} x + 2 \, {\left (c d^{2} - b d e\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 47, normalized size = 1.04 \[ {\left (c d^{2} - b d e\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (c x^{2} e - 2 \, c d x + 2 \, b x e\right )} e^{\left (-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 52, normalized size = 1.16 \[ \frac {c \,x^{2}}{2 e}-\frac {b d \ln \left (e x +d \right )}{e^{2}}+\frac {b x}{e}+\frac {c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {c d x}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 45, normalized size = 1.00 \[ \frac {c e x^{2} - 2 \, {\left (c d - b e\right )} x}{2 \, e^{2}} + \frac {{\left (c d^{2} - b d e\right )} \log \left (e x + d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 46, normalized size = 1.02 \[ x\,\left (\frac {b}{e}-\frac {c\,d}{e^2}\right )+\frac {c\,x^2}{2\,e}+\frac {\ln \left (d+e\,x\right )\,\left (c\,d^2-b\,d\,e\right )}{e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 37, normalized size = 0.82 \[ \frac {c x^{2}}{2 e} - \frac {d \left (b e - c d\right ) \log {\left (d + e x \right )}}{e^{3}} + x \left (\frac {b}{e} - \frac {c d}{e^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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