Optimal. Leaf size=45 \[ -\frac {(c d-b e) x}{e^2}+\frac {c x^2}{2 e}+\frac {d (c d-b e) \log (d+e x)}{e^3} \]
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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 = {712}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
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 \begin {gather*} \frac {e x (-2 c d+2 b e+c e x)+2 d (c d-b e) \log (d+e x)}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 43, normalized size = 0.96
method | result | size |
default | \(\frac {\frac {1}{2} c e \,x^{2}+b e x -c d x}{e^{2}}-\frac {d \left (b e -c d \right ) \ln \left (e x +d \right )}{e^{3}}\) | \(43\) |
norman | \(\frac {\left (b e -c d \right ) x}{e^{2}}+\frac {c \,x^{2}}{2 e}-\frac {d \left (b e -c d \right ) \ln \left (e x +d \right )}{e^{3}}\) | \(44\) |
risch | \(\frac {c \,x^{2}}{2 e}+\frac {b x}{e}-\frac {c d x}{e^{2}}-\frac {d \ln \left (e x +d \right ) b}{e^{2}}+\frac {d^{2} \ln \left (e x +d \right ) c}{e^{3}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 47, normalized size = 1.04 \begin {gather*} {\left (c d^{2} - b d e\right )} e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {1}{2} \, {\left (c x^{2} e - 2 \, {\left (c d - b e\right )} x\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.07, size = 47, normalized size = 1.04 \begin {gather*} -\frac {1}{2} \, {\left (2 \, c d x e - {\left (c x^{2} + 2 \, b x\right )} e^{2} - 2 \, {\left (c d^{2} - b d e\right )} \log \left (x e + d\right )\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 37, normalized size = 0.82 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 47, normalized size = 1.04 \begin {gather*} {\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 )} \end {gather*}
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 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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