Integrand size = 19, antiderivative size = 17 \[ \int f^{a+b x+c x^2} (b+2 c x) \, dx=\frac {f^{a+b x+c x^2}}{\log (f)} \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2268} \[ \int f^{a+b x+c x^2} (b+2 c x) \, dx=\frac {f^{a+b x+c x^2}}{\log (f)} \]
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Rule 2268
Rubi steps \begin{align*} \text {integral}& = \frac {f^{a+b x+c x^2}}{\log (f)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int f^{a+b x+c x^2} (b+2 c x) \, dx=\frac {f^{a+b x+c x^2}}{\log (f)} \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06
method | result | size |
gosper | \(\frac {f^{c \,x^{2}+b x +a}}{\ln \left (f \right )}\) | \(18\) |
derivativedivides | \(\frac {f^{c \,x^{2}+b x +a}}{\ln \left (f \right )}\) | \(18\) |
default | \(\frac {f^{c \,x^{2}+b x +a}}{\ln \left (f \right )}\) | \(18\) |
risch | \(\frac {f^{c \,x^{2}+b x +a}}{\ln \left (f \right )}\) | \(18\) |
parallelrisch | \(\frac {f^{c \,x^{2}+b x +a}}{\ln \left (f \right )}\) | \(18\) |
norman | \(\frac {{\mathrm e}^{\left (c \,x^{2}+b x +a \right ) \ln \left (f \right )}}{\ln \left (f \right )}\) | \(20\) |
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none
Time = 0.34 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int f^{a+b x+c x^2} (b+2 c x) \, dx=\frac {f^{c x^{2} + b x + a}}{\log \left (f\right )} \]
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Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int f^{a+b x+c x^2} (b+2 c x) \, dx=\begin {cases} \frac {f^{a + b x + c x^{2}}}{\log {\left (f \right )}} & \text {for}\: \log {\left (f \right )} \neq 0 \\b x + c x^{2} & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int f^{a+b x+c x^2} (b+2 c x) \, dx=\frac {f^{c x^{2} + b x + a}}{\log \left (f\right )} \]
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none
Time = 0.31 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int f^{a+b x+c x^2} (b+2 c x) \, dx=\frac {f^{c x^{2} + b x + a}}{\log \left (f\right )} \]
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Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int f^{a+b x+c x^2} (b+2 c x) \, dx=\frac {f^{c\,x^2+b\,x+a}}{\ln \left (f\right )} \]
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