Optimal. Leaf size=43 \[ \frac{(b+2 c x)^2 f^{b x+c x^2}}{\log (f)}-\frac{4 c f^{b x+c x^2}}{\log ^2(f)} \]
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Rubi [A] time = 0.0433466, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2237, 2236} \[ \frac{(b+2 c x)^2 f^{b x+c x^2}}{\log (f)}-\frac{4 c f^{b x+c x^2}}{\log ^2(f)} \]
Antiderivative was successfully verified.
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Rule 2237
Rule 2236
Rubi steps
\begin{align*} \int f^{b x+c x^2} (b+2 c x)^3 \, dx &=\frac{f^{b x+c x^2} (b+2 c x)^2}{\log (f)}-\frac{(4 c) \int f^{b x+c x^2} (b+2 c x) \, dx}{\log (f)}\\ &=-\frac{4 c f^{b x+c x^2}}{\log ^2(f)}+\frac{f^{b x+c x^2} (b+2 c x)^2}{\log (f)}\\ \end{align*}
Mathematica [A] time = 0.0946798, size = 29, normalized size = 0.67 \[ \frac{f^{x (b+c x)} \left (\log (f) (b+2 c x)^2-4 c\right )}{\log ^2(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 44, normalized size = 1. \begin{align*}{\frac{ \left ( 4\,\ln \left ( f \right ){c}^{2}{x}^{2}+4\,bcx\ln \left ( f \right ) +\ln \left ( f \right ){b}^{2}-4\,c \right ){f}^{c{x}^{2}+bx}}{ \left ( \ln \left ( f \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.34087, size = 724, normalized size = 16.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52657, size = 93, normalized size = 2.16 \begin{align*} \frac{{\left ({\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (f\right ) - 4 \, c\right )} f^{c x^{2} + b x}}{\log \left (f\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.149724, size = 83, normalized size = 1.93 \begin{align*} \begin{cases} \frac{f^{b x + c x^{2}} \left (b^{2} \log{\left (f \right )} + 4 b c x \log{\left (f \right )} + 4 c^{2} x^{2} \log{\left (f \right )} - 4 c\right )}{\log{\left (f \right )}^{2}} & \text{for}\: \log{\left (f \right )}^{2} \neq 0 \\b^{3} x + 3 b^{2} c x^{2} + 4 b c^{2} x^{3} + 2 c^{3} x^{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31869, size = 54, normalized size = 1.26 \begin{align*} \frac{{\left (c^{2}{\left (2 \, x + \frac{b}{c}\right )}^{2} \log \left (f\right ) - 4 \, c\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right )\right )}}{\log \left (f\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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