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 = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 2236
Rule 2237
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*}
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Mathematica [A] time = 0.10, 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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fricas [A] time = 0.41, size = 40, normalized size = 0.93 \[ \frac {{\left ({\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \relax (f) - 4 \, c\right )} f^{c x^{2} + b x}}{\log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.53, size = 726, normalized size = 16.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 1.02 \[ \frac {\left (4 c^{2} x^{2} \ln \relax (f )+4 b c x \ln \relax (f )+b^{2} \ln \relax (f )-4 c \right ) f^{c \,x^{2}+b x}}{\ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.17, size = 536, normalized size = 12.47 \[ \frac {\sqrt {\pi } b^{3} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f)}}\right )}{2 \, \sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} - \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{2}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {3}{2}}} - \frac {2 \, c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)}{\left (c \log \relax (f)\right )^{\frac {3}{2}}}\right )} b^{2} c}{2 \, \sqrt {c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} + \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{3}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{3}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}\right )^{\frac {3}{2}} \left (c \log \relax (f)\right )^{\frac {5}{2}}} - \frac {4 \, b c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)^{2}}{\left (c \log \relax (f)\right )^{\frac {5}{2}}}\right )} b c^{2}}{2 \, \sqrt {c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{3} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{4}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {7}{2}}} - \frac {12 \, {\left (2 \, c x + b\right )}^{3} b \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{4}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}\right )^{\frac {3}{2}} \left (c \log \relax (f)\right )^{\frac {7}{2}}} - \frac {6 \, b^{2} c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)^{3}}{\left (c \log \relax (f)\right )^{\frac {7}{2}}} + \frac {8 \, c^{2} \Gamma \left (2, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{2}}{\left (c \log \relax (f)\right )^{\frac {7}{2}}}\right )} c^{3}}{2 \, \sqrt {c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 43, normalized size = 1.00 \[ \frac {f^{c\,x^2+b\,x}\,\left (\ln \relax (f)\,b^2+4\,\ln \relax (f)\,b\,c\,x+4\,\ln \relax (f)\,c^2\,x^2-4\,c\right )}{{\ln \relax (f)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 83, normalized size = 1.93 \[ \begin {cases} \frac {f^{b x + c x^{2}} \left (b^{2} \log {\relax (f )} + 4 b c x \log {\relax (f )} + 4 c^{2} x^{2} \log {\relax (f )} - 4 c\right )}{\log {\relax (f )}^{2}} & \text {for}\: \log {\relax (f )}^{2} \neq 0 \\b^{3} x + 3 b^{2} c x^{2} + 4 b c^{2} x^{3} + 2 c^{3} x^{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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