Optimal. Leaf size=75 \[ \frac {(b+2 c x) f^{b x+c x^2}}{\log (f)}-\frac {\sqrt {\pi } \sqrt {c} f^{-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{\log ^{\frac {3}{2}}(f)} \]
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Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2237, 2234, 2204} \[ \frac {(b+2 c x) f^{b x+c x^2}}{\log (f)}-\frac {\sqrt {\pi } \sqrt {c} f^{-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{\log ^{\frac {3}{2}}(f)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2237
Rubi steps
\begin {align*} \int f^{b x+c x^2} (b+2 c x)^2 \, dx &=\frac {f^{b x+c x^2} (b+2 c x)}{\log (f)}-\frac {(2 c) \int f^{b x+c x^2} \, dx}{\log (f)}\\ &=\frac {f^{b x+c x^2} (b+2 c x)}{\log (f)}-\frac {\left (2 c f^{-\frac {b^2}{4 c}}\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{\log (f)}\\ &=-\frac {\sqrt {c} f^{-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{\log ^{\frac {3}{2}}(f)}+\frac {f^{b x+c x^2} (b+2 c x)}{\log (f)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 84, normalized size = 1.12 \[ \frac {f^{-\frac {b^2}{4 c}} \left (\sqrt {\log (f)} (b+2 c x) f^{\frac {(b+2 c x)^2}{4 c}}-\sqrt {\pi } \sqrt {c} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )\right )}{\log ^{\frac {3}{2}}(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 68, normalized size = 0.91 \[ \frac {{\left (2 \, c x + b\right )} f^{c x^{2} + b x} \log \relax (f) + \frac {\sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{f^{\frac {b^{2}}{4 \, c}}}}{\log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 77, normalized size = 1.03 \[ \frac {c {\left (2 \, x + \frac {b}{c}\right )} e^{\left (c x^{2} \log \relax (f) + b x \log \relax (f)\right )}}{\log \relax (f)} + \frac {\sqrt {\pi } c \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right )}{\sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 90, normalized size = 1.20 \[ \frac {2 c x \,f^{b x} f^{c \,x^{2}}}{\ln \relax (f )}+\frac {b \,f^{b x} f^{c \,x^{2}}}{\ln \relax (f )}+\frac {\sqrt {\pi }\, c \,f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{\sqrt {-c \ln \relax (f )}\, \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.04, size = 329, normalized size = 4.39 \[ \frac {\sqrt {\pi } b^{2} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f)}}\right )}{2 \, \sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{2}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {3}{2}}} - \frac {2 \, c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)}{\left (c \log \relax (f)\right )^{\frac {3}{2}}}\right )} b c}{\sqrt {c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{3}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{3}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}\right )^{\frac {3}{2}} \left (c \log \relax (f)\right )^{\frac {5}{2}}} - \frac {4 \, b c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)^{2}}{\left (c \log \relax (f)\right )^{\frac {5}{2}}}\right )} c^{2}}{2 \, \sqrt {c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 86, normalized size = 1.15 \[ \frac {b\,f^{c\,x^2}\,f^{b\,x}}{\ln \relax (f)}+\frac {2\,c\,f^{c\,x^2}\,f^{b\,x}\,x}{\ln \relax (f)}-\frac {c\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \relax (f)}{2}+c\,x\,\ln \relax (f)}{\sqrt {c\,\ln \relax (f)}}\right )}{f^{\frac {b^2}{4\,c}}\,\ln \relax (f)\,\sqrt {c\,\ln \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{b x + c x^{2}} \left (b + 2 c x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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