Optimal. Leaf size=68 \[ \frac {f^{c (a+b x)^2}}{2 b^2 c \log (f)}-\frac {\sqrt {\pi } a \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b^2 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2226, 2204, 2209} \[ \frac {f^{c (a+b x)^2}}{2 b^2 c \log (f)}-\frac {\sqrt {\pi } a \text {Erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b^2 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2209
Rule 2226
Rubi steps
\begin {align*} \int f^{c (a+b x)^2} x \, dx &=\int \left (-\frac {a f^{c (a+b x)^2}}{b}+\frac {f^{c (a+b x)^2} (a+b x)}{b}\right ) \, dx\\ &=\frac {\int f^{c (a+b x)^2} (a+b x) \, dx}{b}-\frac {a \int f^{c (a+b x)^2} \, dx}{b}\\ &=\frac {f^{c (a+b x)^2}}{2 b^2 c \log (f)}-\frac {a \sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b^2 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 0.93 \[ \frac {f^{c (a+b x)^2}-\sqrt {\pi } a \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b^2 c \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 72, normalized size = 1.06 \[ \frac {\sqrt {\pi } \sqrt {-b^{2} c \log \relax (f)} a \operatorname {erf}\left (\frac {\sqrt {-b^{2} c \log \relax (f)} {\left (b x + a\right )}}{b}\right ) + b f^{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{2 \, b^{3} c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 77, normalized size = 1.13 \[ \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-\sqrt {-c \log \relax (f)} b {\left (x + \frac {a}{b}\right )}\right )}{\sqrt {-c \log \relax (f)} b} + \frac {e^{\left (b^{2} c x^{2} \log \relax (f) + 2 \, a b c x \log \relax (f) + a^{2} c \log \relax (f)\right )}}{b c \log \relax (f)}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 80, normalized size = 1.18 \[ \frac {\sqrt {\pi }\, a \erf \left (\frac {a c \ln \relax (f )}{\sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, b x \right )}{2 \sqrt {-c \ln \relax (f )}\, b^{2}}+\frac {f^{a^{2} c} f^{b^{2} c \,x^{2}} f^{2 a b c x}}{2 b^{2} c \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.27, size = 131, normalized size = 1.93 \[ -\frac {\frac {\sqrt {\pi } {\left (b^{2} c x + a b c\right )} a c {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b^{2} c x + a b c\right )}^{2} \log \relax (f)}{b^{2} c}}\right ) - 1\right )} \log \relax (f)^{2}}{\left (c \log \relax (f)\right )^{\frac {3}{2}} b^{2} \sqrt {-\frac {{\left (b^{2} c x + a b c\right )}^{2} \log \relax (f)}{b^{2} c}}} - \frac {c f^{\frac {{\left (b^{2} c x + a b c\right )}^{2}}{b^{2} c}} \log \relax (f)}{\left (c \log \relax (f)\right )^{\frac {3}{2}} b}}{2 \, \sqrt {c \log \relax (f)} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.48, size = 66, normalized size = 0.97 \[ \frac {f^{b^2\,c\,x^2}\,f^{a^2\,c}\,f^{2\,a\,b\,c\,x}}{2\,b^2\,c\,\ln \relax (f)}-\frac {a\,\sqrt {\pi }\,\mathrm {erfi}\left (\sqrt {c\,\ln \relax (f)}\,\left (a+b\,x\right )\right )}{2\,b^2\,\sqrt {c\,\ln \relax (f)}} \]
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 \[ \int f^{c \left (a + b x\right )^{2}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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