Optimal. Leaf size=41 \[ \frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2204} \[ \frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rubi steps
\begin {align*} \int f^{c (a+b x)^2} \, dx &=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 41, normalized size = 1.00 \[ \frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 45, normalized size = 1.10 \[ -\frac {\sqrt {\pi } \sqrt {-b^{2} c \log \relax (f)} \operatorname {erf}\left (\frac {\sqrt {-b^{2} c \log \relax (f)} {\left (b x + a\right )}}{b}\right )}{2 \, b^{2} c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 33, normalized size = 0.80 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f)} b {\left (x + \frac {a}{b}\right )}\right )}{2 \, \sqrt {-c \log \relax (f)} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 41, normalized size = 1.00 \[ -\frac {\sqrt {\pi }\, \erf \left (\frac {a c \ln \relax (f )}{\sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, b x \right )}{2 \sqrt {-c \ln \relax (f )}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 40, normalized size = 0.98 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-c \log \relax (f)} b x - \frac {a c \log \relax (f)}{\sqrt {-c \log \relax (f)}}\right )}{2 \, \sqrt {-c \log \relax (f)} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 45, normalized size = 1.10 \[ -\frac {\sqrt {\pi }\,\mathrm {erf}\left (\frac {1{}\mathrm {i}\,c\,x\,\ln \relax (f)\,b^2+1{}\mathrm {i}\,a\,c\,\ln \relax (f)\,b}{\sqrt {b^2\,c\,\ln \relax (f)}}\right )\,1{}\mathrm {i}}{2\,\sqrt {b^2\,c\,\ln \relax (f)}} \]
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 \[ \int f^{c \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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