Integrand size = 15, antiderivative size = 140 \[ \int f^{c (a+b x)^2} x^2 \, dx=-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{4 b^3 c^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {a f^{c (a+b x)^2}}{b^3 c \log (f)}+\frac {f^{c (a+b x)^2} (a+b x)}{2 b^3 c \log (f)}+\frac {a^2 \sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b^3 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2258, 2235, 2240, 2243} \[ \int f^{c (a+b x)^2} x^2 \, dx=\frac {\sqrt {\pi } a^2 \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b^3 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{4 b^3 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {(a+b x) f^{c (a+b x)^2}}{2 b^3 c \log (f)}-\frac {a f^{c (a+b x)^2}}{b^3 c \log (f)} \]
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Rule 2235
Rule 2240
Rule 2243
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 f^{c (a+b x)^2}}{b^2}-\frac {2 a f^{c (a+b x)^2} (a+b x)}{b^2}+\frac {f^{c (a+b x)^2} (a+b x)^2}{b^2}\right ) \, dx \\ & = \frac {\int f^{c (a+b x)^2} (a+b x)^2 \, dx}{b^2}-\frac {(2 a) \int f^{c (a+b x)^2} (a+b x) \, dx}{b^2}+\frac {a^2 \int f^{c (a+b x)^2} \, dx}{b^2} \\ & = -\frac {a f^{c (a+b x)^2}}{b^3 c \log (f)}+\frac {f^{c (a+b x)^2} (a+b x)}{2 b^3 c \log (f)}+\frac {a^2 \sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b^3 \sqrt {c} \sqrt {\log (f)}}-\frac {\int f^{c (a+b x)^2} \, dx}{2 b^2 c \log (f)} \\ & = -\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{4 b^3 c^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {a f^{c (a+b x)^2}}{b^3 c \log (f)}+\frac {f^{c (a+b x)^2} (a+b x)}{2 b^3 c \log (f)}+\frac {a^2 \sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b^3 \sqrt {c} \sqrt {\log (f)}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.59 \[ \int f^{c (a+b x)^2} x^2 \, dx=\frac {-2 \sqrt {c} f^{c (a+b x)^2} (a-b x) \sqrt {\log (f)}+\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right ) \left (-1+2 a^2 c \log (f)\right )}{4 b^3 c^{3/2} \log ^{\frac {3}{2}}(f)} \]
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Time = 0.03 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.20
method | result | size |
risch | \(\frac {x \,f^{b^{2} c \,x^{2}} f^{2 a b c x} f^{a^{2} c}}{2 b^{2} c \ln \left (f \right )}-\frac {a \,f^{b^{2} c \,x^{2}} f^{2 a b c x} f^{a^{2} c}}{2 b^{3} c \ln \left (f \right )}-\frac {a^{2} \sqrt {\pi }\, \operatorname {erf}\left (-b \sqrt {-c \ln \left (f \right )}\, x +\frac {a c \ln \left (f \right )}{\sqrt {-c \ln \left (f \right )}}\right )}{2 b^{3} \sqrt {-c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (-b \sqrt {-c \ln \left (f \right )}\, x +\frac {a c \ln \left (f \right )}{\sqrt {-c \ln \left (f \right )}}\right )}{4 b^{3} c \ln \left (f \right ) \sqrt {-c \ln \left (f \right )}}\) | \(168\) |
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Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.68 \[ \int f^{c (a+b x)^2} x^2 \, dx=-\frac {\sqrt {\pi } {\left (2 \, a^{2} c \log \left (f\right ) - 1\right )} \sqrt {-b^{2} c \log \left (f\right )} \operatorname {erf}\left (\frac {\sqrt {-b^{2} c \log \left (f\right )} {\left (b x + a\right )}}{b}\right ) - 2 \, {\left (b^{2} c x - a b c\right )} f^{b^{2} c x^{2} + 2 \, a b c x + a^{2} c} \log \left (f\right )}{4 \, b^{4} c^{2} \log \left (f\right )^{2}} \]
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\[ \int f^{c (a+b x)^2} x^2 \, dx=\int f^{c \left (a + b x\right )^{2}} x^{2}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.56 \[ \int f^{c (a+b x)^2} x^2 \, dx=\frac {\frac {\sqrt {\pi } {\left (b^{2} c x + a b c\right )} a^{2} c^{2} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}}\right ) - 1\right )} \log \left (f\right )^{3}}{\left (c \log \left (f\right )\right )^{\frac {5}{2}} b^{3} \sqrt {-\frac {{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}}} - \frac {2 \, a c^{2} f^{\frac {{\left (b^{2} c x + a b c\right )}^{2}}{b^{2} c}} \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac {5}{2}} b^{2}} - \frac {{\left (b^{2} c x + a b c\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}\right ) \log \left (f\right )^{3}}{\left (c \log \left (f\right )\right )^{\frac {5}{2}} b^{5} \left (-\frac {{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}\right )^{\frac {3}{2}}}}{2 \, \sqrt {c \log \left (f\right )} b} \]
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Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.76 \[ \int f^{c (a+b x)^2} x^2 \, dx=-\frac {\frac {\sqrt {\pi } {\left (2 \, a^{2} c \log \left (f\right ) - 1\right )} \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} b {\left (x + \frac {a}{b}\right )}\right )}{\sqrt {-c \log \left (f\right )} b c \log \left (f\right )} - \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (b^{2} c x^{2} \log \left (f\right ) + 2 \, a b c x \log \left (f\right ) + a^{2} c \log \left (f\right )\right )}}{b c \log \left (f\right )}}{4 \, b^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int f^{c (a+b x)^2} x^2 \, dx=\frac {\sqrt {\pi }\,\mathrm {erfi}\left (\sqrt {c\,\ln \left (f\right )}\,\left (a+b\,x\right )\right )\,\left (\frac {a^2}{b^3}-\frac {1}{2\,b^3\,c\,\ln \left (f\right )}\right )}{2\,\sqrt {c\,\ln \left (f\right )}}-\frac {a\,f^{b^2\,c\,x^2}\,f^{a^2\,c}\,f^{2\,a\,b\,c\,x}}{2\,b^3\,c\,\ln \left (f\right )}+\frac {f^{b^2\,c\,x^2}\,f^{a^2\,c}\,f^{2\,a\,b\,c\,x}\,x}{2\,b^2\,c\,\ln \left (f\right )} \]
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