Integrand size = 31, antiderivative size = 128 \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^2} \, dx=-\frac {e^{\frac {a}{b n}-\frac {1}{4 b^2 f n^2 \log (F)}} \sqrt {\pi } \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\frac {1}{n}-2 a b f \log (F)-2 b^2 f \log (F) \log \left (c (d+e x)^n\right )}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^2 n (d+e x) \sqrt {\log (F)}} \]
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Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2314, 2308, 2266, 2235} \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^2} \, dx=-\frac {\sqrt {\pi } \left (c (d+e x)^n\right )^{\frac {1}{n}} e^{\frac {a}{b n}-\frac {1}{4 b^2 f n^2 \log (F)}} \text {erfi}\left (\frac {-2 a b f \log (F)-2 b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac {1}{n}}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^2 n \sqrt {\log (F)} (d+e x)} \]
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Rule 2235
Rule 2266
Rule 2308
Rule 2314
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d+e x)^{2-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)}\right ) \int F^{a^2 f+b^2 f \log ^2\left (c (d+e x)^n\right )} (d+e x)^{-2+2 a b f n \log (F)} \, dx}{(d g+e g x)^2} \\ & = \frac {\left ((d+e x) \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {-1+2 a b f n \log (F)}{n}}\right ) \text {Subst}\left (\int \exp \left (a^2 f \log (F)+b^2 f x^2 \log (F)+\frac {x (-1+2 a b f n \log (F))}{n}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n (d g+e g x)^2} \\ & = \frac {\left (e^{\frac {a}{b n}-\frac {1}{4 b^2 f n^2 \log (F)}} (d+e x) \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {-1+2 a b f n \log (F)}{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {\left (2 b^2 f x \log (F)+\frac {-1+2 a b f n \log (F)}{n}\right )^2}{4 b^2 f \log (F)}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n (d g+e g x)^2} \\ & = -\frac {e^{\frac {a}{b n}-\frac {1}{4 b^2 f n^2 \log (F)}} \sqrt {\pi } \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\frac {1}{n}-2 a b f \log (F)-2 b^2 f \log (F) \log \left (c (d+e x)^n\right )}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^2 n (d+e x) \sqrt {\log (F)}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98 \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^2} \, dx=\frac {e^{\frac {-1+4 a b f n \log (F)}{4 b^2 f n^2 \log (F)}} \sqrt {\pi } \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {-1+2 b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 b \sqrt {f} n \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^2 n (d+e x) \sqrt {\log (F)}} \]
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\[\int \frac {F^{f {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}}{\left (e g x +d g \right )^{2}}d x\]
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none
Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.05 \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^2} \, dx=-\frac {\sqrt {\pi } \sqrt {-b^{2} f n^{2} \log \left (F\right )} \operatorname {erf}\left (\frac {{\left (2 \, b^{2} f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 2 \, a b f n \log \left (F\right ) - 1\right )} \sqrt {-b^{2} f n^{2} \log \left (F\right )}}{2 \, b^{2} f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {4 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 4 \, a b f n \log \left (F\right ) - 1}{4 \, b^{2} f n^{2} \log \left (F\right )}\right )}}{2 \, b e g^{2} n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (112) = 224\).
Time = 151.13 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.31 \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^2} \, dx=\begin {cases} - \frac {2 F^{a^{2} f + 2 a b f \log {\left (c \left (d + e x\right )^{n} \right )} + b^{2} f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} a b f n \log {\left (F \right )}}{d e g^{2} + e^{2} g^{2} x} - \frac {2 F^{a^{2} f + 2 a b f \log {\left (c \left (d + e x\right )^{n} \right )} + b^{2} f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} b^{2} f n^{2} \log {\left (F \right )}}{d e g^{2} + e^{2} g^{2} x} - \frac {2 F^{a^{2} f + 2 a b f \log {\left (c \left (d + e x\right )^{n} \right )} + b^{2} f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} b^{2} f n \log {\left (F \right )} \log {\left (c \left (d + e x\right )^{n} \right )}}{d e g^{2} + e^{2} g^{2} x} - \frac {F^{a^{2} f + 2 a b f \log {\left (c \left (d + e x\right )^{n} \right )} + b^{2} f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}}{d e g^{2} + e^{2} g^{2} x} & \text {for}\: e \neq 0 \\\frac {F^{f \left (a + b \log {\left (c d^{n} \right )}\right )^{2}} x}{d^{2} g^{2}} & \text {otherwise} \end {cases} \]
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\[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^2} \, dx=\int { \frac {F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f}}{{\left (e g x + d g\right )}^{2}} \,d x } \]
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\[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^2} \, dx=\int { \frac {F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f}}{{\left (e g x + d g\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^2} \, dx=\int \frac {{\mathrm {e}}^{f\,\ln \left (F\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}}{{\left (d\,g+e\,g\,x\right )}^2} \,d x \]
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