Integrand size = 28, antiderivative size = 535 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^3 \, dx=\frac {3 e^{-\frac {1+2 a b f n \log (F)}{b^2 f n^2 \log (F)}} h (e g-d h)^2 \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\frac {1}{n}+a b f \log (F)+b^2 f \log (F) \log \left (c (d+e x)^n\right )}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {4 (1+a b f n \log (F))}{b^2 f n^2 \log (F)}} h^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {\frac {2}{n}+a b f \log (F)+b^2 f \log (F) \log \left (c (d+e x)^n\right )}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {1+4 a b f n \log (F)}{4 b^2 f n^2 \log (F)}} (e g-d h)^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\frac {1}{n}+2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {3 e^{-\frac {3 (3+4 a b f n \log (F))}{4 b^2 f n^2 \log (F)}} h^2 (e g-d h) \sqrt {\pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\frac {3}{n}+2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}} \]
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Time = 0.72 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2315, 2312, 2308, 2266, 2235, 2314} \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^3 \, dx=\frac {3 \sqrt {\pi } h^2 (d+e x)^3 (e g-d h) \left (c (d+e x)^n\right )^{-3/n} \exp \left (-\frac {3 (4 a b f n \log (F)+3)}{4 b^2 f n^2 \log (F)}\right ) \text {erfi}\left (\frac {2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac {3}{n}}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {3 \sqrt {\pi } h (d+e x)^2 (e g-d h)^2 \left (c (d+e x)^n\right )^{-2/n} e^{-\frac {2 a b f n \log (F)+1}{b^2 f n^2 \log (F)}} \text {erfi}\left (\frac {a b f \log (F)+b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac {1}{n}}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } (d+e x) (e g-d h)^3 \left (c (d+e x)^n\right )^{-1/n} e^{-\frac {4 a b f n \log (F)+1}{4 b^2 f n^2 \log (F)}} \text {erfi}\left (\frac {2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac {1}{n}}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } h^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} e^{-\frac {4 (a b f n \log (F)+1)}{b^2 f n^2 \log (F)}} \text {erfi}\left (\frac {a b f \log (F)+b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac {2}{n}}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}} \]
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Rule 2235
Rule 2266
Rule 2308
Rule 2312
Rule 2314
Rule 2315
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (e^3 F^{f \left (a+b \log \left (c x^n\right )\right )^2} g^3 \left (1-\frac {d h \left (3 e^2 g^2-3 d e g h+d^2 h^2\right )}{e^3 g^3}\right )+3 e^2 F^{f \left (a+b \log \left (c x^n\right )\right )^2} g^2 h \left (1+\frac {d h (-2 e g+d h)}{e^2 g^2}\right ) x+3 e F^{f \left (a+b \log \left (c x^n\right )\right )^2} g h^2 \left (1-\frac {d h}{e g}\right ) x^2+F^{f \left (a+b \log \left (c x^n\right )\right )^2} h^3 x^3\right ) \, dx,x,d+e x\right )}{e^4} \\ & = \frac {h^3 \text {Subst}\left (\int F^{f \left (a+b \log \left (c x^n\right )\right )^2} x^3 \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 h^2 (e g-d h)\right ) \text {Subst}\left (\int F^{f \left (a+b \log \left (c x^n\right )\right )^2} x^2 \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 h (e g-d h)^2\right ) \text {Subst}\left (\int F^{f \left (a+b \log \left (c x^n\right )\right )^2} x \, dx,x,d+e x\right )}{e^4}+\frac {(e g-d h)^3 \text {Subst}\left (\int F^{f \left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{e^4} \\ & = \frac {\left (h^3 (d+e x)^{-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)}\right ) \text {Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^{3+2 a b f n \log (F)} \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 h^2 (e g-d h) (d+e x)^{-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)}\right ) \text {Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^{2+2 a b f n \log (F)} \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 h (e g-d h)^2 (d+e x)^{-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)}\right ) \text {Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^{1+2 a b f n \log (F)} \, dx,x,d+e x\right )}{e^4}+\frac {\left ((e g-d h)^3 (d+e x)^{-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)}\right ) \text {Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^{2 a b f n \log (F)} \, dx,x,d+e x\right )}{e^4} \\ & = \frac {\left ((e g-d h)^3 (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^4 n}+\frac {\left (3 h (e g-d h)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {2+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 (2+2 a b f n \log (F))}{n}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac {\left (3 h^2 (e g-d h) (d+e x)^3 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {3+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 (3+2 a b f n \log (F))}{n}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac {\left (h^3 (d+e x)^4 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {4+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 (4+2 a b f n \log (F))}{n}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n} \\ & = \frac {\left (\exp \left (a^2 f \log (F)-\frac {(1+2 a b f n \log (F))^2}{4 b^2 f n^2 \log (F)}\right ) (e g-d h)^3 (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^4 n}+\frac {\left (3 \exp \left (a^2 f \log (F)-\frac {(2+2 a b f n \log (F))^2}{4 b^2 f n^2 \log (F)}\right ) h (e g-d h)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {2+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 {2+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^4 n}+\frac {\left (3 \exp \left (a^2 f \log (F)-\frac {(3+2 a b f n \log (F))^2}{4 b^2 f n^2 \log (F)}\right ) h^2 (e g-d h) (d+e x)^3 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {3+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 {3+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^4 n}+\frac {\left (\exp \left (a^2 f \log (F)-\frac {(4+2 a b f n \log (F))^2}{4 b^2 f n^2 \log (F)}\right ) h^3 (d+e x)^4 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {4+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 {4+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^4 n} \\ & = \frac {3 e^{-\frac {1+2 a b f n \log (F)}{b^2 f n^2 \log (F)}} h (e g-d h)^2 \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\frac {1}{n}+a b f \log (F)+b^2 f \log (F) \log \left (c (d+e x)^n\right )}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {4 (1+a b f n \log (F))}{b^2 f n^2 \log (F)}} h^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {\frac {2}{n}+a b f \log (F)+b^2 f \log (F) \log \left (c (d+e x)^n\right )}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {1+4 a b f n \log (F)}{4 b^2 f n^2 \log (F)}} (e g-d h)^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-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^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {3 \exp \left (-\frac {3 (3+4 a b f n \log (F))}{4 b^2 f n^2 \log (F)}\right ) h^2 (e g-d h) \sqrt {\pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\frac {3}{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^4 \sqrt {f} n \sqrt {\log (F)}} \\ \end{align*}
Time = 2.13 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.81 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^3 \, dx=\frac {e^{-\frac {4 (1+a b f n \log (F))}{b^2 f n^2 \log (F)}} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-4/n} \left (3 e^{\frac {3+2 a b f n \log (F)}{b^2 f n^2 \log (F)}} h (e g-d h)^2 (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {1+b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )}{b \sqrt {f} n \sqrt {\log (F)}}\right )+h^3 (d+e x)^3 \text {erfi}\left (\frac {2+b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )}{b \sqrt {f} n \sqrt {\log (F)}}\right )+e^{\frac {7+4 a b f n \log (F)}{4 b^2 f n^2 \log (F)}} (e g-d h) \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (e^{\frac {2+2 a b f n \log (F)}{b^2 f n^2 \log (F)}} (e g-d h)^2 \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 b \sqrt {f} n \sqrt {\log (F)}}\right )+3 h^2 (d+e x)^2 \text {erfi}\left (\frac {3+2 b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 b \sqrt {f} n \sqrt {\log (F)}}\right )\right )\right )}{2 b e^4 \sqrt {f} n \sqrt {\log (F)}} \]
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\[\int F^{f {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}} \left (h x +g \right )^{3}d x\]
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Time = 0.31 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.05 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^3 \, dx=-\frac {\sqrt {\pi } \sqrt {-b^{2} f n^{2} \log \left (F\right )} h^{3} \operatorname {erf}\left (\frac {{\left (b^{2} f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b^{2} f n \log \left (F\right ) \log \left (c\right ) + a b f n \log \left (F\right ) + 2\right )} \sqrt {-b^{2} f n^{2} \log \left (F\right )}}{b^{2} f n^{2} \log \left (F\right )}\right ) e^{\left (-\frac {4 \, {\left (b^{2} f n \log \left (F\right ) \log \left (c\right ) + a b f n \log \left (F\right ) + 1\right )}}{b^{2} f n^{2} \log \left (F\right )}\right )} + 3 \, \sqrt {\pi } \sqrt {-b^{2} f n^{2} \log \left (F\right )} {\left (e g h^{2} - d h^{3}\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 ) + 3\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 {3 \, {\left (4 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 4 \, a b f n \log \left (F\right ) + 3\right )}}{4 \, b^{2} f n^{2} \log \left (F\right )}\right )} + \sqrt {\pi } {\left (e^{3} g^{3} - 3 \, d e^{2} g^{2} h + 3 \, d^{2} e g h^{2} - d^{3} h^{3}\right )} \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 )} + 3 \, \sqrt {\pi } \sqrt {-b^{2} f n^{2} \log \left (F\right )} {\left (e^{2} g^{2} h - 2 \, d e g h^{2} + d^{2} h^{3}\right )} \operatorname {erf}\left (\frac {{\left (b^{2} f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b^{2} f n \log \left (F\right ) \log \left (c\right ) + a b f n \log \left (F\right ) + 1\right )} \sqrt {-b^{2} f n^{2} \log \left (F\right )}}{b^{2} f n^{2} \log \left (F\right )}\right ) e^{\left (-\frac {2 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 2 \, a b f n \log \left (F\right ) + 1}{b^{2} f n^{2} \log \left (F\right )}\right )}}{2 \, b e^{4} n} \]
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Timed out. \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^3 \, dx=\text {Timed out} \]
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\[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^3 \, dx=\int { {\left (h x + g\right )}^{3} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f} \,d x } \]
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\[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^3 \, dx=\int { {\left (h x + g\right )}^{3} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f} \,d x } \]
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Timed out. \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^3 \, dx=\int {\mathrm {e}}^{f\,\ln \left (F\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}\,{\left (g+h\,x\right )}^3 \,d x \]
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