Integrand size = 23, antiderivative size = 374 \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=-\frac {b e \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^3 f (2+m) (3+m) (4+m) (5+m) \sqrt {c^2 x^2}}-\frac {b e^2 x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c f^3 (4+m) (5+m) \sqrt {c^2 x^2}}+\frac {d^2 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {b \left (c^4 d^2 (2+m) (3+m) (4+m) (5+m)+e (1+m)^2 \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right )\right ) x (f x)^{1+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{c^3 f (1+m)^2 (2+m) (3+m) (4+m) (5+m) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}} \]
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Time = 0.32 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {276, 5346, 12, 1281, 470, 372, 371} \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {d^2 (f x)^{m+1} \left (a+b \sec ^{-1}(c x)\right )}{f (m+1)}+\frac {2 d e (f x)^{m+3} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (m+5)}-\frac {b e^2 x \sqrt {c^2 x^2-1} (f x)^{m+3}}{c f^3 (m+4) (m+5) \sqrt {c^2 x^2}}-\frac {b c x \sqrt {1-c^2 x^2} (f x)^{m+1} \left (\frac {e \left (2 c^2 d \left (m^2+9 m+20\right )+e (m+3)^2\right )}{c^4 (m+2) (m+3) (m+4) (m+5)}+\frac {d^2}{(m+1)^2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right )}{f \sqrt {c^2 x^2} \sqrt {c^2 x^2-1}}-\frac {b e x \sqrt {c^2 x^2-1} (f x)^{m+1} \left (2 c^2 d \left (m^2+9 m+20\right )+e (m+3)^2\right )}{c^3 f (m+2) (m+3) (m+4) (m+5) \sqrt {c^2 x^2}} \]
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Rule 12
Rule 276
Rule 371
Rule 372
Rule 470
Rule 1281
Rule 5346
Rubi steps \begin{align*} \text {integral}& = \frac {d^2 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {(b c x) \int \frac {(f x)^m \left (d^2 \left (15+8 m+m^2\right )+2 d e \left (5+6 m+m^2\right ) x^2+e^2 \left (3+4 m+m^2\right ) x^4\right )}{(1+m) (3+m) (5+m) \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {d^2 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {(b c x) \int \frac {(f x)^m \left (d^2 \left (15+8 m+m^2\right )+2 d e \left (5+6 m+m^2\right ) x^2+e^2 \left (3+4 m+m^2\right ) x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{\left (15+23 m+9 m^2+m^3\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b e^2 x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c f^3 (4+m) (5+m) \sqrt {c^2 x^2}}+\frac {d^2 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {(b x) \int \frac {(f x)^m \left (c^2 d^2 (3+m) (4+m) (5+m)+e (1+m) \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c (4+m) \left (15+23 m+9 m^2+m^3\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b e \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^3 f (2+m) (4+m) \left (15+8 m+m^2\right ) \sqrt {c^2 x^2}}-\frac {b e^2 x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c f^3 (4+m) (5+m) \sqrt {c^2 x^2}}+\frac {d^2 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}--\frac {\left (b \left (-c^4 d^2 (2+m) (3+m) (4+m) (5+m)-e (1+m)^2 \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right )\right ) x\right ) \int \frac {(f x)^m}{\sqrt {-1+c^2 x^2}} \, dx}{c^3 (2+m) (4+m) \left (15+23 m+9 m^2+m^3\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b e \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^3 f (2+m) (4+m) \left (15+8 m+m^2\right ) \sqrt {c^2 x^2}}-\frac {b e^2 x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c f^3 (4+m) (5+m) \sqrt {c^2 x^2}}+\frac {d^2 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}--\frac {\left (b \left (-c^4 d^2 (2+m) (3+m) (4+m) (5+m)-e (1+m)^2 \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right )\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^m}{\sqrt {1-c^2 x^2}} \, dx}{c^3 (2+m) (4+m) \left (15+23 m+9 m^2+m^3\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}} \\ & = -\frac {b e \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^3 f (2+m) (4+m) \left (15+8 m+m^2\right ) \sqrt {c^2 x^2}}-\frac {b e^2 x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c f^3 (4+m) (5+m) \sqrt {c^2 x^2}}+\frac {d^2 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {b \left (c^4 d^2 (2+m) (3+m) (4+m) (5+m)+e (1+m)^2 \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right )\right ) x (f x)^{1+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{c^3 f (1+m) (2+m) (4+m) \left (15+23 m+9 m^2+m^3\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.78 \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=x (f x)^m \left (\frac {a d^2}{1+m}+\frac {2 a d e x^2}{3+m}+\frac {a e^2 x^4}{5+m}+\frac {b d^2 \sec ^{-1}(c x)}{1+m}+\frac {2 b d e x^2 \sec ^{-1}(c x)}{3+m}+\frac {b e^2 x^4 \sec ^{-1}(c x)}{5+m}+\frac {b c d^2 \sqrt {1-\frac {1}{c^2 x^2}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{(1+m)^2 \sqrt {1-c^2 x^2}}+\frac {2 b c d e \sqrt {1-\frac {1}{c^2 x^2}} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{(3+m)^2 \sqrt {1-c^2 x^2}}+\frac {b c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5+m}{2},\frac {7+m}{2},c^2 x^2\right )}{(5+m)^2 \sqrt {1-c^2 x^2}}\right ) \]
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\[\int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )d x\]
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\[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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\[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int \left (f x\right )^{m} \left (a + b \operatorname {asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
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\[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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\[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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Timed out. \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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