Integrand size = 23, antiderivative size = 353 \[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{c^3 f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{c f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {b \left (\frac {c^4 d^2 (3+m) (5+m)}{1+m}+\frac {e (2+m) \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{(3+m) (5+m)}\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{c^3 f^2 (2+m) (3+m) (5+m) \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.40 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {276, 5958, 12, 534, 1281, 470, 372, 371} \[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}+\frac {b e^2 \left (1-c^2 x^2\right ) (f x)^{m+4}}{c f^4 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {1-c^2 x^2} (f x)^{m+2} \left (\frac {e \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^4 (m+3)^2 (m+5)^2}+\frac {d^2}{m^2+3 m+2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{f^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e \left (1-c^2 x^2\right ) (f x)^{m+2} \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^3 f^2 (m+3)^2 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 12
Rule 276
Rule 371
Rule 372
Rule 470
Rule 534
Rule 1281
Rule 5958
Rubi steps \begin{align*} \text {integral}& = \frac {d^2 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-(b c) \int \frac {(f x)^{1+m} \left (\frac {d^2}{1+m}+\frac {2 d e x^2}{3+m}+\frac {e^2 x^4}{5+m}\right )}{f \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {d^2 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {(b c) \int \frac {(f x)^{1+m} \left (\frac {d^2}{1+m}+\frac {2 d e x^2}{3+m}+\frac {e^2 x^4}{5+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f} \\ & = \frac {d^2 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {d^2}{1+m}+\frac {2 d e x^2}{3+m}+\frac {e^2 x^4}{5+m}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{f \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b e^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{c f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {c^2 d^2 (5+m)}{1+m}+\frac {e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) x^2}{(3+m) (5+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c f (5+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{c^3 f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{c f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {\left (b \left (\frac {c^2 d^2 (5+m)}{1+m}+\frac {e (2+m) \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{c^2 (3+m)^2 (5+m)}\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {-1+c^2 x^2}} \, dx}{c f (5+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{c^3 f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{c f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {\left (b \left (\frac {c^2 d^2 (5+m)}{1+m}+\frac {e (2+m) \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{c^2 (3+m)^2 (5+m)}\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{c f (5+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{c^3 f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{c f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {b \left (\frac {c^4 d^2}{2+3 m+m^2}+\frac {e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{(3+m)^2 (5+m)^2}\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{c^3 f^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.83 \[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=x (f x)^m \left (\frac {d^2 (a+b \text {arccosh}(c x))}{1+m}+\frac {2 d e x^2 (a+b \text {arccosh}(c x))}{3+m}+\frac {e^2 x^4 (a+b \text {arccosh}(c x))}{5+m}-\frac {b c d^2 x \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d e x^3 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )}{\left (12+7 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c e^2 x^5 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},c^2 x^2\right )}{(5+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}\right ) \]
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\[\int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]
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\[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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\[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int \left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
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\[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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\[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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Timed out. \[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^2 \,d x \]
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