Integrand size = 30, antiderivative size = 98 \[ \int \frac {(f x)^{3/2} (a+b \text {arccosh}(c x))}{\sqrt {1-c^2 x^2}} \, dx=\frac {2 (f x)^{5/2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )}{5 f}+\frac {4 b c (f x)^{7/2} \sqrt {-1+c x} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{35 f^2 \sqrt {1-c x}} \]
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Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {5948} \[ \int \frac {(f x)^{3/2} (a+b \text {arccosh}(c x))}{\sqrt {1-c^2 x^2}} \, dx=\frac {4 b c \sqrt {c x-1} (f x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{35 f^2 \sqrt {1-c x}}+\frac {2 (f x)^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{5 f} \]
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Rule 5948
Rubi steps \begin{align*} \text {integral}& = \frac {2 (f x)^{5/2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )}{5 f}+\frac {4 b c (f x)^{7/2} \sqrt {-1+c x} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{35 f^2 \sqrt {1-c x}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02 \[ \int \frac {(f x)^{3/2} (a+b \text {arccosh}(c x))}{\sqrt {1-c^2 x^2}} \, dx=\frac {2}{35} x (f x)^{3/2} \left (7 (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )+\frac {2 b c x \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{\sqrt {1-c^2 x^2}}\right ) \]
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\[\int \frac {\left (f x \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{\sqrt {-c^{2} x^{2}+1}}d x\]
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\[ \int \frac {(f x)^{3/2} (a+b \text {arccosh}(c x))}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {(f x)^{3/2} (a+b \text {arccosh}(c x))}{\sqrt {1-c^2 x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(f x)^{3/2} (a+b \text {arccosh}(c x))}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
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\[ \int \frac {(f x)^{3/2} (a+b \text {arccosh}(c x))}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {(f x)^{3/2} (a+b \text {arccosh}(c x))}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^{3/2}}{\sqrt {1-c^2\,x^2}} \,d x \]
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