Integrand size = 29, antiderivative size = 300 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{d f \sqrt {d-c^2 d x^2}}-\frac {m (f x)^{1+m} \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{d f (1+m) \sqrt {d-c^2 d x^2}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{d f^2 (2+m) \sqrt {d-c^2 d x^2}}-\frac {b c m (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{d f^2 (1+m) (2+m) \sqrt {d-c^2 d x^2}} \]
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Time = 0.19 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {5936, 5948, 74, 371} \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c m \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{d f^2 (m+1) (m+2) \sqrt {d-c^2 d x^2}}-\frac {m \sqrt {1-c^2 x^2} (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{d f (m+1) \sqrt {d-c^2 d x^2}}+\frac {(f x)^{m+1} (a+b \text {arccosh}(c x))}{d f \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{d f^2 (m+2) \sqrt {d-c^2 d x^2}} \]
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Rule 74
Rule 371
Rule 5936
Rule 5948
Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{d f \sqrt {d-c^2 d x^2}}-\frac {m \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx}{d}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f x)^{1+m}}{(-1+c x) (1+c x)} \, dx}{d f \sqrt {d-c^2 d x^2}} \\ & = \frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{d f \sqrt {d-c^2 d x^2}}-\frac {m (f x)^{1+m} \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{d f (1+m) \sqrt {d-c^2 d x^2}}-\frac {b c m (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{d f^2 (1+m) (2+m) \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f x)^{1+m}}{-1+c^2 x^2} \, dx}{d f \sqrt {d-c^2 d x^2}} \\ & = \frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{d f \sqrt {d-c^2 d x^2}}-\frac {m (f x)^{1+m} \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{d f (1+m) \sqrt {d-c^2 d x^2}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{d f^2 (2+m) \sqrt {d-c^2 d x^2}}-\frac {b c m (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{d f^2 (1+m) (2+m) \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.72 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x (f x)^m \left (-m (2+m) \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )+(1+m) \left ((2+m) (a+b \text {arccosh}(c x))+b c x \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )\right )-b c m x \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )\right )}{d (1+m) (2+m) \sqrt {d-c^2 d x^2}} \]
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\[\int \frac {\left (f x \right )^{m} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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