Optimal. Leaf size=98 \[ \frac{4 b c \sqrt{c x-1} (f x)^{7/2} \text{HypergeometricPFQ}\left (\left \{1,\frac{7}{4},\frac{7}{4}\right \},\left \{\frac{9}{4},\frac{11}{4}\right \},c^2 x^2\right )}{35 f^2 \sqrt{1-c x}}+\frac{2 (f x)^{5/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{5 f} \]
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Rubi [A] time = 0.357704, antiderivative size = 111, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {5798, 5763} \[ \frac{4 b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{7/2} \, _3F_2\left (1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right )}{35 f^2 \sqrt{1-c^2 x^2}}+\frac{2 (f x)^{5/2} \, _2F_1\left (\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{5 f} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5763
Rubi steps
\begin{align*} \int \frac{(f x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{(f x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{2 (f x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\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} \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^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.106495, size = 100, normalized size = 1.02 \[ \frac{2}{35} x (f x)^{3/2} \left (\frac{2 b c x \sqrt{c x-1} \sqrt{c x+1} \text{HypergeometricPFQ}\left (\left \{1,\frac{7}{4},\frac{7}{4}\right \},\left \{\frac{9}{4},\frac{11}{4}\right \},c^2 x^2\right )}{\sqrt{1-c^2 x^2}}+7 \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.277, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\rm arccosh} \left (cx\right )) \left ( fx \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1}{\left (b f x \operatorname{arcosh}\left (c x\right ) + a f x\right )} \sqrt{f x}}{c^{2} x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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