Optimal. Leaf size=141 \[ \frac{4 b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{7/2} \text{HypergeometricPFQ}\left (\left \{1,\frac{7}{4},\frac{7}{4}\right \},\left \{\frac{9}{4},\frac{11}{4}\right \},c^2 x^2\right )}{35 f^2 \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{1-c^2 x^2} (f x)^{5/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{5 f \sqrt{d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.382549, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {5798, 5763} \[ \frac{4 b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{7/2} \, _3F_2\left (1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right )}{35 f^2 \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{1-c^2 x^2} (f x)^{5/2} \, _2F_1\left (\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{5 f \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5763
Rubi steps
\begin{align*} \int \frac{(f x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{d-c^2 d 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{d-c^2 d x^2}}\\ &=\frac{2 (f x)^{5/2} \sqrt{1-c^2 x^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 \sqrt{d-c^2 d x^2}}+\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{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.044996, size = 115, normalized size = 0.82 \[ \frac{2 x (f x)^{3/2} \left (2 b c x \sqrt{c x-1} \sqrt{c x+1} \text{HypergeometricPFQ}\left (\left \{1,\frac{7}{4},\frac{7}{4}\right \},\left \{\frac{9}{4},\frac{11}{4}\right \},c^2 x^2\right )+7 \sqrt{1-c^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )\right )}{35 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.363, 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}d{x}^{2}+d}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b f x \operatorname{arcosh}\left (c x\right ) + a f x\right )} \sqrt{f x}}{c^{2} d x^{2} - d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]