Optimal. Leaf size=182 \[ \frac{2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 b \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{3 d^2 \sqrt{e} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{3 d \left (c^2 d+e\right ) \sqrt{d+e x^2}} \]
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Rubi [A] time = 0.182742, antiderivative size = 190, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {192, 191, 5705, 12, 519, 571, 78, 63, 217, 206} \[ \frac{2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{2 b \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{3 d^2 \sqrt{e} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \left (1-c^2 x^2\right )}{3 d \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right ) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rule 5705
Rule 12
Rule 519
Rule 571
Rule 78
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-(b c) \int \frac{x \left (3 d+2 e x^2\right )}{3 d^2 \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{(b c) \int \frac{x \left (3 d+2 e x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x \left (3 d+2 e x^2\right )}{\sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{3 d+2 e x}{\sqrt{-1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \left (1-c^2 x^2\right )}{3 d \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \sqrt{d+e x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \left (1-c^2 x^2\right )}{3 d \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \sqrt{d+e x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{\left (2 b \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+\frac{e}{c^2}+\frac{e x^2}{c^2}}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{3 c d^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \left (1-c^2 x^2\right )}{3 d \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \sqrt{d+e x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{\left (2 b \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{-1+c^2 x^2}}{\sqrt{d+e x^2}}\right )}{3 c d^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \left (1-c^2 x^2\right )}{3 d \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \sqrt{d+e x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{2 b \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{3 d^2 \sqrt{e} \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [C] time = 2.24151, size = 633, normalized size = 3.48 \[ \frac{\frac{4 b (c x-1)^{3/2} \left (d+e x^2\right ) \sqrt{\frac{(c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )}{(c x-1) \left (c \sqrt{d}+i \sqrt{e}\right )}} \left (\frac{c \left (\sqrt{e}-i c \sqrt{d}\right ) \left (\sqrt{e} x+i \sqrt{d}\right ) \sqrt{\frac{\frac{i c \sqrt{d}}{\sqrt{e}}+c (-x)+\frac{i \sqrt{e} x}{\sqrt{d}}+1}{1-c x}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{c \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{2-2 c x}}\right ),\frac{4 i c \sqrt{d} \sqrt{e}}{\left (c \sqrt{d}+i \sqrt{e}\right )^2}\right )}{c x-1}+c \sqrt{d} \left (-c \sqrt{d}+i \sqrt{e}\right ) \sqrt{\frac{\left (c^2 d+e\right ) \left (d+e x^2\right )}{d e (c x-1)^2}} \sqrt{-\frac{c \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}} \Pi \left (\frac{2 c \sqrt{d}}{\sqrt{d} c+i \sqrt{e}};\sin ^{-1}\left (\sqrt{-\frac{\frac{i \sqrt{e} x}{\sqrt{d}}+c \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )-1}{2-2 c x}}\right )|\frac{4 i c \sqrt{d} \sqrt{e}}{\left (\sqrt{d} c+i \sqrt{e}\right )^2}\right )\right )}{c d^2 \sqrt{c x+1} \left (c^2 d+e\right ) \sqrt{-\frac{c \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}}}+\frac{a x \left (3 d+2 e x^2\right )}{d^2}-\frac{b c \sqrt{c x+1} \sqrt{c x-1} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}+\frac{b x \cosh ^{-1}(c x) \left (3 d+2 e x^2\right )}{d^2}}{3 \left (d+e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.806, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\rm arccosh} \left (cx\right )) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, 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*} \frac{1}{3} \, a{\left (\frac{2 \, x}{\sqrt{e x^{2} + d} d^{2}} + \frac{x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} d}\right )} + b \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.52482, size = 1503, normalized size = 8.26 \begin{align*} \left [\frac{{\left (b c^{2} d^{3} +{\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \,{\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt{e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \,{\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} - 4 \,{\left (2 \, c^{3} e x^{2} + c^{3} d - c e\right )} \sqrt{c^{2} x^{2} - 1} \sqrt{e x^{2} + d} \sqrt{e} + e^{2}\right ) + 2 \,{\left (2 \,{\left (b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \,{\left (b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 2 \,{\left (2 \,{\left (a c^{2} d e^{2} + a e^{3}\right )} x^{3} + 3 \,{\left (a c^{2} d^{2} e + a d e^{2}\right )} x -{\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt{c^{2} x^{2} - 1}\right )} \sqrt{e x^{2} + d}}{6 \,{\left (c^{2} d^{5} e + d^{4} e^{2} +{\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \,{\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}, \frac{{\left (b c^{2} d^{3} +{\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \,{\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt{-e} \arctan \left (\frac{{\left (2 \, c^{2} e x^{2} + c^{2} d - e\right )} \sqrt{c^{2} x^{2} - 1} \sqrt{e x^{2} + d} \sqrt{-e}}{2 \,{\left (c^{3} e^{2} x^{4} - c d e +{\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) +{\left (2 \,{\left (b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \,{\left (b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (2 \,{\left (a c^{2} d e^{2} + a e^{3}\right )} x^{3} + 3 \,{\left (a c^{2} d^{2} e + a d e^{2}\right )} x -{\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt{c^{2} x^{2} - 1}\right )} \sqrt{e x^{2} + d}}{3 \,{\left (c^{2} d^{5} e + d^{4} e^{2} +{\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \,{\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}\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{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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