Optimal. Leaf size=158 \[ -\frac{a+b \cosh ^{-1}(c x)}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 c^4 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b x \sqrt{d-c^2 d x^2}}{6 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{5 b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^4 d^3 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.447681, antiderivative size = 243, normalized size of antiderivative = 1.54, number of steps used = 5, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5798, 94, 89, 21, 37, 5733, 12, 385, 206} \[ \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{(1-c x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 (c x+1) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{c^4 d^2 (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{6 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{5 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{6 c^4 d^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 94
Rule 89
Rule 21
Rule 37
Rule 5733
Rule 12
Rule 385
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{c^4 d^2 (1+c x) \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{(1-c x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{2-3 c^2 x^2}{3 c^4 \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{c^4 d^2 (1+c x) \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{(1-c x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{2-3 c^2 x^2}{\left (1-c^2 x^2\right )^2} \, dx}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{6 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{c^4 d^2 (1+c x) \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{(1-c x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (5 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{1-c^2 x^2} \, dx}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{6 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{c^4 d^2 (1+c x) \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{(1-c x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 (1+c x) \sqrt{d-c^2 d x^2}}-\frac{5 b \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.121852, size = 122, normalized size = 0.77 \[ \frac{-6 a c^2 x^2+4 a+b \left (4-6 c^2 x^2\right ) \cosh ^{-1}(c x)-5 b \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2-1\right ) \tanh ^{-1}(c x)-b c x \sqrt{c x-1} \sqrt{c x+1}}{6 c^4 d^2 \left (c^2 x^2-1\right ) \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.208, size = 313, normalized size = 2. \begin{align*}{\frac{a{x}^{2}}{{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,a}{3\,d{c}^{4}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}+{\frac{b{x}^{2}{\rm arccosh} \left (cx\right )}{{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}{c}^{2}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bx}{6\,{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}{c}^{3}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}-{\frac{2\,b{\rm arccosh} \left (cx\right )}{3\,{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}{c}^{4}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{5\,b}{6\,{c}^{4}{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( cx+\sqrt{cx-1}\sqrt{cx+1}-1 \right ) }+{\frac{5\,b}{6\,{c}^{4}{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.27614, size = 236, normalized size = 1.49 \begin{align*} \frac{1}{12} \, b c{\left (\frac{2 \, \sqrt{-d} x}{c^{6} d^{3} x^{2} - c^{4} d^{3}} + \frac{5 \, \sqrt{-d} \log \left (c x + 1\right )}{c^{5} d^{3}} - \frac{5 \, \sqrt{-d} \log \left (c x - 1\right )}{c^{5} d^{3}}\right )} + \frac{1}{3} \, b{\left (\frac{3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d} - \frac{2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{4} d}\right )} \operatorname{arcosh}\left (c x\right ) + \frac{1}{3} \, a{\left (\frac{3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d} - \frac{2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{4} d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58097, size = 1013, normalized size = 6.41 \begin{align*} \left [\frac{4 \, \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} b c x + 8 \,{\left (3 \, b c^{2} x^{2} - 2 \, b\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - 5 \,{\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt{-d} \log \left (-\frac{c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \,{\left (c^{3} x^{3} + c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} \sqrt{-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) + 8 \,{\left (3 \, a c^{2} x^{2} - 2 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{24 \,{\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}}, \frac{2 \, \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} b c x + 5 \,{\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt{d} \arctan \left (\frac{2 \, \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} c \sqrt{d} x}{c^{4} d x^{4} - d}\right ) + 4 \,{\left (3 \, b c^{2} x^{2} - 2 \, b\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 4 \,{\left (3 \, a c^{2} x^{2} - 2 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{12 \,{\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{5}{2}}}\, dx \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 (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (-c^{2} d 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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