Optimal. Leaf size=133 \[ \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{6 c^3 d \left (d-c^2 d x^2\right )^{3/2}} \]
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Rubi [A] time = 0.404869, antiderivative size = 160, normalized size of antiderivative = 1.2, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {5798, 5724, 266, 43} \[ \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{6 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )}{6 c^3 d^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5724
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 \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^2 \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{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (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{x^3}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (-1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2 \left (-1+c^2 x\right )^2}+\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b \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{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{-1+c x} \sqrt{1+c x} \log \left (1-c^2 x^2\right )}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.198563, size = 101, normalized size = 0.76 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (\frac{b \left (\frac{1}{1-c^2 x^2}+\log \left (1-c^2 x^2\right )\right )}{c^3}-\frac{2 x^3 \left (a+b \cosh ^{-1}(c x)\right )}{(c x-1)^{3/2} (c x+1)^{3/2}}\right )}{6 d^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.202, size = 1228, normalized size = 9.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28269, size = 228, normalized size = 1.71 \begin{align*} \frac{1}{6} \, b c{\left (\frac{\sqrt{-d}}{c^{6} d^{3} x^{2} - c^{4} d^{3}} - \frac{\sqrt{-d} \log \left (c x + 1\right )}{c^{4} d^{3}} - \frac{\sqrt{-d} \log \left (c x - 1\right )}{c^{4} d^{3}}\right )} - \frac{1}{3} \, b{\left (\frac{x}{\sqrt{-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d}\right )} \operatorname{arcosh}\left (c x\right ) - \frac{1}{3} \, a{\left (\frac{x}{\sqrt{-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d}\right )} \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} d x^{2} + d}{\left (b x^{2} \operatorname{arcosh}\left (c x\right ) + a x^{2}\right )}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\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^{2} \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^{2}}{{\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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