Optimal. Leaf size=202 \[ \frac{1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac{b d^3 \left (c^2 x^2+1\right )^{9/2}}{81 c^3}+\frac{b d^3 \left (c^2 x^2+1\right )^{7/2}}{441 c^3}+\frac{2 b d^3 \left (c^2 x^2+1\right )^{5/2}}{525 c^3}+\frac{8 b d^3 \left (c^2 x^2+1\right )^{3/2}}{945 c^3}+\frac{16 b d^3 \sqrt{c^2 x^2+1}}{315 c^3} \]
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Rubi [A] time = 0.24858, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {270, 5730, 12, 1799, 1620} \[ \frac{1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac{b d^3 \left (c^2 x^2+1\right )^{9/2}}{81 c^3}+\frac{b d^3 \left (c^2 x^2+1\right )^{7/2}}{441 c^3}+\frac{2 b d^3 \left (c^2 x^2+1\right )^{5/2}}{525 c^3}+\frac{8 b d^3 \left (c^2 x^2+1\right )^{3/2}}{945 c^3}+\frac{16 b d^3 \sqrt{c^2 x^2+1}}{315 c^3} \]
Antiderivative was successfully verified.
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Rule 270
Rule 5730
Rule 12
Rule 1799
Rule 1620
Rubi steps
\begin{align*} \int x^2 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac{d^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )}{315 \sqrt{1+c^2 x^2}} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{315} \left (b c d^3\right ) \int \frac{x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{630} \left (b c d^3\right ) \operatorname{Subst}\left (\int \frac{x \left (105+189 c^2 x+135 c^4 x^2+35 c^6 x^3\right )}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{630} \left (b c d^3\right ) \operatorname{Subst}\left (\int \left (-\frac{16}{c^2 \sqrt{1+c^2 x}}-\frac{8 \sqrt{1+c^2 x}}{c^2}-\frac{6 \left (1+c^2 x\right )^{3/2}}{c^2}-\frac{5 \left (1+c^2 x\right )^{5/2}}{c^2}+\frac{35 \left (1+c^2 x\right )^{7/2}}{c^2}\right ) \, dx,x,x^2\right )\\ &=\frac{16 b d^3 \sqrt{1+c^2 x^2}}{315 c^3}+\frac{8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{945 c^3}+\frac{2 b d^3 \left (1+c^2 x^2\right )^{5/2}}{525 c^3}+\frac{b d^3 \left (1+c^2 x^2\right )^{7/2}}{441 c^3}-\frac{b d^3 \left (1+c^2 x^2\right )^{9/2}}{81 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.103453, size = 135, normalized size = 0.67 \[ \frac{d^3 \left (315 a c^3 x^3 \left (35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right )-b \sqrt{c^2 x^2+1} \left (1225 c^8 x^8+4675 c^6 x^6+6297 c^4 x^4+2629 c^2 x^2-5258\right )+315 b c^3 x^3 \left (35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right ) \sinh ^{-1}(c x)\right )}{99225 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 187, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{3}} \left ({d}^{3}a \left ({\frac{{c}^{9}{x}^{9}}{9}}+{\frac{3\,{c}^{7}{x}^{7}}{7}}+{\frac{3\,{c}^{5}{x}^{5}}{5}}+{\frac{{c}^{3}{x}^{3}}{3}} \right ) +{d}^{3}b \left ({\frac{{\it Arcsinh} \left ( cx \right ){c}^{9}{x}^{9}}{9}}+{\frac{3\,{\it Arcsinh} \left ( cx \right ){c}^{7}{x}^{7}}{7}}+{\frac{3\,{\it Arcsinh} \left ( cx \right ){c}^{5}{x}^{5}}{5}}+{\frac{{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{3}}{3}}-{\frac{{c}^{8}{x}^{8}}{81}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{187\,{c}^{6}{x}^{6}}{3969}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{2099\,{c}^{4}{x}^{4}}{33075}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{2629\,{c}^{2}{x}^{2}}{99225}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{5258}{99225}\sqrt{{c}^{2}{x}^{2}+1}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1779, size = 524, normalized size = 2.59 \begin{align*} \frac{1}{9} \, a c^{6} d^{3} x^{9} + \frac{3}{7} \, a c^{4} d^{3} x^{7} + \frac{1}{2835} \,{\left (315 \, x^{9} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{35 \, \sqrt{c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac{40 \, \sqrt{c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac{48 \, \sqrt{c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac{64 \, \sqrt{c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac{128 \, \sqrt{c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{6} d^{3} + \frac{3}{5} \, a c^{2} d^{3} x^{5} + \frac{3}{245} \,{\left (35 \, x^{7} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{5 \, \sqrt{c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac{6 \, \sqrt{c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac{16 \, \sqrt{c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{4} d^{3} + \frac{1}{25} \,{\left (15 \, x^{5} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{3 \, \sqrt{c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac{4 \, \sqrt{c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{2} d^{3} + \frac{1}{3} \, a d^{3} x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \operatorname{arsinh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac{2 \, \sqrt{c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53317, size = 448, normalized size = 2.22 \begin{align*} \frac{11025 \, a c^{9} d^{3} x^{9} + 42525 \, a c^{7} d^{3} x^{7} + 59535 \, a c^{5} d^{3} x^{5} + 33075 \, a c^{3} d^{3} x^{3} + 315 \,{\left (35 \, b c^{9} d^{3} x^{9} + 135 \, b c^{7} d^{3} x^{7} + 189 \, b c^{5} d^{3} x^{5} + 105 \, b c^{3} d^{3} x^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (1225 \, b c^{8} d^{3} x^{8} + 4675 \, b c^{6} d^{3} x^{6} + 6297 \, b c^{4} d^{3} x^{4} + 2629 \, b c^{2} d^{3} x^{2} - 5258 \, b d^{3}\right )} \sqrt{c^{2} x^{2} + 1}}{99225 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 23.7592, size = 265, normalized size = 1.31 \begin{align*} \begin{cases} \frac{a c^{6} d^{3} x^{9}}{9} + \frac{3 a c^{4} d^{3} x^{7}}{7} + \frac{3 a c^{2} d^{3} x^{5}}{5} + \frac{a d^{3} x^{3}}{3} + \frac{b c^{6} d^{3} x^{9} \operatorname{asinh}{\left (c x \right )}}{9} - \frac{b c^{5} d^{3} x^{8} \sqrt{c^{2} x^{2} + 1}}{81} + \frac{3 b c^{4} d^{3} x^{7} \operatorname{asinh}{\left (c x \right )}}{7} - \frac{187 b c^{3} d^{3} x^{6} \sqrt{c^{2} x^{2} + 1}}{3969} + \frac{3 b c^{2} d^{3} x^{5} \operatorname{asinh}{\left (c x \right )}}{5} - \frac{2099 b c d^{3} x^{4} \sqrt{c^{2} x^{2} + 1}}{33075} + \frac{b d^{3} x^{3} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{2629 b d^{3} x^{2} \sqrt{c^{2} x^{2} + 1}}{99225 c} + \frac{5258 b d^{3} \sqrt{c^{2} x^{2} + 1}}{99225 c^{3}} & \text{for}\: c \neq 0 \\\frac{a d^{3} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.83501, size = 501, normalized size = 2.48 \begin{align*} \frac{1}{9} \, a c^{6} d^{3} x^{9} + \frac{3}{7} \, a c^{4} d^{3} x^{7} + \frac{1}{2835} \,{\left (315 \, x^{9} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{35 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{9}{2}} - 180 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{7}{2}} + 378 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 420 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 315 \, \sqrt{c^{2} x^{2} + 1}}{c^{9}}\right )} b c^{6} d^{3} + \frac{3}{5} \, a c^{2} d^{3} x^{5} + \frac{3}{245} \,{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{7}{2}} - 21 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 35 \, \sqrt{c^{2} x^{2} + 1}}{c^{7}}\right )} b c^{4} d^{3} + \frac{1}{25} \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} + 1}}{c^{5}}\right )} b c^{2} d^{3} + \frac{1}{3} \, a d^{3} x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{c^{2} x^{2} + 1}}{c^{3}}\right )} b d^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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