Optimal. Leaf size=312 \[ \frac{6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 b e^2 \left (c^2 x^2+1\right )^{5/2} \left (63 c^4 d^2-90 c^2 d e+35 e^2\right )}{525 c^9}-\frac{4 b e \left (c^2 x^2+1\right )^{3/2} \left (-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right )}{945 c^9}-\frac{b \sqrt{c^2 x^2+1} \left (378 c^4 d^2 e^2-420 c^6 d^3 e+315 c^8 d^4-180 c^2 d e^3+35 e^4\right )}{315 c^9}-\frac{4 b e^3 \left (c^2 x^2+1\right )^{7/2} \left (9 c^2 d-7 e\right )}{441 c^9}-\frac{b e^4 \left (c^2 x^2+1\right )^{9/2}}{81 c^9} \]
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Rubi [A] time = 0.350649, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {194, 5704, 12, 1799, 1850} \[ \frac{6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 b e^2 \left (c^2 x^2+1\right )^{5/2} \left (63 c^4 d^2-90 c^2 d e+35 e^2\right )}{525 c^9}-\frac{4 b e \left (c^2 x^2+1\right )^{3/2} \left (-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right )}{945 c^9}-\frac{b \sqrt{c^2 x^2+1} \left (378 c^4 d^2 e^2-420 c^6 d^3 e+315 c^8 d^4-180 c^2 d e^3+35 e^4\right )}{315 c^9}-\frac{4 b e^3 \left (c^2 x^2+1\right )^{7/2} \left (9 c^2 d-7 e\right )}{441 c^9}-\frac{b e^4 \left (c^2 x^2+1\right )^{9/2}}{81 c^9} \]
Antiderivative was successfully verified.
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Rule 194
Rule 5704
Rule 12
Rule 1799
Rule 1850
Rubi steps
\begin{align*} \int \left (d+e x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac{x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{315 \sqrt{1+c^2 x^2}} \, dx\\ &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{315} (b c) \int \frac{x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{630} (b c) \operatorname{Subst}\left (\int \frac{315 d^4+420 d^3 e x+378 d^2 e^2 x^2+180 d e^3 x^3+35 e^4 x^4}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )\\ &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{630} (b c) \operatorname{Subst}\left (\int \left (\frac{315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4}{c^8 \sqrt{1+c^2 x}}+\frac{4 e \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \sqrt{1+c^2 x}}{c^8}+\frac{6 e^2 \left (63 c^4 d^2-90 c^2 d e+35 e^2\right ) \left (1+c^2 x\right )^{3/2}}{c^8}+\frac{20 \left (9 c^2 d-7 e\right ) e^3 \left (1+c^2 x\right )^{5/2}}{c^8}+\frac{35 e^4 \left (1+c^2 x\right )^{7/2}}{c^8}\right ) \, dx,x,x^2\right )\\ &=-\frac{b \left (315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4\right ) \sqrt{1+c^2 x^2}}{315 c^9}-\frac{4 b e \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \left (1+c^2 x^2\right )^{3/2}}{945 c^9}-\frac{2 b e^2 \left (63 c^4 d^2-90 c^2 d e+35 e^2\right ) \left (1+c^2 x^2\right )^{5/2}}{525 c^9}-\frac{4 b \left (9 c^2 d-7 e\right ) e^3 \left (1+c^2 x^2\right )^{7/2}}{441 c^9}-\frac{b e^4 \left (1+c^2 x^2\right )^{9/2}}{81 c^9}+d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.343265, size = 260, normalized size = 0.83 \[ \frac{315 a x \left (378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4+180 d e^3 x^6+35 e^4 x^8\right )-\frac{b \sqrt{c^2 x^2+1} \left (c^8 \left (23814 d^2 e^2 x^4+44100 d^3 e x^2+99225 d^4+8100 d e^3 x^6+1225 e^4 x^8\right )-8 c^6 e \left (3969 d^2 e x^2+11025 d^3+1215 d e^2 x^4+175 e^3 x^6\right )+48 c^4 e^2 \left (1323 d^2+270 d e x^2+35 e^2 x^4\right )-320 c^2 e^3 \left (81 d+7 e x^2\right )+4480 e^4\right )}{c^9}+315 b x \sinh ^{-1}(c x) \left (378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4+180 d e^3 x^6+35 e^4 x^8\right )}{99225} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 451, normalized size = 1.5 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{8}} \left ({\frac{{e}^{4}{c}^{9}{x}^{9}}{9}}+{\frac{4\,{c}^{9}d{e}^{3}{x}^{7}}{7}}+{\frac{6\,{c}^{9}{d}^{2}{e}^{2}{x}^{5}}{5}}+{\frac{4\,{c}^{9}{d}^{3}e{x}^{3}}{3}}+{c}^{9}{d}^{4}x \right ) }+{\frac{b}{{c}^{8}} \left ({\frac{{\it Arcsinh} \left ( cx \right ){e}^{4}{c}^{9}{x}^{9}}{9}}+{\frac{4\,{\it Arcsinh} \left ( cx \right ){c}^{9}d{e}^{3}{x}^{7}}{7}}+{\frac{6\,{\it Arcsinh} \left ( cx \right ){c}^{9}{d}^{2}{e}^{2}{x}^{5}}{5}}+{\frac{4\,{\it Arcsinh} \left ( cx \right ){c}^{9}{d}^{3}e{x}^{3}}{3}}+{\it Arcsinh} \left ( cx \right ){c}^{9}{d}^{4}x-{\frac{{e}^{4}}{9} \left ({\frac{{c}^{8}{x}^{8}}{9}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{8\,{c}^{6}{x}^{6}}{63}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{16\,{c}^{4}{x}^{4}}{105}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{64\,{c}^{2}{x}^{2}}{315}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{128}{315}\sqrt{{c}^{2}{x}^{2}+1}} \right ) }-{\frac{4\,{c}^{2}d{e}^{3}}{7} \left ({\frac{{c}^{6}{x}^{6}}{7}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{6\,{c}^{4}{x}^{4}}{35}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{8\,{c}^{2}{x}^{2}}{35}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{16}{35}\sqrt{{c}^{2}{x}^{2}+1}} \right ) }-{\frac{6\,{c}^{4}{d}^{2}{e}^{2}}{5} \left ({\frac{{c}^{4}{x}^{4}}{5}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{4\,{c}^{2}{x}^{2}}{15}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{8}{15}\sqrt{{c}^{2}{x}^{2}+1}} \right ) }-{\frac{4\,{c}^{6}{d}^{3}e}{3} \left ({\frac{{c}^{2}{x}^{2}}{3}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{{c}^{2}{x}^{2}+1}} \right ) }-{c}^{8}{d}^{4}\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 [A] time = 1.22702, size = 560, normalized size = 1.79 \begin{align*} \frac{1}{9} \, a e^{4} x^{9} + \frac{4}{7} \, a d e^{3} x^{7} + \frac{6}{5} \, a d^{2} e^{2} x^{5} + \frac{4}{3} \, a d^{3} e x^{3} + \frac{4}{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} e + \frac{2}{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 d^{2} e^{2} + \frac{4}{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 d e^{3} + \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 e^{4} + a d^{4} x + \frac{{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} b d^{4}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48825, size = 803, normalized size = 2.57 \begin{align*} \frac{11025 \, a c^{9} e^{4} x^{9} + 56700 \, a c^{9} d e^{3} x^{7} + 119070 \, a c^{9} d^{2} e^{2} x^{5} + 132300 \, a c^{9} d^{3} e x^{3} + 99225 \, a c^{9} d^{4} x + 315 \,{\left (35 \, b c^{9} e^{4} x^{9} + 180 \, b c^{9} d e^{3} x^{7} + 378 \, b c^{9} d^{2} e^{2} x^{5} + 420 \, b c^{9} d^{3} e x^{3} + 315 \, b c^{9} d^{4} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (1225 \, b c^{8} e^{4} x^{8} + 99225 \, b c^{8} d^{4} - 88200 \, b c^{6} d^{3} e + 63504 \, b c^{4} d^{2} e^{2} - 25920 \, b c^{2} d e^{3} + 100 \,{\left (81 \, b c^{8} d e^{3} - 14 \, b c^{6} e^{4}\right )} x^{6} + 4480 \, b e^{4} + 6 \,{\left (3969 \, b c^{8} d^{2} e^{2} - 1620 \, b c^{6} d e^{3} + 280 \, b c^{4} e^{4}\right )} x^{4} + 4 \,{\left (11025 \, b c^{8} d^{3} e - 7938 \, b c^{6} d^{2} e^{2} + 3240 \, b c^{4} d e^{3} - 560 \, b c^{2} e^{4}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} + 1}}{99225 \, c^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.2907, size = 593, normalized size = 1.9 \begin{align*} \begin{cases} a d^{4} x + \frac{4 a d^{3} e x^{3}}{3} + \frac{6 a d^{2} e^{2} x^{5}}{5} + \frac{4 a d e^{3} x^{7}}{7} + \frac{a e^{4} x^{9}}{9} + b d^{4} x \operatorname{asinh}{\left (c x \right )} + \frac{4 b d^{3} e x^{3} \operatorname{asinh}{\left (c x \right )}}{3} + \frac{6 b d^{2} e^{2} x^{5} \operatorname{asinh}{\left (c x \right )}}{5} + \frac{4 b d e^{3} x^{7} \operatorname{asinh}{\left (c x \right )}}{7} + \frac{b e^{4} x^{9} \operatorname{asinh}{\left (c x \right )}}{9} - \frac{b d^{4} \sqrt{c^{2} x^{2} + 1}}{c} - \frac{4 b d^{3} e x^{2} \sqrt{c^{2} x^{2} + 1}}{9 c} - \frac{6 b d^{2} e^{2} x^{4} \sqrt{c^{2} x^{2} + 1}}{25 c} - \frac{4 b d e^{3} x^{6} \sqrt{c^{2} x^{2} + 1}}{49 c} - \frac{b e^{4} x^{8} \sqrt{c^{2} x^{2} + 1}}{81 c} + \frac{8 b d^{3} e \sqrt{c^{2} x^{2} + 1}}{9 c^{3}} + \frac{8 b d^{2} e^{2} x^{2} \sqrt{c^{2} x^{2} + 1}}{25 c^{3}} + \frac{24 b d e^{3} x^{4} \sqrt{c^{2} x^{2} + 1}}{245 c^{3}} + \frac{8 b e^{4} x^{6} \sqrt{c^{2} x^{2} + 1}}{567 c^{3}} - \frac{16 b d^{2} e^{2} \sqrt{c^{2} x^{2} + 1}}{25 c^{5}} - \frac{32 b d e^{3} x^{2} \sqrt{c^{2} x^{2} + 1}}{245 c^{5}} - \frac{16 b e^{4} x^{4} \sqrt{c^{2} x^{2} + 1}}{945 c^{5}} + \frac{64 b d e^{3} \sqrt{c^{2} x^{2} + 1}}{245 c^{7}} + \frac{64 b e^{4} x^{2} \sqrt{c^{2} x^{2} + 1}}{2835 c^{7}} - \frac{128 b e^{4} \sqrt{c^{2} x^{2} + 1}}{2835 c^{9}} & \text{for}\: c \neq 0 \\a \left (d^{4} x + \frac{4 d^{3} e x^{3}}{3} + \frac{6 d^{2} e^{2} x^{5}}{5} + \frac{4 d e^{3} x^{7}}{7} + \frac{e^{4} x^{9}}{9}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.84178, size = 547, normalized size = 1.75 \begin{align*}{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{c}\right )} b d^{4} + a d^{4} x + \frac{1}{2835} \,{\left (315 \, a x^{9} +{\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\right )} e^{4} + \frac{4}{245} \,{\left (35 \, a d x^{7} +{\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 d\right )} e^{3} + \frac{2}{25} \,{\left (15 \, a d^{2} x^{5} +{\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 d^{2}\right )} e^{2} + \frac{4}{9} \,{\left (3 \, a d^{3} x^{3} +{\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}\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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