Optimal. Leaf size=150 \[ d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \left (15 c^4 d^2+10 c^2 d e+3 e^2\right )}{15 c^5}-\frac{2 b e \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+3 e\right )}{45 c^5}+\frac{b e^2 \left (1-c^2 x^2\right )^{5/2}}{25 c^5} \]
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Rubi [A] time = 0.136157, antiderivative size = 150, 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, 4665, 12, 1247, 698} \[ d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \left (15 c^4 d^2+10 c^2 d e+3 e^2\right )}{15 c^5}-\frac{2 b e \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+3 e\right )}{45 c^5}+\frac{b e^2 \left (1-c^2 x^2\right )^{5/2}}{25 c^5} \]
Antiderivative was successfully verified.
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Rule 194
Rule 4665
Rule 12
Rule 1247
Rule 698
Rubi steps
\begin{align*} \int \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{15 \sqrt{1-c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{15} (b c) \int \frac{x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{30} (b c) \operatorname{Subst}\left (\int \frac{15 d^2+10 d e x+3 e^2 x^2}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{30} (b c) \operatorname{Subst}\left (\int \left (\frac{15 c^4 d^2+10 c^2 d e+3 e^2}{c^4 \sqrt{1-c^2 x}}-\frac{2 e \left (5 c^2 d+3 e\right ) \sqrt{1-c^2 x}}{c^4}+\frac{3 e^2 \left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )\\ &=\frac{b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt{1-c^2 x^2}}{15 c^5}-\frac{2 b e \left (5 c^2 d+3 e\right ) \left (1-c^2 x^2\right )^{3/2}}{45 c^5}+\frac{b e^2 \left (1-c^2 x^2\right )^{5/2}}{25 c^5}+d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.151411, size = 125, normalized size = 0.83 \[ \frac{1}{225} \left (15 a x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+\frac{b \sqrt{1-c^2 x^2} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )+4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )}{c^5}+15 b x \sin ^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 209, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{5}{x}^{5}}{5}}+{\frac{2\,{c}^{5}ed{x}^{3}}{3}}+{d}^{2}{c}^{5}x \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{\arcsin \left ( cx \right ){e}^{2}{c}^{5}{x}^{5}}{5}}+{\frac{2\,\arcsin \left ( cx \right ){c}^{5}ed{x}^{3}}{3}}+\arcsin \left ( cx \right ){d}^{2}{c}^{5}x-{\frac{{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{2\,{c}^{2}ed}{3} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }+{d}^{2}{c}^{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 = 1.45678, size = 246, normalized size = 1.64 \begin{align*} \frac{1}{5} \, a e^{2} x^{5} + \frac{2}{3} \, a d e x^{3} + \frac{2}{9} \,{\left (3 \, x^{3} \arcsin \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 e + \frac{1}{75} \,{\left (15 \, x^{5} \arcsin \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 e^{2} + a d^{2} x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d^{2}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0348, size = 348, normalized size = 2.32 \begin{align*} \frac{45 \, a c^{5} e^{2} x^{5} + 150 \, a c^{5} d e x^{3} + 225 \, a c^{5} d^{2} x + 15 \,{\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x\right )} \arcsin \left (c x\right ) +{\left (9 \, b c^{4} e^{2} x^{4} + 225 \, b c^{4} d^{2} + 100 \, b c^{2} d e + 24 \, b e^{2} + 2 \,{\left (25 \, b c^{4} d e + 6 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{-c^{2} x^{2} + 1}}{225 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.85913, size = 240, normalized size = 1.6 \begin{align*} \begin{cases} a d^{2} x + \frac{2 a d e x^{3}}{3} + \frac{a e^{2} x^{5}}{5} + b d^{2} x \operatorname{asin}{\left (c x \right )} + \frac{2 b d e x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{b e^{2} x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{b d^{2} \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{2 b d e x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{b e^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{25 c} + \frac{4 b d e \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{4 b e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{75 c^{3}} + \frac{8 b e^{2} \sqrt{- c^{2} x^{2} + 1}}{75 c^{5}} & \text{for}\: c \neq 0 \\a \left (d^{2} x + \frac{2 d e x^{3}}{3} + \frac{e^{2} x^{5}}{5}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24982, size = 355, normalized size = 2.37 \begin{align*} \frac{1}{5} \, a x^{5} e^{2} + \frac{2}{3} \, a d x^{3} e + b d^{2} x \arcsin \left (c x\right ) + a d^{2} x + \frac{2 \,{\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac{2 \, b d x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d^{2}}{c} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} - \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d e}{9 \, c^{3}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b d e}{3 \, c^{3}} + \frac{b x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b e^{2}}{25 \, c^{5}} - \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b e^{2}}{15 \, c^{5}} + \frac{\sqrt{-c^{2} x^{2} + 1} b e^{2}}{5 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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