Optimal. Leaf size=120 \[ \frac{1}{3} d x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \cos ^{-1}(c x)\right )+\frac{b \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+6 e\right )}{45 c^5}-\frac{b \sqrt{1-c^2 x^2} \left (5 c^2 d+3 e\right )}{15 c^5}-\frac{b e \left (1-c^2 x^2\right )^{5/2}}{25 c^5} \]
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Rubi [A] time = 0.12064, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {14, 4732, 12, 446, 77} \[ \frac{1}{3} d x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \cos ^{-1}(c x)\right )+\frac{b \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+6 e\right )}{45 c^5}-\frac{b \sqrt{1-c^2 x^2} \left (5 c^2 d+3 e\right )}{15 c^5}-\frac{b e \left (1-c^2 x^2\right )^{5/2}}{25 c^5} \]
Antiderivative was successfully verified.
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Rule 14
Rule 4732
Rule 12
Rule 446
Rule 77
Rubi steps
\begin{align*} \int x^2 \left (d+e x^2\right ) \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \cos ^{-1}(c x)\right )+(b c) \int \frac{x^3 \left (5 d+3 e x^2\right )}{15 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{3} d x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{15} (b c) \int \frac{x^3 \left (5 d+3 e x^2\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{3} d x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{30} (b c) \operatorname{Subst}\left (\int \frac{x (5 d+3 e x)}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} d x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{30} (b c) \operatorname{Subst}\left (\int \left (\frac{5 c^2 d+3 e}{c^4 \sqrt{1-c^2 x}}+\frac{\left (-5 c^2 d-6 e\right ) \sqrt{1-c^2 x}}{c^4}+\frac{3 e \left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{b \left (5 c^2 d+3 e\right ) \sqrt{1-c^2 x^2}}{15 c^5}+\frac{b \left (5 c^2 d+6 e\right ) \left (1-c^2 x^2\right )^{3/2}}{45 c^5}-\frac{b e \left (1-c^2 x^2\right )^{5/2}}{25 c^5}+\frac{1}{3} d x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \cos ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.102891, size = 125, normalized size = 1.04 \[ \frac{1}{3} a d x^3+\frac{1}{5} a e x^5+b d \left (-\frac{2}{9 c^3}-\frac{x^2}{9 c}\right ) \sqrt{1-c^2 x^2}+b e \sqrt{1-c^2 x^2} \left (-\frac{4 x^2}{75 c^3}-\frac{8}{75 c^5}-\frac{x^4}{25 c}\right )+\frac{1}{3} b d x^3 \cos ^{-1}(c x)+\frac{1}{5} b e x^5 \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 161, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{5}{x}^{5}}{5}}+{\frac{{c}^{5}d{x}^{3}}{3}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{\arccos \left ( cx \right ) e{c}^{5}{x}^{5}}{5}}+{\frac{\arccos \left ( cx \right ) d{c}^{5}{x}^{3}}{3}}+{\frac{e}{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{{c}^{2}d}{3} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.75077, size = 194, normalized size = 1.62 \begin{align*} \frac{1}{5} \, a e x^{5} + \frac{1}{3} \, a d x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \arccos \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 + \frac{1}{75} \,{\left (15 \, x^{5} \arccos \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23938, size = 250, normalized size = 2.08 \begin{align*} \frac{45 \, a c^{5} e x^{5} + 75 \, a c^{5} d x^{3} + 15 \,{\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3}\right )} \arccos \left (c x\right ) -{\left (9 \, b c^{4} e x^{4} + 50 \, b c^{2} d +{\left (25 \, b c^{4} d + 12 \, b c^{2} e\right )} x^{2} + 24 \, b e\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 = 2.63005, size = 177, normalized size = 1.48 \begin{align*} \begin{cases} \frac{a d x^{3}}{3} + \frac{a e x^{5}}{5} + \frac{b d x^{3} \operatorname{acos}{\left (c x \right )}}{3} + \frac{b e x^{5} \operatorname{acos}{\left (c x \right )}}{5} - \frac{b d x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} - \frac{b e x^{4} \sqrt{- c^{2} x^{2} + 1}}{25 c} - \frac{2 b d \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} - \frac{4 b e x^{2} \sqrt{- c^{2} x^{2} + 1}}{75 c^{3}} - \frac{8 b e \sqrt{- c^{2} x^{2} + 1}}{75 c^{5}} & \text{for}\: c \neq 0 \\\left (a + \frac{\pi b}{2}\right ) \left (\frac{d x^{3}}{3} + \frac{e 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 = 1.18421, size = 197, normalized size = 1.64 \begin{align*} \frac{1}{5} \, b x^{5} \arccos \left (c x\right ) e + \frac{1}{5} \, a x^{5} e + \frac{1}{3} \, b d x^{3} \arccos \left (c x\right ) - \frac{\sqrt{-c^{2} x^{2} + 1} b x^{4} e}{25 \, c} + \frac{1}{3} \, a d x^{3} - \frac{\sqrt{-c^{2} x^{2} + 1} b d x^{2}}{9 \, c} - \frac{4 \, \sqrt{-c^{2} x^{2} + 1} b x^{2} e}{75 \, c^{3}} - \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b d}{9 \, c^{3}} - \frac{8 \, \sqrt{-c^{2} x^{2} + 1} b e}{75 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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