Optimal. Leaf size=122 \[ \frac{\left (d+e x^2\right )^2 \left (a+b \cos ^{-1}(c x)\right )}{4 e}+\frac{b \left (8 c^4 d^2+8 c^2 d e+3 e^2\right ) \sin ^{-1}(c x)}{32 c^4 e}-\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{16 c}-\frac{3 b x \sqrt{1-c^2 x^2} \left (2 c^2 d+e\right )}{32 c^3} \]
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Rubi [A] time = 0.0858386, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {4730, 416, 388, 216} \[ \frac{\left (d+e x^2\right )^2 \left (a+b \cos ^{-1}(c x)\right )}{4 e}+\frac{b \left (8 c^4 d^2+8 c^2 d e+3 e^2\right ) \sin ^{-1}(c x)}{32 c^4 e}-\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{16 c}-\frac{3 b x \sqrt{1-c^2 x^2} \left (2 c^2 d+e\right )}{32 c^3} \]
Antiderivative was successfully verified.
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Rule 4730
Rule 416
Rule 388
Rule 216
Rubi steps
\begin{align*} \int x \left (d+e x^2\right ) \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^2 \left (a+b \cos ^{-1}(c x)\right )}{4 e}+\frac{(b c) \int \frac{\left (d+e x^2\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{4 e}\\ &=-\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{16 c}+\frac{\left (d+e x^2\right )^2 \left (a+b \cos ^{-1}(c x)\right )}{4 e}-\frac{b \int \frac{-d \left (4 c^2 d+e\right )-3 e \left (2 c^2 d+e\right ) x^2}{\sqrt{1-c^2 x^2}} \, dx}{16 c e}\\ &=-\frac{3 b \left (2 c^2 d+e\right ) x \sqrt{1-c^2 x^2}}{32 c^3}-\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{16 c}+\frac{\left (d+e x^2\right )^2 \left (a+b \cos ^{-1}(c x)\right )}{4 e}+\frac{\left (b \left (8 c^4 d^2+8 c^2 d e+3 e^2\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{32 c^3 e}\\ &=-\frac{3 b \left (2 c^2 d+e\right ) x \sqrt{1-c^2 x^2}}{32 c^3}-\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{16 c}+\frac{\left (d+e x^2\right )^2 \left (a+b \cos ^{-1}(c x)\right )}{4 e}+\frac{b \left (8 c^4 d^2+8 c^2 d e+3 e^2\right ) \sin ^{-1}(c x)}{32 c^4 e}\\ \end{align*}
Mathematica [A] time = 0.103258, size = 131, normalized size = 1.07 \[ \frac{1}{2} a d x^2+\frac{1}{4} a e x^4-\frac{b d x \sqrt{1-c^2 x^2}}{4 c}+\frac{b d \sin ^{-1}(c x)}{4 c^2}+b e \sqrt{1-c^2 x^2} \left (-\frac{3 x}{32 c^3}-\frac{x^3}{16 c}\right )+\frac{3 b e \sin ^{-1}(c x)}{32 c^4}+\frac{1}{2} b d x^2 \cos ^{-1}(c x)+\frac{1}{4} b e x^4 \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 137, normalized size = 1.1 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{4}{x}^{4}}{4}}+{\frac{{x}^{2}{c}^{4}d}{2}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{\arccos \left ( cx \right ) e{c}^{4}{x}^{4}}{4}}+{\frac{\arccos \left ( cx \right ) d{c}^{4}{x}^{2}}{2}}+{\frac{e}{4} \left ( -{\frac{{c}^{3}{x}^{3}}{4}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,cx}{8}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{3\,\arcsin \left ( cx \right ) }{8}} \right ) }+{\frac{{c}^{2}d}{2} \left ( -{\frac{cx}{2}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{\arcsin \left ( cx \right ) }{2}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74605, size = 200, normalized size = 1.64 \begin{align*} \frac{1}{4} \, a e x^{4} + \frac{1}{2} \, a d x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \arccos \left (c x\right ) - c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d + \frac{1}{32} \,{\left (8 \, x^{4} \arccos \left (c x\right ) -{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13934, size = 234, normalized size = 1.92 \begin{align*} \frac{8 \, a c^{4} e x^{4} + 16 \, a c^{4} d x^{2} +{\left (8 \, b c^{4} e x^{4} + 16 \, b c^{4} d x^{2} - 8 \, b c^{2} d - 3 \, b e\right )} \arccos \left (c x\right ) -{\left (2 \, b c^{3} e x^{3} +{\left (8 \, b c^{3} d + 3 \, b c e\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{32 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.53018, size = 158, normalized size = 1.3 \begin{align*} \begin{cases} \frac{a d x^{2}}{2} + \frac{a e x^{4}}{4} + \frac{b d x^{2} \operatorname{acos}{\left (c x \right )}}{2} + \frac{b e x^{4} \operatorname{acos}{\left (c x \right )}}{4} - \frac{b d x \sqrt{- c^{2} x^{2} + 1}}{4 c} - \frac{b e x^{3} \sqrt{- c^{2} x^{2} + 1}}{16 c} - \frac{b d \operatorname{acos}{\left (c x \right )}}{4 c^{2}} - \frac{3 b e x \sqrt{- c^{2} x^{2} + 1}}{32 c^{3}} - \frac{3 b e \operatorname{acos}{\left (c x \right )}}{32 c^{4}} & \text{for}\: c \neq 0 \\\left (a + \frac{\pi b}{2}\right ) \left (\frac{d x^{2}}{2} + \frac{e x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18315, size = 170, normalized size = 1.39 \begin{align*} \frac{1}{4} \, b x^{4} \arccos \left (c x\right ) e + \frac{1}{4} \, a x^{4} e + \frac{1}{2} \, b d x^{2} \arccos \left (c x\right ) - \frac{\sqrt{-c^{2} x^{2} + 1} b x^{3} e}{16 \, c} + \frac{1}{2} \, a d x^{2} - \frac{\sqrt{-c^{2} x^{2} + 1} b d x}{4 \, c} - \frac{b d \arccos \left (c x\right )}{4 \, c^{2}} - \frac{3 \, \sqrt{-c^{2} x^{2} + 1} b x e}{32 \, c^{3}} - \frac{3 \, b \arccos \left (c x\right ) e}{32 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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