Optimal. Leaf size=183 \[ \frac{\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}+\frac{b x \sqrt{1-c^2 x^2} \left (44 c^4 d^2+44 c^2 d e+15 e^2\right )}{288 c^5}-\frac{b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sin ^{-1}(c x)}{96 c^6 e}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}+\frac{5 b x \sqrt{1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{144 c^3} \]
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Rubi [A] time = 0.175834, antiderivative size = 183, 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 = {4729, 416, 528, 388, 216} \[ \frac{\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}+\frac{b x \sqrt{1-c^2 x^2} \left (44 c^4 d^2+44 c^2 d e+15 e^2\right )}{288 c^5}-\frac{b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sin ^{-1}(c x)}{96 c^6 e}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}+\frac{5 b x \sqrt{1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{144 c^3} \]
Antiderivative was successfully verified.
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Rule 4729
Rule 416
Rule 528
Rule 388
Rule 216
Rubi steps
\begin{align*} \int x \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}-\frac{(b c) \int \frac{\left (d+e x^2\right )^3}{\sqrt{1-c^2 x^2}} \, dx}{6 e}\\ &=\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}+\frac{\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}+\frac{b \int \frac{\left (d+e x^2\right ) \left (-d \left (6 c^2 d+e\right )-5 e \left (2 c^2 d+e\right ) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{36 c e}\\ &=\frac{5 b \left (2 c^2 d+e\right ) x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{144 c^3}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}+\frac{\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}-\frac{b \int \frac{d \left (24 c^4 d^2+14 c^2 d e+5 e^2\right )+e \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x^2}{\sqrt{1-c^2 x^2}} \, dx}{144 c^3 e}\\ &=\frac{b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \sqrt{1-c^2 x^2}}{288 c^5}+\frac{5 b \left (2 c^2 d+e\right ) x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{144 c^3}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}+\frac{\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}-\frac{\left (b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{96 c^5 e}\\ &=\frac{b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \sqrt{1-c^2 x^2}}{288 c^5}+\frac{5 b \left (2 c^2 d+e\right ) x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{144 c^3}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2}{36 c}-\frac{b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sin ^{-1}(c x)}{96 c^6 e}+\frac{\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{6 e}\\ \end{align*}
Mathematica [A] time = 0.14344, size = 159, normalized size = 0.87 \[ \frac{c x \left (48 a c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )+b \sqrt{1-c^2 x^2} \left (4 c^4 \left (18 d^2+9 d e x^2+2 e^2 x^4\right )+2 c^2 e \left (27 d+5 e x^2\right )+15 e^2\right )\right )+3 b \sin ^{-1}(c x) \left (16 c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )-24 c^4 d^2-18 c^2 d e-5 e^2\right )}{288 c^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 243, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{6}{x}^{6}}{6}}+{\frac{{c}^{6}ed{x}^{4}}{2}}+{\frac{{x}^{2}{c}^{6}{d}^{2}}{2}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{\arcsin \left ( cx \right ){e}^{2}{c}^{6}{x}^{6}}{6}}+{\frac{\arcsin \left ( cx \right ){c}^{6}ed{x}^{4}}{2}}+{\frac{\arcsin \left ( cx \right ){d}^{2}{c}^{6}{x}^{2}}{2}}-{\frac{{e}^{2}}{6} \left ( -{\frac{{c}^{5}{x}^{5}}{6}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5\,{c}^{3}{x}^{3}}{24}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5\,cx}{16}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{5\,\arcsin \left ( cx \right ) }{16}} \right ) }-{\frac{{c}^{2}ed}{2} \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{{d}^{2}{c}^{4}}{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 = 1.46077, size = 350, normalized size = 1.91 \begin{align*} \frac{1}{6} \, a e^{2} x^{6} + \frac{1}{2} \, a d e x^{4} + \frac{1}{2} \, a d^{2} x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \arcsin \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^{2} + \frac{1}{16} \,{\left (8 \, x^{4} \arcsin \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 d e + \frac{1}{288} \,{\left (48 \, x^{6} \arcsin \left (c x\right ) +{\left (\frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac{10 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{6}} - \frac{15 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11304, size = 414, normalized size = 2.26 \begin{align*} \frac{48 \, a c^{6} e^{2} x^{6} + 144 \, a c^{6} d e x^{4} + 144 \, a c^{6} d^{2} x^{2} + 3 \,{\left (16 \, b c^{6} e^{2} x^{6} + 48 \, b c^{6} d e x^{4} + 48 \, b c^{6} d^{2} x^{2} - 24 \, b c^{4} d^{2} - 18 \, b c^{2} d e - 5 \, b e^{2}\right )} \arcsin \left (c x\right ) +{\left (8 \, b c^{5} e^{2} x^{5} + 2 \,{\left (18 \, b c^{5} d e + 5 \, b c^{3} e^{2}\right )} x^{3} + 3 \,{\left (24 \, b c^{5} d^{2} + 18 \, b c^{3} d e + 5 \, b c e^{2}\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{288 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.86242, size = 299, normalized size = 1.63 \begin{align*} \begin{cases} \frac{a d^{2} x^{2}}{2} + \frac{a d e x^{4}}{2} + \frac{a e^{2} x^{6}}{6} + \frac{b d^{2} x^{2} \operatorname{asin}{\left (c x \right )}}{2} + \frac{b d e x^{4} \operatorname{asin}{\left (c x \right )}}{2} + \frac{b e^{2} x^{6} \operatorname{asin}{\left (c x \right )}}{6} + \frac{b d^{2} x \sqrt{- c^{2} x^{2} + 1}}{4 c} + \frac{b d e x^{3} \sqrt{- c^{2} x^{2} + 1}}{8 c} + \frac{b e^{2} x^{5} \sqrt{- c^{2} x^{2} + 1}}{36 c} - \frac{b d^{2} \operatorname{asin}{\left (c x \right )}}{4 c^{2}} + \frac{3 b d e x \sqrt{- c^{2} x^{2} + 1}}{16 c^{3}} + \frac{5 b e^{2} x^{3} \sqrt{- c^{2} x^{2} + 1}}{144 c^{3}} - \frac{3 b d e \operatorname{asin}{\left (c x \right )}}{16 c^{4}} + \frac{5 b e^{2} x \sqrt{- c^{2} x^{2} + 1}}{96 c^{5}} - \frac{5 b e^{2} \operatorname{asin}{\left (c x \right )}}{96 c^{6}} & \text{for}\: c \neq 0 \\a \left (\frac{d^{2} x^{2}}{2} + \frac{d e x^{4}}{2} + \frac{e^{2} x^{6}}{6}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.35075, size = 571, normalized size = 3.12 \begin{align*} \frac{\sqrt{-c^{2} x^{2} + 1} b d^{2} x}{4 \, c} + \frac{{\left (c^{2} x^{2} - 1\right )} b d^{2} \arcsin \left (c x\right )}{2 \, c^{2}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d x e}{8 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )} a d^{2}}{2 \, c^{2}} + \frac{b d^{2} \arcsin \left (c x\right )}{4 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b d \arcsin \left (c x\right ) e}{2 \, c^{4}} + \frac{5 \, \sqrt{-c^{2} x^{2} + 1} b d x e}{16 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a d e}{2 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )} b d \arcsin \left (c x\right ) e}{c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b x e^{2}}{36 \, c^{5}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b \arcsin \left (c x\right ) e^{2}}{6 \, c^{6}} + \frac{{\left (c^{2} x^{2} - 1\right )} a d e}{c^{4}} + \frac{5 \, b d \arcsin \left (c x\right ) e}{16 \, c^{4}} - \frac{13 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b x e^{2}}{144 \, c^{5}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} a e^{2}}{6 \, c^{6}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b \arcsin \left (c x\right ) e^{2}}{2 \, c^{6}} + \frac{11 \, \sqrt{-c^{2} x^{2} + 1} b x e^{2}}{96 \, c^{5}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a e^{2}}{2 \, c^{6}} + \frac{{\left (c^{2} x^{2} - 1\right )} b \arcsin \left (c x\right ) e^{2}}{2 \, c^{6}} + \frac{{\left (c^{2} x^{2} - 1\right )} a e^{2}}{2 \, c^{6}} + \frac{11 \, b \arcsin \left (c x\right ) e^{2}}{96 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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