Integrand size = 19, antiderivative size = 120 \[ \int x^2 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=-\frac {b \left (5 c^2 d+3 e\right ) \sqrt {1-c^2 x^2}}{15 c^5}+\frac {b \left (5 c^2 d+6 e\right ) \left (1-c^2 x^2\right )^{3/2}}{45 c^5}-\frac {b e \left (1-c^2 x^2\right )^{5/2}}{25 c^5}+\frac {1}{3} d x^3 (a+b \arccos (c x))+\frac {1}{5} e x^5 (a+b \arccos (c x)) \]
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Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 4816, 12, 457, 78} \[ \int x^2 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {1}{3} d x^3 (a+b \arccos (c x))+\frac {1}{5} e x^5 (a+b \arccos (c x))+\frac {b \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+6 e\right )}{45 c^5}-\frac {b \sqrt {1-c^2 x^2} \left (5 c^2 d+3 e\right )}{15 c^5}-\frac {b e \left (1-c^2 x^2\right )^{5/2}}{25 c^5} \]
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Rule 12
Rule 14
Rule 78
Rule 457
Rule 4816
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} d x^3 (a+b \arccos (c x))+\frac {1}{5} e x^5 (a+b \arccos (c x))+(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 (a+b \arccos (c x))+\frac {1}{5} e x^5 (a+b \arccos (c x))+\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 (a+b \arccos (c x))+\frac {1}{5} e x^5 (a+b \arccos (c x))+\frac {1}{30} (b c) \text {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 (a+b \arccos (c x))+\frac {1}{5} e x^5 (a+b \arccos (c x))+\frac {1}{30} (b c) \text {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 (a+b \arccos (c x))+\frac {1}{5} e x^5 (a+b \arccos (c x)) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.04 \[ \int x^2 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {1}{3} a d x^3+\frac {1}{5} a e x^5+b d \left (-\frac {2}{9 c^3}-\frac {x^2}{9 c}\right ) \sqrt {1-c^2 x^2}+b e \sqrt {1-c^2 x^2} \left (-\frac {8}{75 c^5}-\frac {4 x^2}{75 c^3}-\frac {x^4}{25 c}\right )+\frac {1}{3} b d x^3 \arccos (c x)+\frac {1}{5} b e x^5 \arccos (c x) \]
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Time = 0.76 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.28
method | result | size |
parts | \(a \left (\frac {1}{5} e \,x^{5}+\frac {1}{3} d \,x^{3}\right )+\frac {b \left (\frac {c^{3} \arccos \left (c x \right ) x^{5} e}{5}+\frac {\arccos \left (c x \right ) c^{3} x^{3} d}{3}+\frac {3 e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )+5 d \,c^{2} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{15 c^{2}}\right )}{c^{3}}\) | \(154\) |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \left (\frac {\arccos \left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\arccos \left (c x \right ) e \,c^{5} x^{5}}{5}+\frac {e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+\frac {d \,c^{2} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{2}}}{c^{3}}\) | \(161\) |
default | \(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \left (\frac {\arccos \left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\arccos \left (c x \right ) e \,c^{5} x^{5}}{5}+\frac {e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+\frac {d \,c^{2} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{2}}}{c^{3}}\) | \(161\) |
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Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.90 \[ \int x^2 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {45 \, a c^{5} e x^{5} + 75 \, a c^{5} d x^{3} + 15 \, {\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3}\right )} \arccos \left (c x\right ) - {\left (9 \, b c^{4} e x^{4} + 50 \, b c^{2} d + {\left (25 \, b c^{4} d + 12 \, b c^{2} e\right )} x^{2} + 24 \, b e\right )} \sqrt {-c^{2} x^{2} + 1}}{225 \, c^{5}} \]
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Time = 0.43 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.48 \[ \int x^2 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\begin {cases} \frac {a d x^{3}}{3} + \frac {a e x^{5}}{5} + \frac {b d x^{3} \operatorname {acos}{\left (c x \right )}}{3} + \frac {b e x^{5} \operatorname {acos}{\left (c x \right )}}{5} - \frac {b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {b e x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} - \frac {2 b d \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} - \frac {4 b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} - \frac {8 b e \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right ) \left (\frac {d x^{3}}{3} + \frac {e x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.20 \[ \int x^2 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {1}{5} \, a e x^{5} + \frac {1}{3} \, a d x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d + \frac {1}{75} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b e \]
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Time = 0.29 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.18 \[ \int x^2 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {1}{5} \, b e x^{5} \arccos \left (c x\right ) + \frac {1}{5} \, a e x^{5} + \frac {1}{3} \, b d x^{3} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b e x^{4}}{25 \, c} + \frac {1}{3} \, a d x^{3} - \frac {\sqrt {-c^{2} x^{2} + 1} b d x^{2}}{9 \, c} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b e x^{2}}{75 \, c^{3}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d}{9 \, c^{3}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b e}{75 \, c^{5}} \]
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Timed out. \[ \int x^2 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
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