Optimal. Leaf size=120 \[ \frac {1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+6 e\right )}{45 c^5}+\frac {b \sqrt {1-c^2 x^2} \left (5 c^2 d+3 e\right )}{15 c^5}+\frac {b e \left (1-c^2 x^2\right )^{5/2}}{25 c^5} \]
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Rubi [A] time = 0.13, antiderivative size = 120, normalized size of antiderivative = 1.00, 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, 4731, 12, 446, 77} \[ \frac {1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+6 e\right )}{45 c^5}+\frac {b \sqrt {1-c^2 x^2} \left (5 c^2 d+3 e\right )}{15 c^5}+\frac {b e \left (1-c^2 x^2\right )^{5/2}}{25 c^5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 77
Rule 446
Rule 4731
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \sin ^{-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 \sin ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \sin ^{-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 \sin ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \sin ^{-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 \sin ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \sin ^{-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 \sin ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 96, normalized size = 0.80 \[ \frac {1}{225} \left (15 a x^3 \left (5 d+3 e x^2\right )+\frac {b \sqrt {1-c^2 x^2} \left (c^4 \left (25 d x^2+9 e x^4\right )+2 c^2 \left (25 d+6 e x^2\right )+24 e\right )}{c^5}+15 b x^3 \sin ^{-1}(c x) \left (5 d+3 e x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 107, normalized size = 0.89 \[ \frac {45 \, a c^{5} e x^{5} + 75 \, a c^{5} d x^{3} + 15 \, {\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3}\right )} \arcsin \left (c x\right ) + {\left (9 \, b c^{4} e x^{4} + 50 \, b c^{2} d + {\left (25 \, b c^{4} d + 12 \, b c^{2} e\right )} x^{2} + 24 \, b e\right )} \sqrt {-c^{2} x^{2} + 1}}{225 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 217, normalized size = 1.81 \[ \frac {1}{5} \, a x^{5} e + \frac {1}{3} \, a d x^{3} + \frac {{\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b d x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b x \arcsin \left (c x\right ) e}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right ) e}{5 \, c^{4}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d}{9 \, c^{3}} + \frac {b x \arcsin \left (c x\right ) e}{5 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d}{3 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e}{25 \, c^{5}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e}{15 \, c^{5}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e}{5 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 161, normalized size = 1.34 \[ \frac {\frac {a \left (\frac {1}{5} e \,c^{5} x^{5}+\frac {1}{3} c^{5} d \,x^{3}\right )}{c^{2}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e \,c^{5} x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{5} d \,x^{3}}{3}-\frac {e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {c^{2} d \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{2}}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 142, normalized size = 1.18 \[ \frac {1}{5} \, a e x^{5} + \frac {1}{3} \, a d x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b e \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.05, size = 172, normalized size = 1.43 \[ \begin {cases} \frac {a d x^{3}}{3} + \frac {a e x^{5}}{5} + \frac {b d x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b e x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b e x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {2 b d \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {4 b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {8 b e \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} & \text {for}\: c \neq 0 \\a \left (\frac {d x^{3}}{3} + \frac {e x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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