Optimal. Leaf size=84 \[ \frac {1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac {b \left (1-c^2 x^4\right )^{5/2}}{50 c^5}-\frac {b \left (1-c^2 x^4\right )^{3/2}}{15 c^5}+\frac {b \sqrt {1-c^2 x^4}}{10 c^5} \]
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Rubi [A] time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4842, 12, 266, 43} \[ \frac {1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac {b \left (1-c^2 x^4\right )^{5/2}}{50 c^5}-\frac {b \left (1-c^2 x^4\right )^{3/2}}{15 c^5}+\frac {b \sqrt {1-c^2 x^4}}{10 c^5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 4842
Rubi steps
\begin {align*} \int x^9 \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{10} b \int \frac {2 c x^{11}}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{5} (b c) \int \frac {x^{11}}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{20} (b c) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x}} \, dx,x,x^4\right )\\ &=\frac {1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{20} (b c) \operatorname {Subst}\left (\int \left (\frac {1}{c^4 \sqrt {1-c^2 x}}-\frac {2 \sqrt {1-c^2 x}}{c^4}+\frac {\left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^4\right )\\ &=\frac {b \sqrt {1-c^2 x^4}}{10 c^5}-\frac {b \left (1-c^2 x^4\right )^{3/2}}{15 c^5}+\frac {b \left (1-c^2 x^4\right )^{5/2}}{50 c^5}+\frac {1}{10} x^{10} \left (a+b \sin ^{-1}\left (c x^2\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 60, normalized size = 0.71 \[ \frac {1}{150} \left (15 a x^{10}+\frac {b \sqrt {1-c^2 x^4} \left (3 c^4 x^8+4 c^2 x^4+8\right )}{c^5}+15 b x^{10} \sin ^{-1}\left (c x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 65, normalized size = 0.77 \[ \frac {15 \, b c^{5} x^{10} \arcsin \left (c x^{2}\right ) + 15 \, a c^{5} x^{10} + {\left (3 \, b c^{4} x^{8} + 4 \, b c^{2} x^{4} + 8 \, b\right )} \sqrt {-c^{2} x^{4} + 1}}{150 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 140, normalized size = 1.67 \[ \frac {15 \, a c x^{10} + {\left (\frac {15 \, {\left (c^{2} x^{4} - 1\right )}^{2} x^{2} \arcsin \left (c x^{2}\right )}{c^{3}} + \frac {30 \, {\left (c^{2} x^{4} - 1\right )} x^{2} \arcsin \left (c x^{2}\right )}{c^{3}} + \frac {15 \, x^{2} \arcsin \left (c x^{2}\right )}{c^{3}} + \frac {3 \, {\left (c^{2} x^{4} - 1\right )}^{2} \sqrt {-c^{2} x^{4} + 1}}{c^{4}} - \frac {10 \, {\left (-c^{2} x^{4} + 1\right )}^{\frac {3}{2}}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{4} + 1}}{c^{4}}\right )} b}{150 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 71, normalized size = 0.85 \[ \frac {x^{10} a}{10}+b \left (\frac {x^{10} \arcsin \left (c \,x^{2}\right )}{10}-\frac {\left (c \,x^{2}-1\right ) \left (c \,x^{2}+1\right ) \left (3 c^{4} x^{8}+4 c^{2} x^{4}+8\right )}{150 c^{5} \sqrt {-c^{2} x^{4}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 76, normalized size = 0.90 \[ \frac {1}{10} \, a x^{10} + \frac {1}{150} \, {\left (15 \, x^{10} \arcsin \left (c x^{2}\right ) + c {\left (\frac {3 \, {\left (-c^{2} x^{4} + 1\right )}^{\frac {5}{2}}}{c^{6}} - \frac {10 \, {\left (-c^{2} x^{4} + 1\right )}^{\frac {3}{2}}}{c^{6}} + \frac {15 \, \sqrt {-c^{2} x^{4} + 1}}{c^{6}}\right )}\right )} b \]
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 \[ \int x^9\,\left (a+b\,\mathrm {asin}\left (c\,x^2\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.77, size = 90, normalized size = 1.07 \[ \begin {cases} \frac {a x^{10}}{10} + \frac {b x^{10} \operatorname {asin}{\left (c x^{2} \right )}}{10} + \frac {b x^{8} \sqrt {- c^{2} x^{4} + 1}}{50 c} + \frac {2 b x^{4} \sqrt {- c^{2} x^{4} + 1}}{75 c^{3}} + \frac {4 b \sqrt {- c^{2} x^{4} + 1}}{75 c^{5}} & \text {for}\: c \neq 0 \\\frac {a x^{10}}{10} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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