Optimal. Leaf size=184 \[ \frac {1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {73 b d^2 \sin ^{-1}(c x)}{3072 c^4}-\frac {43 b c d^2 x^5 \sqrt {1-c^2 x^2}}{1152}+\frac {73 b d^2 x^3 \sqrt {1-c^2 x^2}}{4608 c}+\frac {73 b d^2 x \sqrt {1-c^2 x^2}}{3072 c^3}+\frac {1}{64} b c^3 d^2 x^7 \sqrt {1-c^2 x^2} \]
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Rubi [A] time = 0.17, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {266, 43, 4687, 12, 1267, 459, 321, 216} \[ \frac {1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{64} b c^3 d^2 x^7 \sqrt {1-c^2 x^2}-\frac {43 b c d^2 x^5 \sqrt {1-c^2 x^2}}{1152}+\frac {73 b d^2 x^3 \sqrt {1-c^2 x^2}}{4608 c}+\frac {73 b d^2 x \sqrt {1-c^2 x^2}}{3072 c^3}-\frac {73 b d^2 \sin ^{-1}(c x)}{3072 c^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 216
Rule 266
Rule 321
Rule 459
Rule 1267
Rule 4687
Rubi steps
\begin {align*} \int x^3 \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^2 x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{24} \left (b c d^2\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{64} b c^3 d^2 x^7 \sqrt {1-c^2 x^2}+\frac {1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (b d^2\right ) \int \frac {x^4 \left (-48 c^2+43 c^4 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{192 c}\\ &=-\frac {43 b c d^2 x^5 \sqrt {1-c^2 x^2}}{1152}+\frac {1}{64} b c^3 d^2 x^7 \sqrt {1-c^2 x^2}+\frac {1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (73 b c d^2\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{1152}\\ &=\frac {73 b d^2 x^3 \sqrt {1-c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1-c^2 x^2}}{1152}+\frac {1}{64} b c^3 d^2 x^7 \sqrt {1-c^2 x^2}+\frac {1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (73 b d^2\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{1536 c}\\ &=\frac {73 b d^2 x \sqrt {1-c^2 x^2}}{3072 c^3}+\frac {73 b d^2 x^3 \sqrt {1-c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1-c^2 x^2}}{1152}+\frac {1}{64} b c^3 d^2 x^7 \sqrt {1-c^2 x^2}+\frac {1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (73 b d^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{3072 c^3}\\ &=\frac {73 b d^2 x \sqrt {1-c^2 x^2}}{3072 c^3}+\frac {73 b d^2 x^3 \sqrt {1-c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1-c^2 x^2}}{1152}+\frac {1}{64} b c^3 d^2 x^7 \sqrt {1-c^2 x^2}-\frac {73 b d^2 \sin ^{-1}(c x)}{3072 c^4}+\frac {1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 115, normalized size = 0.62 \[ \frac {d^2 \left (384 a c^4 x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )+3 b \left (384 c^8 x^8-1024 c^6 x^6+768 c^4 x^4-73\right ) \sin ^{-1}(c x)+b c x \sqrt {1-c^2 x^2} \left (144 c^6 x^6-344 c^4 x^4+146 c^2 x^2+219\right )\right )}{9216 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 149, normalized size = 0.81 \[ \frac {1152 \, a c^{8} d^{2} x^{8} - 3072 \, a c^{6} d^{2} x^{6} + 2304 \, a c^{4} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} d^{2} x^{8} - 1024 \, b c^{6} d^{2} x^{6} + 768 \, b c^{4} d^{2} x^{4} - 73 \, b d^{2}\right )} \arcsin \left (c x\right ) + {\left (144 \, b c^{7} d^{2} x^{7} - 344 \, b c^{5} d^{2} x^{5} + 146 \, b c^{3} d^{2} x^{3} + 219 \, b c d^{2} x\right )} \sqrt {-c^{2} x^{2} + 1}}{9216 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 205, normalized size = 1.11 \[ \frac {1}{8} \, a c^{4} d^{2} x^{8} - \frac {1}{3} \, a c^{2} d^{2} x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{2} x}{64 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b d^{2} \arcsin \left (c x\right )}{8 \, c^{4}} + \frac {11 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2} x}{1152 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d^{2} \arcsin \left (c x\right )}{6 \, c^{4}} + \frac {55 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} x}{4608 \, c^{3}} + \frac {55 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} x}{3072 \, c^{3}} + \frac {55 \, b d^{2} \arcsin \left (c x\right )}{3072 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 160, normalized size = 0.87 \[ \frac {d^{2} a \left (\frac {1}{8} c^{8} x^{8}-\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{1152}+\frac {73 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {-c^{2} x^{2}+1}}{3072}-\frac {73 \arcsin \left (c x \right )}{3072}\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 298, normalized size = 1.62 \[ \frac {1}{8} \, a c^{4} d^{2} x^{8} - \frac {1}{3} \, a c^{2} d^{2} x^{6} + \frac {1}{3072} \, {\left (384 \, x^{8} \arcsin \left (c x\right ) + {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{4} \, a d^{2} x^{4} - \frac {1}{144} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} \]
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 \[ \int x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.59, size = 218, normalized size = 1.18 \[ \begin {cases} \frac {a c^{4} d^{2} x^{8}}{8} - \frac {a c^{2} d^{2} x^{6}}{3} + \frac {a d^{2} x^{4}}{4} + \frac {b c^{4} d^{2} x^{8} \operatorname {asin}{\left (c x \right )}}{8} + \frac {b c^{3} d^{2} x^{7} \sqrt {- c^{2} x^{2} + 1}}{64} - \frac {b c^{2} d^{2} x^{6} \operatorname {asin}{\left (c x \right )}}{3} - \frac {43 b c d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{1152} + \frac {b d^{2} x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {73 b d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{4608 c} + \frac {73 b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{3072 c^{3}} - \frac {73 b d^{2} \operatorname {asin}{\left (c x \right )}}{3072 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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