Optimal. Leaf size=207 \[ \frac {16 b d^3 \sqrt {1-c^2 x^2}}{315 c^3}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{945 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2}}{525 c^3}+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{441 c^3}-\frac {b d^3 \left (1-c^2 x^2\right )^{9/2}}{81 c^3}+\frac {1}{3} d^3 x^3 (a+b \text {ArcSin}(c x))-\frac {3}{5} c^2 d^3 x^5 (a+b \text {ArcSin}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {ArcSin}(c x))-\frac {1}{9} c^6 d^3 x^9 (a+b \text {ArcSin}(c x)) \]
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Rubi [A]
time = 0.17, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {276, 4777, 12,
1813, 1634} \begin {gather*} -\frac {1}{9} c^6 d^3 x^9 (a+b \text {ArcSin}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {ArcSin}(c x))-\frac {3}{5} c^2 d^3 x^5 (a+b \text {ArcSin}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {ArcSin}(c x))-\frac {b d^3 \left (1-c^2 x^2\right )^{9/2}}{81 c^3}+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{441 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2}}{525 c^3}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{945 c^3}+\frac {16 b d^3 \sqrt {1-c^2 x^2}}{315 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 276
Rule 1634
Rule 1813
Rule 4777
Rubi steps
\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^3 x^3 \left (105-189 c^2 x^2+135 c^4 x^4-35 c^6 x^6\right )}{315 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{315} \left (b c d^3\right ) \int \frac {x^3 \left (105-189 c^2 x^2+135 c^4 x^4-35 c^6 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{630} \left (b c d^3\right ) \text {Subst}\left (\int \frac {x \left (105-189 c^2 x+135 c^4 x^2-35 c^6 x^3\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{630} \left (b c d^3\right ) \text {Subst}\left (\int \left (\frac {16}{c^2 \sqrt {1-c^2 x}}+\frac {8 \sqrt {1-c^2 x}}{c^2}+\frac {6 \left (1-c^2 x\right )^{3/2}}{c^2}+\frac {5 \left (1-c^2 x\right )^{5/2}}{c^2}-\frac {35 \left (1-c^2 x\right )^{7/2}}{c^2}\right ) \, dx,x,x^2\right )\\ &=\frac {16 b d^3 \sqrt {1-c^2 x^2}}{315 c^3}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{945 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2}}{525 c^3}+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{441 c^3}-\frac {b d^3 \left (1-c^2 x^2\right )^{9/2}}{81 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 135, normalized size = 0.65 \begin {gather*} \frac {d^3 \left (-315 a c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (5258+2629 c^2 x^2-6297 c^4 x^4+4675 c^6 x^6-1225 c^8 x^8\right )-315 b c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right ) \text {ArcSin}(c x)\right )}{99225 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 194, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {-d^{3} a \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {2099 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {5258 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(194\) |
default | \(\frac {-d^{3} a \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {2099 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {5258 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 398 vs.
\(2 (179) = 358\).
time = 0.48, size = 398, normalized size = 1.92 \begin {gather*} -\frac {1}{9} \, a c^{6} d^{3} x^{9} + \frac {3}{7} \, a c^{4} d^{3} x^{7} - \frac {1}{2835} \, {\left (315 \, x^{9} \arcsin \left (c x\right ) + {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{6} d^{3} - \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {3}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{4} d^{3} - \frac {1}{25} \, {\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 c^{2} d^{3} + \frac {1}{3} \, a d^{3} 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^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.19, size = 177, normalized size = 0.86 \begin {gather*} -\frac {11025 \, a c^{9} d^{3} x^{9} - 42525 \, a c^{7} d^{3} x^{7} + 59535 \, a c^{5} d^{3} x^{5} - 33075 \, a c^{3} d^{3} x^{3} + 315 \, {\left (35 \, b c^{9} d^{3} x^{9} - 135 \, b c^{7} d^{3} x^{7} + 189 \, b c^{5} d^{3} x^{5} - 105 \, b c^{3} d^{3} x^{3}\right )} \arcsin \left (c x\right ) + {\left (1225 \, b c^{8} d^{3} x^{8} - 4675 \, b c^{6} d^{3} x^{6} + 6297 \, b c^{4} d^{3} x^{4} - 2629 \, b c^{2} d^{3} x^{2} - 5258 \, b d^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{99225 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.61, size = 265, normalized size = 1.28 \begin {gather*} \begin {cases} - \frac {a c^{6} d^{3} x^{9}}{9} + \frac {3 a c^{4} d^{3} x^{7}}{7} - \frac {3 a c^{2} d^{3} x^{5}}{5} + \frac {a d^{3} x^{3}}{3} - \frac {b c^{6} d^{3} x^{9} \operatorname {asin}{\left (c x \right )}}{9} - \frac {b c^{5} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81} + \frac {3 b c^{4} d^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {187 b c^{3} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{3969} - \frac {3 b c^{2} d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} - \frac {2099 b c d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{33075} + \frac {b d^{3} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2629 b d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c} + \frac {5258 b d^{3} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 296, normalized size = 1.43 \begin {gather*} -\frac {1}{9} \, a c^{6} d^{3} x^{9} + \frac {3}{7} \, a c^{4} d^{3} x^{7} - \frac {3}{5} \, a c^{2} d^{3} x^{5} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b d^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} + \frac {1}{3} \, a d^{3} x^{3} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d^{3} x \arcsin \left (c x\right )}{63 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{3} x \arcsin \left (c x\right )}{105 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{81 \, c^{3}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )} b d^{3} x \arcsin \left (c x\right )}{315 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{441 \, c^{3}} + \frac {16 \, b d^{3} x \arcsin \left (c x\right )}{315 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{525 \, c^{3}} + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{3}}{945 \, c^{3}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b d^{3}}{315 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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