Optimal. Leaf size=186 \[ \frac {8 b d^2 \sqrt {1-c^2 x^2}}{315 c^5}+\frac {4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{945 c^5}+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2}}{525 c^5}-\frac {10 b d^2 \left (1-c^2 x^2\right )^{7/2}}{441 c^5}+\frac {b d^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^5}+\frac {1}{5} d^2 x^5 (a+b \text {ArcSin}(c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \text {ArcSin}(c x))+\frac {1}{9} c^4 d^2 x^9 (a+b \text {ArcSin}(c x)) \]
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Rubi [A]
time = 0.14, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {276, 4777, 12,
1265, 911, 1167} \begin {gather*} \frac {1}{9} c^4 d^2 x^9 (a+b \text {ArcSin}(c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \text {ArcSin}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {ArcSin}(c x))+\frac {b d^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^5}-\frac {10 b d^2 \left (1-c^2 x^2\right )^{7/2}}{441 c^5}+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2}}{525 c^5}+\frac {4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{945 c^5}+\frac {8 b d^2 \sqrt {1-c^2 x^2}}{315 c^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 276
Rule 911
Rule 1167
Rule 1265
Rule 4777
Rubi steps
\begin {align*} \int x^4 \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^2 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{315} \left (b c d^2\right ) \int \frac {x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{630} \left (b c d^2\right ) \text {Subst}\left (\int \frac {x^2 \left (63-90 c^2 x+35 c^4 x^2\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (b d^2\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {x^2}{c^2}\right )^2 \left (8+20 x^2+35 x^4\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{315 c}\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (b d^2\right ) \text {Subst}\left (\int \left (\frac {8}{c^4}+\frac {4 x^2}{c^4}+\frac {3 x^4}{c^4}-\frac {50 x^6}{c^4}+\frac {35 x^8}{c^4}\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{315 c}\\ &=\frac {8 b d^2 \sqrt {1-c^2 x^2}}{315 c^5}+\frac {4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{945 c^5}+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2}}{525 c^5}-\frac {10 b d^2 \left (1-c^2 x^2\right )^{7/2}}{441 c^5}+\frac {b d^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^5}+\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 119, normalized size = 0.64 \begin {gather*} \frac {d^2 \left (315 a c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )+b \sqrt {1-c^2 x^2} \left (2104+1052 c^2 x^2+789 c^4 x^4-2650 c^6 x^6+1225 c^8 x^8\right )+315 b c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right ) \text {ArcSin}(c x)\right )}{99225 c^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 172, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {d^{2} a \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {263 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}+\frac {2104 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{5}}\) | \(172\) |
default | \(\frac {d^{2} a \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {263 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}+\frac {2104 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{5}}\) | \(172\) |
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. 328 vs.
\(2 (160) = 320\).
time = 0.49, size = 328, normalized size = 1.76 \begin {gather*} \frac {1}{9} \, a c^{4} d^{2} x^{9} - \frac {2}{7} \, a c^{2} d^{2} 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^{4} d^{2} + \frac {1}{5} \, a d^{2} x^{5} - \frac {2}{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^{2} d^{2} + \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 d^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.79, size = 153, normalized size = 0.82 \begin {gather*} \frac {11025 \, a c^{9} d^{2} x^{9} - 28350 \, a c^{7} d^{2} x^{7} + 19845 \, a c^{5} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} d^{2} x^{9} - 90 \, b c^{7} d^{2} x^{7} + 63 \, b c^{5} d^{2} x^{5}\right )} \arcsin \left (c x\right ) + {\left (1225 \, b c^{8} d^{2} x^{8} - 2650 \, b c^{6} d^{2} x^{6} + 789 \, b c^{4} d^{2} x^{4} + 1052 \, b c^{2} d^{2} x^{2} + 2104 \, b d^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{99225 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.51, size = 230, normalized size = 1.24 \begin {gather*} \begin {cases} \frac {a c^{4} d^{2} x^{9}}{9} - \frac {2 a c^{2} d^{2} x^{7}}{7} + \frac {a d^{2} x^{5}}{5} + \frac {b c^{4} d^{2} x^{9} \operatorname {asin}{\left (c x \right )}}{9} + \frac {b c^{3} d^{2} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81} - \frac {2 b c^{2} d^{2} x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {106 b c d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{3969} + \frac {b d^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {263 b d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{33075 c} + \frac {1052 b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{3}} + \frac {2104 b d^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{5}}{5} & \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 = 284, normalized size = 1.53 \begin {gather*} \frac {1}{9} \, a c^{4} d^{2} x^{9} - \frac {2}{7} \, a c^{2} d^{2} x^{7} + \frac {1}{5} \, a d^{2} x^{5} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b d^{2} x \arcsin \left (c x\right )}{9 \, c^{4}} + \frac {10 \, {\left (c^{2} x^{2} - 1\right )}^{3} b d^{2} x \arcsin \left (c x\right )}{63 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} x \arcsin \left (c x\right )}{105 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{81 \, c^{5}} - \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right )}{315 \, c^{4}} + \frac {10 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{441 \, c^{5}} + \frac {8 \, b d^{2} x \arcsin \left (c x\right )}{315 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{525 \, c^{5}} + \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2}}{945 \, c^{5}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d^{2}}{315 \, c^{5}} \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 = -0.01 \begin {gather*} \int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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