Optimal. Leaf size=395 \[ -\frac {4208 b^2 d^2 x}{99225 c^4}-\frac {2104 b^2 d^2 x^3}{297675 c^2}-\frac {526 b^2 d^2 x^5}{165375}+\frac {212 b^2 c^2 d^2 x^7}{27783}-\frac {2}{729} b^2 c^4 d^2 x^9+\frac {128 b d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{4725 c^5}+\frac {64 b d^2 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{4725 c^3}+\frac {16 b d^2 x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{1575 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{189 c^5}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{315 c^5}-\frac {20 b d^2 \left (1-c^2 x^2\right )^{7/2} (a+b \text {ArcSin}(c x))}{441 c^5}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{9/2} (a+b \text {ArcSin}(c x))}{81 c^5}+\frac {8}{315} d^2 x^5 (a+b \text {ArcSin}(c x))^2+\frac {4}{63} d^2 x^5 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2 \]
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Rubi [A]
time = 0.50, antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {4787, 4723,
4795, 4767, 8, 30, 272, 45, 4779, 12, 1167} \begin {gather*} \frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2+\frac {4}{63} d^2 x^5 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2+\frac {16 b d^2 x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{1575 c}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{9/2} (a+b \text {ArcSin}(c x))}{81 c^5}-\frac {20 b d^2 \left (1-c^2 x^2\right )^{7/2} (a+b \text {ArcSin}(c x))}{441 c^5}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{315 c^5}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{189 c^5}+\frac {128 b d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{4725 c^5}+\frac {64 b d^2 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{4725 c^3}+\frac {8}{315} d^2 x^5 (a+b \text {ArcSin}(c x))^2-\frac {2}{729} b^2 c^4 d^2 x^9-\frac {4208 b^2 d^2 x}{99225 c^4}+\frac {212 b^2 c^2 d^2 x^7}{27783}-\frac {2104 b^2 d^2 x^3}{297675 c^2}-\frac {526 b^2 d^2 x^5}{165375} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 30
Rule 45
Rule 272
Rule 1167
Rule 4723
Rule 4767
Rule 4779
Rule 4787
Rule 4795
Rubi steps
\begin {align*} \int x^4 \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} (4 d) \int x^4 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{9} \left (2 b c d^2\right ) \int x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^5}-\frac {4 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{63 c^5}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^5}+\frac {4}{63} d^2 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{63} \left (8 d^2\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{63} \left (8 b c d^2\right ) \int x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac {1}{9} \left (2 b^2 c^2 d^2\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (-8-20 c^2 x^2-35 c^4 x^4\right )}{315 c^6} \, dx\\ &=\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{189 c^5}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^5}-\frac {20 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^5}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (2 b^2 d^2\right ) \int \left (1-c^2 x^2\right )^2 \left (-8-20 c^2 x^2-35 c^4 x^4\right ) \, dx}{2835 c^4}-\frac {1}{315} \left (16 b c d^2\right ) \int \frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{63} \left (8 b^2 c^2 d^2\right ) \int \frac {-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx\\ &=\frac {16 b d^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{1575 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{189 c^5}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^5}-\frac {20 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^5}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (16 b^2 d^2\right ) \int x^4 \, dx}{1575}+\frac {\left (2 b^2 d^2\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+50 c^6 x^6-35 c^8 x^8\right ) \, dx}{2835 c^4}+\frac {\left (8 b^2 d^2\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right ) \, dx}{6615 c^4}-\frac {\left (64 b d^2\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{1575 c}\\ &=-\frac {304 b^2 d^2 x}{19845 c^4}-\frac {152 b^2 d^2 x^3}{59535 c^2}-\frac {526 b^2 d^2 x^5}{165375}+\frac {212 b^2 c^2 d^2 x^7}{27783}-\frac {2}{729} b^2 c^4 d^2 x^9+\frac {64 b d^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{4725 c^3}+\frac {16 b d^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{1575 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{189 c^5}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^5}-\frac {20 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^5}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (128 b d^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{4725 c^3}-\frac {\left (64 b^2 d^2\right ) \int x^2 \, dx}{4725 c^2}\\ &=-\frac {304 b^2 d^2 x}{19845 c^4}-\frac {2104 b^2 d^2 x^3}{297675 c^2}-\frac {526 b^2 d^2 x^5}{165375}+\frac {212 b^2 c^2 d^2 x^7}{27783}-\frac {2}{729} b^2 c^4 d^2 x^9+\frac {128 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{4725 c^5}+\frac {64 b d^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{4725 c^3}+\frac {16 b d^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{1575 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{189 c^5}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^5}-\frac {20 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^5}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (128 b^2 d^2\right ) \int 1 \, dx}{4725 c^4}\\ &=-\frac {4208 b^2 d^2 x}{99225 c^4}-\frac {2104 b^2 d^2 x^3}{297675 c^2}-\frac {526 b^2 d^2 x^5}{165375}+\frac {212 b^2 c^2 d^2 x^7}{27783}-\frac {2}{729} b^2 c^4 d^2 x^9+\frac {128 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{4725 c^5}+\frac {64 b d^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{4725 c^3}+\frac {16 b d^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{1575 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{189 c^5}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^5}-\frac {20 b d^2 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^5}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^5}+\frac {8}{315} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {4}{63} d^2 x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 253, normalized size = 0.64 \begin {gather*} \frac {d^2 \left (99225 a^2 c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )+630 a 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 )-2 b^2 c x \left (662760+110460 c^2 x^2+49707 c^4 x^4-119250 c^6 x^6+42875 c^8 x^8\right )+630 b \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 )\right ) \text {ArcSin}(c x)+99225 b^2 c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right ) \text {ArcSin}(c x)^2\right )}{31255875 c^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 531, normalized size = 1.34
method | result | size |
derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}-\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{2835}-\frac {16 c x}{315}+\frac {16 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {2 \arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {20 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {4 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{3087}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {263 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}+\frac {2104 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{5}}\) | \(531\) |
default | \(\frac {d^{2} a^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}-\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{2835}-\frac {16 c x}{315}+\frac {16 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {2 \arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {20 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {4 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{3087}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {263 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}+\frac {2104 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{5}}\) | \(531\) |
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. 781 vs.
\(2 (349) = 698\).
time = 0.53, size = 781, normalized size = 1.98 \begin {gather*} \frac {1}{9} \, b^{2} c^{4} d^{2} x^{9} \arcsin \left (c x\right )^{2} + \frac {1}{9} \, a^{2} c^{4} d^{2} x^{9} - \frac {2}{7} \, b^{2} c^{2} d^{2} x^{7} \arcsin \left (c x\right )^{2} - \frac {2}{7} \, a^{2} c^{2} d^{2} x^{7} + \frac {1}{5} \, b^{2} d^{2} x^{5} \arcsin \left (c x\right )^{2} + \frac {2}{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 )} a b c^{4} d^{2} + \frac {2}{893025} \, {\left (315 \, {\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 \arcsin \left (c x\right ) - \frac {1225 \, c^{8} x^{9} + 1800 \, c^{6} x^{7} + 3024 \, c^{4} x^{5} + 6720 \, c^{2} x^{3} + 40320 \, x}{c^{8}}\right )} b^{2} c^{4} d^{2} + \frac {1}{5} \, a^{2} d^{2} x^{5} - \frac {4}{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 )} a b c^{2} d^{2} - \frac {4}{25725} \, {\left (105 \, {\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 \arcsin \left (c x\right ) - \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} c^{2} d^{2} + \frac {2}{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 )} a b d^{2} + \frac {2}{1125} \, {\left (15 \, {\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 \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.11, size = 337, normalized size = 0.85 \begin {gather*} \frac {42875 \, {\left (81 \, a^{2} - 2 \, b^{2}\right )} c^{9} d^{2} x^{9} - 2250 \, {\left (3969 \, a^{2} - 106 \, b^{2}\right )} c^{7} d^{2} x^{7} + 189 \, {\left (33075 \, a^{2} - 526 \, b^{2}\right )} c^{5} d^{2} x^{5} - 220920 \, b^{2} c^{3} d^{2} x^{3} - 1325520 \, b^{2} c d^{2} x + 99225 \, {\left (35 \, b^{2} c^{9} d^{2} x^{9} - 90 \, b^{2} c^{7} d^{2} x^{7} + 63 \, b^{2} c^{5} d^{2} x^{5}\right )} \arcsin \left (c x\right )^{2} + 198450 \, {\left (35 \, a b c^{9} d^{2} x^{9} - 90 \, a b c^{7} d^{2} x^{7} + 63 \, a b c^{5} d^{2} x^{5}\right )} \arcsin \left (c x\right ) + 630 \, {\left (1225 \, a b c^{8} d^{2} x^{8} - 2650 \, a b c^{6} d^{2} x^{6} + 789 \, a b c^{4} d^{2} x^{4} + 1052 \, a b c^{2} d^{2} x^{2} + 2104 \, a b d^{2} + {\left (1225 \, b^{2} c^{8} d^{2} x^{8} - 2650 \, b^{2} c^{6} d^{2} x^{6} + 789 \, b^{2} c^{4} d^{2} x^{4} + 1052 \, b^{2} c^{2} d^{2} x^{2} + 2104 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{31255875 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.24, size = 563, normalized size = 1.43 \begin {gather*} \begin {cases} \frac {a^{2} c^{4} d^{2} x^{9}}{9} - \frac {2 a^{2} c^{2} d^{2} x^{7}}{7} + \frac {a^{2} d^{2} x^{5}}{5} + \frac {2 a b c^{4} d^{2} x^{9} \operatorname {asin}{\left (c x \right )}}{9} + \frac {2 a b c^{3} d^{2} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81} - \frac {4 a b c^{2} d^{2} x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {212 a b c d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{3969} + \frac {2 a b d^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {526 a b d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{33075 c} + \frac {2104 a b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{3}} + \frac {4208 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{5}} + \frac {b^{2} c^{4} d^{2} x^{9} \operatorname {asin}^{2}{\left (c x \right )}}{9} - \frac {2 b^{2} c^{4} d^{2} x^{9}}{729} + \frac {2 b^{2} c^{3} d^{2} x^{8} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{81} - \frac {2 b^{2} c^{2} d^{2} x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} + \frac {212 b^{2} c^{2} d^{2} x^{7}}{27783} - \frac {212 b^{2} c d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3969} + \frac {b^{2} d^{2} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {526 b^{2} d^{2} x^{5}}{165375} + \frac {526 b^{2} d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{33075 c} - \frac {2104 b^{2} d^{2} x^{3}}{297675 c^{2}} + \frac {2104 b^{2} d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{99225 c^{3}} - \frac {4208 b^{2} d^{2} x}{99225 c^{4}} + \frac {4208 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{99225 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 702 vs.
\(2 (349) = 698\).
time = 0.45, size = 702, normalized size = 1.78 \begin {gather*} \frac {1}{9} \, a^{2} c^{4} d^{2} x^{9} - \frac {2}{7} \, a^{2} c^{2} d^{2} x^{7} + \frac {1}{5} \, a^{2} d^{2} x^{5} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{9 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} a b d^{2} x \arcsin \left (c x\right )}{9 \, c^{4}} + \frac {10 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{63 \, c^{4}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{2} x}{729 \, c^{4}} + \frac {20 \, {\left (c^{2} x^{2} - 1\right )}^{3} a b d^{2} x \arcsin \left (c x\right )}{63 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{105 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{81 \, c^{5}} - \frac {836 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} x}{250047 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b d^{2} x \arcsin \left (c x\right )}{105 \, c^{4}} - \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{315 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{81 \, c^{5}} + \frac {20 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{441 \, c^{5}} + \frac {33862 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2} x}{10418625 \, c^{4}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )} a b d^{2} x \arcsin \left (c x\right )}{315 \, c^{4}} + \frac {8 \, b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{315 \, c^{4}} + \frac {20 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{441 \, c^{5}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{525 \, c^{5}} - \frac {47248 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} x}{31255875 \, c^{4}} + \frac {16 \, a b d^{2} x \arcsin \left (c x\right )}{315 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{525 \, c^{5}} + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{2} \arcsin \left (c x\right )}{945 \, c^{5}} - \frac {1493104 \, b^{2} d^{2} x}{31255875 \, c^{4}} + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{2}}{945 \, c^{5}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{315 \, c^{5}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} a 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 = -0.00 \begin {gather*} \int x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\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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