Optimal. Leaf size=556 \[ \frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt {1-c^2 x^2}}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt {1-c^2 x^2}}-\frac {359 b^2 d^2 x \sqrt {d-c^2 d x^2}}{36864 c^2}+\frac {209 b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}}{13824}-\frac {1079 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{55296}-\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d-c^2 d x^2}+\frac {359 b^2 d^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{36864 c^3 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 1.11, antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {4699, 4697, 4707, 4641, 4627, 321, 216, 14, 4687, 12, 459, 266, 43, 1267} \[ -\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt {1-c^2 x^2}}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d-c^2 d x^2}+\frac {209 b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}}{13824}-\frac {1079 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{55296}-\frac {359 b^2 d^2 x \sqrt {d-c^2 d x^2}}{36864 c^2}+\frac {359 b^2 d^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{36864 c^3 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 216
Rule 266
Rule 321
Rule 459
Rule 1267
Rule 4627
Rule 4641
Rule 4687
Rule 4697
Rule 4699
Rule 4707
Rubi steps
\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} (5 d) \int x^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{16} \left (5 d^2\right ) \int x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{24 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}\\ &=-\frac {11 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{96 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{32 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^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}{96 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}} \, dx}{24 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d-c^2 d x^2}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (-48 c^2+43 c^4 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{768 \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (5 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{64 c \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{288 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{128 \sqrt {1-c^2 x^2}}\\ &=-\frac {5}{512} b^2 d^2 x^3 \sqrt {d-c^2 d x^2}+\frac {209 b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}}{13824}-\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d-c^2 d x^2}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (15 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{512 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{128 \sqrt {1-c^2 x^2}}+\frac {\left (73 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{4608 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{216 \sqrt {1-c^2 x^2}}\\ &=\frac {5 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1024 c^2}-\frac {1079 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{55296}+\frac {209 b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}}{13824}-\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d-c^2 d x^2}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (73 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{6144 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{288 \sqrt {1-c^2 x^2}}+\frac {\left (15 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{1024 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{256 c^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {359 b^2 d^2 x \sqrt {d-c^2 d x^2}}{36864 c^2}-\frac {1079 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{55296}+\frac {209 b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}}{13824}-\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d-c^2 d x^2}-\frac {5 b^2 d^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1024 c^3 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (73 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{12288 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{576 c^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {359 b^2 d^2 x \sqrt {d-c^2 d x^2}}{36864 c^2}-\frac {1079 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{55296}+\frac {209 b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}}{13824}-\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d-c^2 d x^2}+\frac {359 b^2 d^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{36864 c^3 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 348, normalized size = 0.63 \[ \frac {d^2 \sqrt {d-c^2 d x^2} \left (1440 a^3+3 b \sin ^{-1}(c x) \left (1440 a^2+192 a b c x \sqrt {1-c^2 x^2} \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )+b^2 \left (-1152 c^8 x^8+4352 c^6 x^6-5664 c^4 x^4+1440 c^2 x^2+359\right )\right )+288 a^2 b c x \sqrt {1-c^2 x^2} \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )-96 a b^2 c^2 x^2 \left (36 c^6 x^6-136 c^4 x^4+177 c^2 x^2-45\right )+288 b^2 \sin ^{-1}(c x)^2 \left (15 a+b c x \sqrt {1-c^2 x^2} \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )\right )-b^3 c x \sqrt {1-c^2 x^2} \left (432 c^6 x^6-1672 c^4 x^4+2158 c^2 x^2+1077\right )+1440 b^3 \sin ^{-1}(c x)^3\right )}{110592 b c^3 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c^{4} d^{2} x^{6} - 2 \, a^{2} c^{2} d^{2} x^{4} + a^{2} d^{2} x^{2} + {\left (b^{2} c^{4} d^{2} x^{6} - 2 \, b^{2} c^{2} d^{2} x^{4} + b^{2} d^{2} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{6} - 2 \, a b c^{2} d^{2} x^{4} + a b d^{2} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.76, size = 6108, normalized size = 10.99 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{384} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2}} - \frac {48 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x}{c^{2} d} + \frac {10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x}{c^{2}} + \frac {15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x}{c^{2}} + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} a^{2} + \sqrt {d} \int {\left ({\left (b^{2} c^{4} d^{2} x^{6} - 2 \, b^{2} c^{2} d^{2} x^{4} + b^{2} d^{2} x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{6} - 2 \, a b c^{2} d^{2} x^{4} + a b d^{2} x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]
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 \[ \int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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[Out]
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