Optimal. Leaf size=959 \[ \frac {b c^3 d g^3 \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 \sqrt {d-c^2 d x^2} x^6}{12 \sqrt {1-c^2 x^2}}-\frac {8 b c d g^3 \sqrt {d-c^2 d x^2} x^5}{175 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d f^2 g \sqrt {d-c^2 d x^2} x^5}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f^3 \sqrt {d-c^2 d x^2} x^4}{16 \sqrt {1-c^2 x^2}}-\frac {7 b c d f g^2 \sqrt {d-c^2 d x^2} x^4}{32 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3+\frac {1}{2} d f g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3+\frac {b d g^3 \sqrt {d-c^2 d x^2} x^3}{105 c \sqrt {1-c^2 x^2}}-\frac {2 b c d f^2 g \sqrt {d-c^2 d x^2} x^3}{5 \sqrt {1-c^2 x^2}}-\frac {5 b c d f^3 \sqrt {d-c^2 d x^2} x^2}{16 \sqrt {1-c^2 x^2}}+\frac {3 b d f g^2 \sqrt {d-c^2 d x^2} x^2}{32 c \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{16 c^2}+\frac {1}{4} d f^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x+\frac {2 b d g^3 \sqrt {d-c^2 d x^2} x}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {3 b d f^2 g \sqrt {d-c^2 d x^2} x}{5 c \sqrt {1-c^2 x^2}}+\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}+\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2} \]
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Rubi [A] time = 0.94, antiderivative size = 959, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 17, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {4777, 4763, 4649, 4647, 4641, 30, 14, 4677, 194, 4699, 4697, 4707, 266, 43, 4689, 12, 373} \[ \frac {b c^3 d g^3 \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 \sqrt {d-c^2 d x^2} x^6}{12 \sqrt {1-c^2 x^2}}-\frac {8 b c d g^3 \sqrt {d-c^2 d x^2} x^5}{175 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d f^2 g \sqrt {d-c^2 d x^2} x^5}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f^3 \sqrt {d-c^2 d x^2} x^4}{16 \sqrt {1-c^2 x^2}}-\frac {7 b c d f g^2 \sqrt {d-c^2 d x^2} x^4}{32 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3+\frac {1}{2} d f g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3+\frac {b d g^3 \sqrt {d-c^2 d x^2} x^3}{105 c \sqrt {1-c^2 x^2}}-\frac {2 b c d f^2 g \sqrt {d-c^2 d x^2} x^3}{5 \sqrt {1-c^2 x^2}}-\frac {5 b c d f^3 \sqrt {d-c^2 d x^2} x^2}{16 \sqrt {1-c^2 x^2}}+\frac {3 b d f g^2 \sqrt {d-c^2 d x^2} x^2}{32 c \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{16 c^2}+\frac {1}{4} d f^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x+\frac {2 b d g^3 \sqrt {d-c^2 d x^2} x}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {3 b d f^2 g \sqrt {d-c^2 d x^2} x}{5 c \sqrt {1-c^2 x^2}}+\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}+\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 30
Rule 43
Rule 194
Rule 266
Rule 373
Rule 4641
Rule 4647
Rule 4649
Rule 4677
Rule 4689
Rule 4697
Rule 4699
Rule 4707
Rule 4763
Rule 4777
Rubi steps
\begin {align*} \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int (f+g x)^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (f^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+3 f^2 g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+3 f g^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+g^3 x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d f^2 g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4}+\frac {\left (3 d f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b c d f^3 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (3 b d f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \, dx}{5 c \sqrt {1-c^2 x^2}}+\frac {\left (3 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c d g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{35 c^4} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4}+\frac {\left (3 d f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c d f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (3 b c d f^3 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (3 b d f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt {1-c^2 x^2}}+\frac {\left (3 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (3 b c d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b d g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2 \, dx}{35 c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {3 b d f^2 g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}-\frac {5 b c d f^3 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d f^2 g x^3 \sqrt {d-c^2 d x^2}}{5 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f^3 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2}}{12 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4}+\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}+\frac {\left (3 d f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (3 b d f g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt {1-c^2 x^2}}-\frac {\left (b d g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+8 c^4 x^4-5 c^6 x^6\right ) \, dx}{35 c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {3 b d f^2 g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}+\frac {2 b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}-\frac {5 b c d f^3 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}-\frac {2 b c d f^2 g x^3 \sqrt {d-c^2 d x^2}}{5 \sqrt {1-c^2 x^2}}+\frac {b d g^3 x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {1-c^2 x^2}}+\frac {b c^3 d f^3 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {8 b c d g^3 x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2}}{12 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^3 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4}+\frac {3 d f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}+\frac {3 d f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 463, normalized size = 0.48 \[ \frac {d \sqrt {d-c^2 d x^2} \left (11025 a^2 c f \left (2 c^2 f^2+g^2\right )-210 a b \sqrt {1-c^2 x^2} \left (4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )-2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+32 g^3\right )-210 b \sin ^{-1}(c x) \left (b \sqrt {1-c^2 x^2} \left (4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )-2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+32 g^3\right )-105 a c f \left (2 c^2 f^2+g^2\right )\right )+11025 b^2 c f \left (2 c^2 f^2+g^2\right ) \sin ^{-1}(c x)^2+b^2 c x \left (2 c^6 x^3 \left (3675 f^3+7056 f^2 g x+4900 f g^2 x^2+1200 g^3 x^3\right )-21 c^4 x \left (1750 f^3+2240 f^2 g x+1225 f g^2 x^2+256 g^3 x^3\right )+35 c^2 g \left (2016 f^2+315 f g x+32 g^2 x^2\right )+6720 g^3\right )\right )}{117600 b c^4 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{2} d g^{3} x^{5} + 3 \, a c^{2} d f g^{2} x^{4} - 3 \, a d f^{2} g x - a d f^{3} + {\left (3 \, a c^{2} d f^{2} g - a d g^{3}\right )} x^{3} + {\left (a c^{2} d f^{3} - 3 \, a d f g^{2}\right )} x^{2} + {\left (b c^{2} d g^{3} x^{5} + 3 \, b c^{2} d f g^{2} x^{4} - 3 \, b d f^{2} g x - b d f^{3} + {\left (3 \, b c^{2} d f^{2} g - b d g^{3}\right )} x^{3} + {\left (b c^{2} d f^{3} - 3 \, b d f g^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.37, size = 5085, normalized size = 5.30 \[ \text {output too large to display} \]
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 \[ \frac {1}{8} \, {\left (2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-c^{2} d x^{2} + d} d x + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c}\right )} a f^{3} - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a g^{3} + \frac {1}{16} \, a f g^{2} {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{2}} - \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2} d} + \frac {3 \, \sqrt {-c^{2} d x^{2} + d} d x}{c^{2}} + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a f^{2} g}{5 \, c^{2} d} + \sqrt {d} \int -{\left (b c^{2} d g^{3} x^{5} + 3 \, b c^{2} d f g^{2} x^{4} - 3 \, b d f^{2} g x - b d f^{3} + {\left (3 \, b c^{2} d f^{2} g - b d g^{3}\right )} x^{3} + {\left (b c^{2} d f^{3} - 3 \, b d f g^{2}\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\,{d x} \]
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 \[ \int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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