Optimal. Leaf size=1281 \[ \frac {b c^5 d^2 g^3 \sqrt {d-c^2 d x^2} x^9}{81 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f g^2 \sqrt {d-c^2 d x^2} x^8}{64 \sqrt {1-c^2 x^2}}-\frac {19 b c^3 d^2 g^3 \sqrt {d-c^2 d x^2} x^7}{441 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 \sqrt {d-c^2 d x^2} x^6}{96 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g^3 \sqrt {d-c^2 d x^2} x^5}{21 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g \sqrt {d-c^2 d x^2} x^5}{35 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^3 \sqrt {d-c^2 d x^2} x^4}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 f g^2 \sqrt {d-c^2 d x^2} x^4}{256 \sqrt {1-c^2 x^2}}+\frac {15}{64} d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {3}{8} d^2 f g^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {5}{16} d^2 f g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3-\frac {b d^2 g^3 \sqrt {d-c^2 d x^2} x^3}{189 c \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g \sqrt {d-c^2 d x^2} x^3}{7 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^3 \sqrt {d-c^2 d x^2} x^2}{96 \sqrt {1-c^2 x^2}}-\frac {15 b d^2 f g^2 \sqrt {d-c^2 d x^2} x^2}{256 c \sqrt {1-c^2 x^2}}+\frac {5}{16} d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x}{128 c^2}+\frac {1}{6} d^2 f^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x+\frac {5}{24} d^2 f^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {2 b d^2 g^3 \sqrt {d-c^2 d x^2} x}{63 c^3 \sqrt {1-c^2 x^2}}-\frac {3 b d^2 f^2 g \sqrt {d-c^2 d x^2} x}{7 c \sqrt {1-c^2 x^2}}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1-c^2 x^2}}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.18, antiderivative size = 1281, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {4778, 4764, 4650, 4648, 4642, 30, 14, 261, 4678, 194, 4700, 4698, 4708, 266, 43, 4690, 12, 373} \[ \frac {b c^5 d^2 g^3 \sqrt {d-c^2 d x^2} x^9}{81 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f g^2 \sqrt {d-c^2 d x^2} x^8}{64 \sqrt {1-c^2 x^2}}-\frac {19 b c^3 d^2 g^3 \sqrt {d-c^2 d x^2} x^7}{441 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 \sqrt {d-c^2 d x^2} x^6}{96 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g^3 \sqrt {d-c^2 d x^2} x^5}{21 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g \sqrt {d-c^2 d x^2} x^5}{35 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^3 \sqrt {d-c^2 d x^2} x^4}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 f g^2 \sqrt {d-c^2 d x^2} x^4}{256 \sqrt {1-c^2 x^2}}+\frac {15}{64} d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {3}{8} d^2 f g^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {5}{16} d^2 f g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3-\frac {b d^2 g^3 \sqrt {d-c^2 d x^2} x^3}{189 c \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g \sqrt {d-c^2 d x^2} x^3}{7 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^3 \sqrt {d-c^2 d x^2} x^2}{96 \sqrt {1-c^2 x^2}}-\frac {15 b d^2 f g^2 \sqrt {d-c^2 d x^2} x^2}{256 c \sqrt {1-c^2 x^2}}+\frac {5}{16} d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x}{128 c^2}+\frac {1}{6} d^2 f^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x+\frac {5}{24} d^2 f^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {2 b d^2 g^3 \sqrt {d-c^2 d x^2} x}{63 c^3 \sqrt {1-c^2 x^2}}-\frac {3 b d^2 f^2 g \sqrt {d-c^2 d x^2} x}{7 c \sqrt {1-c^2 x^2}}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1-c^2 x^2}}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 30
Rule 43
Rule 194
Rule 261
Rule 266
Rule 373
Rule 4642
Rule 4648
Rule 4650
Rule 4678
Rule 4690
Rule 4698
Rule 4700
Rule 4708
Rule 4764
Rule 4778
Rubi steps
\begin {align*} \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (f+g x)^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (f^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )+3 f^2 g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )+3 f g^2 x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )+g^3 x^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{6 \sqrt {1-c^2 x^2}}+\frac {\left (b c d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 \, dx}{6 \sqrt {1-c^2 x^2}}-\frac {\left (3 b d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \, dx}{7 c \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 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 \cos ^{-1}(c x)\right ) \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (3 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (b c d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 c^4} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{24 \sqrt {1-c^2 x^2}}-\frac {\left (3 b d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-3 c^2 x^2+3 c^4 x^4-c^6 x^6\right ) \, dx}{7 c \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (3 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int x \left (1-c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (b d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3 \, dx}{63 c^3 \sqrt {1-c^2 x^2}}\\ &=-\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{24 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (3 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (x-2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt {1-c^2 x^2}}+\frac {\left (15 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (b d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+15 c^4 x^4-19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt {1-c^2 x^2}}\\ &=-\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2}}{256 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (15 b d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {1-c^2 x^2}}\\ &=-\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2}}{256 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{9 c^4}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 7.99, size = 1144, normalized size = 0.89 \[ \frac {d^2 \left (-3175200 b c f \left (8 c^2 f^2+3 g^2\right ) \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)^2+504 b \sqrt {d-c^2 d x^2} \left (503424 f^2 g x^2 \sqrt {1-c^2 x^2} c^4+75600 f^3 \sin \left (2 \cos ^{-1}(c x)\right ) c^3-15120 f^3 \sin \left (4 \cos ^{-1}(c x)\right ) c^3+1680 f^3 \sin \left (6 \cos ^{-1}(c x)\right ) c^3-40320 f^2 g \sin \left (3 \cos ^{-1}(c x)\right ) c^2-24192 f^2 g \sin \left (5 \cos ^{-1}(c x)\right ) c^2-120576 g^3 x^2 \sqrt {1-c^2 x^2} c^2-261504 f^2 g \sqrt {1-c^2 x^2} c^2+15120 f g^2 \sin \left (2 \cos ^{-1}(c x)\right ) c+7560 f g^2 \sin \left (4 \cos ^{-1}(c x)\right ) c-5040 f g^2 \sin \left (6 \cos ^{-1}(c x)\right ) c+945 f g^2 \sin \left (8 \cos ^{-1}(c x)\right ) c-41472 g \left (3 c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right )^{3/2} \cos \left (2 \cos ^{-1}(c x)\right )-5760 g \left (3 c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right )^{3/2} \cos \left (4 \cos ^{-1}(c x)\right )+6720 g^3 \sin \left (3 \cos ^{-1}(c x)\right )+6048 g^3 \sin \left (5 \cos ^{-1}(c x)\right )+900 g^3 \sin \left (7 \cos ^{-1}(c x)\right )+140 g^3 \sin \left (9 \cos ^{-1}(c x)\right )+62616 g^3 \sqrt {1-c^2 x^2}\right ) \cos ^{-1}(c x)-6350400 a c \sqrt {d} f \left (8 c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+\sqrt {d-c^2 d x^2} \left (18063360 a g^3 x^8 \sqrt {1-c^2 x^2} c^8+60963840 a f g^2 x^7 \sqrt {1-c^2 x^2} c^8+69672960 a f^2 g x^6 \sqrt {1-c^2 x^2} c^8+27095040 a f^3 x^5 \sqrt {1-c^2 x^2} c^8-49029120 a g^3 x^6 \sqrt {1-c^2 x^2} c^6-172730880 a f g^2 x^5 \sqrt {1-c^2 x^2} c^6-209018880 a f^2 g x^4 \sqrt {1-c^2 x^2} c^6-88058880 a f^3 x^3 \sqrt {1-c^2 x^2} c^6+38707200 a g^3 x^4 \sqrt {1-c^2 x^2} c^4+149869440 a f g^2 x^3 \sqrt {1-c^2 x^2} c^4+209018880 a f^2 g x^2 \sqrt {1-c^2 x^2} c^4+111767040 a f^3 x \sqrt {1-c^2 x^2} c^4-38102400 b f^2 g x c^3-1905120 b f^3 \cos \left (4 \cos ^{-1}(c x)\right ) c^3+141120 b f^3 \cos \left (6 \cos ^{-1}(c x)\right ) c^3-1524096 b f^2 g \cos \left (5 \cos ^{-1}(c x)\right ) c^2+155520 b f^2 g \cos \left (7 \cos ^{-1}(c x)\right ) c^2-2580480 a g^3 x^2 \sqrt {1-c^2 x^2} c^2-69672960 a f^2 g \sqrt {1-c^2 x^2} c^2-19051200 a f g^2 x \sqrt {1-c^2 x^2} c^2-3810240 b g^3 x c+3810240 b f \left (5 c^2 f^2+g^2\right ) \cos \left (2 \cos ^{-1}(c x)\right ) c+952560 b f g^2 \cos \left (4 \cos ^{-1}(c x)\right ) c-423360 b f g^2 \cos \left (6 \cos ^{-1}(c x)\right ) c+59535 b f g^2 \cos \left (8 \cos ^{-1}(c x)\right ) c+282240 b g \left (27 c^2 f^2+2 g^2\right ) \cos \left (3 \cos ^{-1}(c x)\right )-38880 b g^3 \cos \left (7 \cos ^{-1}(c x)\right )+7840 b g^3 \cos \left (9 \cos ^{-1}(c x)\right )-5160960 a g^3 \sqrt {1-c^2 x^2}\right )\right )}{162570240 c^4 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} g^{3} x^{7} + 3 \, a c^{4} d^{2} f g^{2} x^{6} + 3 \, a d^{2} f^{2} g x + a d^{2} f^{3} + {\left (3 \, a c^{4} d^{2} f^{2} g - 2 \, a c^{2} d^{2} g^{3}\right )} x^{5} + {\left (a c^{4} d^{2} f^{3} - 6 \, a c^{2} d^{2} f g^{2}\right )} x^{4} - {\left (6 \, a c^{2} d^{2} f^{2} g - a d^{2} g^{3}\right )} x^{3} - {\left (2 \, a c^{2} d^{2} f^{3} - 3 \, a d^{2} f g^{2}\right )} x^{2} + {\left (b c^{4} d^{2} g^{3} x^{7} + 3 \, b c^{4} d^{2} f g^{2} x^{6} + 3 \, b d^{2} f^{2} g x + b d^{2} f^{3} + {\left (3 \, b c^{4} d^{2} f^{2} g - 2 \, b c^{2} d^{2} g^{3}\right )} x^{5} + {\left (b c^{4} d^{2} f^{3} - 6 \, b c^{2} d^{2} f g^{2}\right )} x^{4} - {\left (6 \, b c^{2} d^{2} f^{2} g - b d^{2} g^{3}\right )} x^{3} - {\left (2 \, b c^{2} d^{2} f^{3} - 3 \, b d^{2} f g^{2}\right )} x^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
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 = 2.03, size = 4046, normalized size = 3.16 \[ \text {output too large to display} \]
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 \[ \frac {1}{48} \, {\left (8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x + 10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x + 15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c}\right )} a f^{3} + \frac {1}{128} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2}} - \frac {48 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x}{c^{2} d} + \frac {10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x}{c^{2}} + \frac {15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x}{c^{2}} + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} a f g^{2} - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a g^{3} - \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a f^{2} g}{7 \, c^{2} d} + \sqrt {d} \int {\left (b c^{4} d^{2} g^{3} x^{7} + 3 \, b c^{4} d^{2} f g^{2} x^{6} + 3 \, b d^{2} f^{2} g x + b d^{2} f^{3} + {\left (3 \, b c^{4} d^{2} f^{2} g - 2 \, b c^{2} d^{2} g^{3}\right )} x^{5} + {\left (b c^{4} d^{2} f^{3} - 6 \, b c^{2} d^{2} f g^{2}\right )} x^{4} - {\left (6 \, b c^{2} d^{2} f^{2} g - b d^{2} g^{3}\right )} x^{3} - {\left (2 \, b c^{2} d^{2} f^{3} - 3 \, b d^{2} f g^{2}\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )\,{d x} \]
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 \[ \int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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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