Optimal. Leaf size=940 \[ \frac {b c^5 d^2 g^2 \sqrt {d-c^2 d x^2} x^8}{64 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 g^2 \sqrt {d-c^2 d x^2} x^6}{288 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g \sqrt {d-c^2 d x^2} x^5}{35 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^2 \sqrt {d-c^2 d x^2} x^4}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 g^2 \sqrt {d-c^2 d x^2} x^4}{768 \sqrt {1-c^2 x^2}}+\frac {5}{64} d^2 g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {1}{8} d^2 g^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {5}{48} d^2 g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {2 b c d^2 f g \sqrt {d-c^2 d x^2} x^3}{7 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^2 \sqrt {d-c^2 d x^2} x^2}{96 \sqrt {1-c^2 x^2}}-\frac {5 b d^2 g^2 \sqrt {d-c^2 d x^2} x^2}{256 c \sqrt {1-c^2 x^2}}+\frac {5}{16} d^2 f^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x}{128 c^2}+\frac {1}{6} d^2 f^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x+\frac {5}{24} d^2 f^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {2 b d^2 f g \sqrt {d-c^2 d x^2} x}{7 c \sqrt {1-c^2 x^2}}-\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c} \]
[Out]
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Rubi [A] time = 0.96, antiderivative size = 940, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {4778, 4764, 4650, 4648, 4642, 30, 14, 261, 4678, 194, 4700, 4698, 4708, 266, 43} \[ \frac {b c^5 d^2 g^2 \sqrt {d-c^2 d x^2} x^8}{64 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 g^2 \sqrt {d-c^2 d x^2} x^6}{288 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g \sqrt {d-c^2 d x^2} x^5}{35 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^2 \sqrt {d-c^2 d x^2} x^4}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 g^2 \sqrt {d-c^2 d x^2} x^4}{768 \sqrt {1-c^2 x^2}}+\frac {5}{64} d^2 g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {1}{8} d^2 g^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {5}{48} d^2 g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x^3+\frac {2 b c d^2 f g \sqrt {d-c^2 d x^2} x^3}{7 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^2 \sqrt {d-c^2 d x^2} x^2}{96 \sqrt {1-c^2 x^2}}-\frac {5 b d^2 g^2 \sqrt {d-c^2 d x^2} x^2}{256 c \sqrt {1-c^2 x^2}}+\frac {5}{16} d^2 f^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x}{128 c^2}+\frac {1}{6} d^2 f^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x+\frac {5}{24} d^2 f^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) x-\frac {2 b d^2 f g \sqrt {d-c^2 d x^2} x}{7 c \sqrt {1-c^2 x^2}}-\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{7 c^2}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 30
Rule 43
Rule 194
Rule 261
Rule 266
Rule 4642
Rule 4648
Rule 4650
Rule 4678
Rule 4698
Rule 4700
Rule 4708
Rule 4764
Rule 4778
Rubi steps
\begin {align*} \int (f+g x)^2 \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)^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 \sqrt {d-c^2 d x^2}\right ) \int \left (f^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )+2 f g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )+g^2 x^2 \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^2 \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 (2 d^2 f 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 (d^2 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 {1}{6} d^2 f^2 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 {1}{8} d^2 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 {2 d^2 f 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 {\left (5 d^2 f^2 \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^2 \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 (2 b d^2 f 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 (5 d^2 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 (b c d^2 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 {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f^2 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}{48} d^2 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^2 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 {1}{8} d^2 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 {2 d^2 f 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 {\left (5 d^2 f^2 \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^2 \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 (2 b d^2 f 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 (5 d^2 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 (b c d^2 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 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{48 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b d^2 f g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 f g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac {5}{64} d^2 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^2 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}{48} d^2 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^2 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 {1}{8} d^2 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 {2 d^2 f 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 {\left (5 d^2 f^2 \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^2 \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^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 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 (b c d^2 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 (5 b c d^2 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 g^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{48 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b d^2 f g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 f g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 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^2 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}{48} d^2 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^2 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 {1}{8} d^2 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 {2 d^2 f 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 {5 d^2 f^2 \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 (5 d^2 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 (5 b d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b d^2 f g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {5 b d^2 g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 f g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}+\frac {2 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 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^2 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}{48} d^2 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^2 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 {1}{8} d^2 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 {2 d^2 f 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 {5 d^2 f^2 \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 {5 d^2 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 = 4.69, size = 794, normalized size = 0.84 \[ \frac {d^2 \left (\sqrt {d-c^2 d x^2} \left (-5160960 a c f g \sqrt {1-c^2 x^2}-705600 a c g^2 x \sqrt {1-c^2 x^2}+3010560 a c^7 f^2 x^5 \sqrt {1-c^2 x^2}+5160960 a c^7 f g x^6 \sqrt {1-c^2 x^2}+2257920 a c^7 g^2 x^7 \sqrt {1-c^2 x^2}-9784320 a c^5 f^2 x^3 \sqrt {1-c^2 x^2}-15482880 a c^5 f g x^4 \sqrt {1-c^2 x^2}-6397440 a c^5 g^2 x^5 \sqrt {1-c^2 x^2}+12418560 a c^3 f^2 x \sqrt {1-c^2 x^2}+15482880 a c^3 f g x^2 \sqrt {1-c^2 x^2}+5550720 a c^3 g^2 x^3 \sqrt {1-c^2 x^2}+141120 b \left (15 c^2 f^2+g^2\right ) \cos \left (2 \cos ^{-1}(c x)\right )-211680 b c^2 f^2 \cos \left (4 \cos ^{-1}(c x)\right )+15680 b c^2 f^2 \cos \left (6 \cos ^{-1}(c x)\right )-2822400 b c^2 f g x+564480 b c f g \cos \left (3 \cos ^{-1}(c x)\right )-112896 b c f g \cos \left (5 \cos ^{-1}(c x)\right )+11520 b c f g \cos \left (7 \cos ^{-1}(c x)\right )+35280 b g^2 \cos \left (4 \cos ^{-1}(c x)\right )-15680 b g^2 \cos \left (6 \cos ^{-1}(c x)\right )+2205 b g^2 \cos \left (8 \cos ^{-1}(c x)\right )\right )-705600 a \sqrt {d} \sqrt {1-c^2 x^2} \left (8 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-352800 b \sqrt {d-c^2 d x^2} \left (8 c^2 f^2+g^2\right ) \cos ^{-1}(c x)^2+168 b \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \left (25200 c^2 f^2 \sin \left (2 \cos ^{-1}(c x)\right )-5040 c^2 f^2 \sin \left (4 \cos ^{-1}(c x)\right )+560 c^2 f^2 \sin \left (6 \cos ^{-1}(c x)\right )-58112 c f g \sqrt {1-c^2 x^2}-27648 c f g \left (1-c^2 x^2\right )^{3/2} \cos \left (2 \cos ^{-1}(c x)\right )-3840 c f g \left (1-c^2 x^2\right )^{3/2} \cos \left (4 \cos ^{-1}(c x)\right )+111872 c^3 f g x^2 \sqrt {1-c^2 x^2}-8960 c f g \sin \left (3 \cos ^{-1}(c x)\right )-5376 c f g \sin \left (5 \cos ^{-1}(c x)\right )+1680 g^2 \sin \left (2 \cos ^{-1}(c x)\right )+840 g^2 \sin \left (4 \cos ^{-1}(c x)\right )-560 g^2 \sin \left (6 \cos ^{-1}(c x)\right )+105 g^2 \sin \left (8 \cos ^{-1}(c x)\right )\right )\right )}{18063360 c^3 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} g^{2} x^{6} + 2 \, a c^{4} d^{2} f g x^{5} - 4 \, a c^{2} d^{2} f g x^{3} + 2 \, a d^{2} f g x + a d^{2} f^{2} + {\left (a c^{4} d^{2} f^{2} - 2 \, a c^{2} d^{2} g^{2}\right )} x^{4} - {\left (2 \, a c^{2} d^{2} f^{2} - a d^{2} g^{2}\right )} x^{2} + {\left (b c^{4} d^{2} g^{2} x^{6} + 2 \, b c^{4} d^{2} f g x^{5} - 4 \, b c^{2} d^{2} f g x^{3} + 2 \, b d^{2} f g x + b d^{2} f^{2} + {\left (b c^{4} d^{2} f^{2} - 2 \, b c^{2} d^{2} g^{2}\right )} x^{4} - {\left (2 \, b c^{2} d^{2} f^{2} - b d^{2} g^{2}\right )} x^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
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 = 1.43, size = 5035, normalized size = 5.36 \[ \text {output too large to display} \]
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 \[ \frac {1}{48} \, {\left (8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x + 10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x + 15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c}\right )} a f^{2} + \frac {1}{384} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2}} - \frac {48 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x}{c^{2} d} + \frac {10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x}{c^{2}} + \frac {15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x}{c^{2}} + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} a g^{2} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a f g}{7 \, c^{2} d} + \sqrt {d} \int {\left (b c^{4} d^{2} g^{2} x^{6} + 2 \, b c^{4} d^{2} f g x^{5} - 4 \, b c^{2} d^{2} f g x^{3} + 2 \, b d^{2} f g x + b d^{2} f^{2} + {\left (b c^{4} d^{2} f^{2} - 2 \, b c^{2} d^{2} g^{2}\right )} x^{4} - {\left (2 \, b c^{2} d^{2} f^{2} - b d^{2} g^{2}\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )\,{d x} \]
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 \[ \int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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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