Optimal. Leaf size=2290 \[ \text {result too large to display} \]
[Out]
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Rubi [A] time = 3.29, antiderivative size = 2290, normalized size of antiderivative = 1.00, number of steps used = 77, number of rules used = 32, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.970, Rules used = {4777, 4763, 4649, 4647, 4641, 4627, 321, 216, 4677, 195, 194, 4645, 12, 1799, 1850, 4699, 4697, 4707, 14, 4687, 459, 266, 43, 1267, 4619, 261, 446, 77, 270, 1251, 897, 1153} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 43
Rule 77
Rule 194
Rule 195
Rule 216
Rule 261
Rule 266
Rule 270
Rule 321
Rule 446
Rule 459
Rule 897
Rule 1153
Rule 1251
Rule 1267
Rule 1799
Rule 1850
Rule 4619
Rule 4627
Rule 4641
Rule 4645
Rule 4647
Rule 4649
Rule 4677
Rule 4687
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 )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, 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 \sin ^{-1}(c x)\right )^2 \, 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 \sin ^{-1}(c x)\right )^2+3 f^2 g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+3 f g^2 x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+g^3 x^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2\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 \sin ^{-1}(c x)\right )^2 \, 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 \sin ^{-1}(c x)\right )^2 \, 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 \sin ^{-1}(c x)\right )^2 \, 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 \sin ^{-1}(c x)\right )^2 \, 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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\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 \sin ^{-1}(c x)\right )^2}{7 c^2}+\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 \sin ^{-1}(c x)\right )^2 \, 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 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt {1-c^2 x^2}}+\frac {\left (6 b d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\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 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, 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 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 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 )^2 \, dx}{9 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{9 \sqrt {1-c^2 x^2}}\\ &=\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}-\frac {3 b c d^2 f g^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 {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{63 \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} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{18 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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\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 \sin ^{-1}(c x)\right )^2}{7 c^2}+\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 \sin ^{-1}(c x)\right )^2 \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{18 \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 ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{12 \sqrt {1-c^2 x^2}}-\frac {\left (6 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt {1-c^2 x^2}} \, dx}{7 \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 \sin ^{-1}(c x)\right )^2 \, dx}{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 ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d^2 f g^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 {\left (5 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{21 \sqrt {1-c^2 x^2}}-\frac {\left (10 b c d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{63 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt {1-c^2 x^2}} \, dx}{9 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}-\frac {11 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {8 b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 \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} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\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 \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {\left (5 d^2 f^3 \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}{16 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{108 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{48 \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 (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (6 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx}{245 \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 \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{64 \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 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{32 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 f g^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}{32 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 f g^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}{8 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{21 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{21 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{2835 \sqrt {1-c^2 x^2}}+\frac {\left (10 b^2 c^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {1-c^2 x^2}} \, dx}{63 \sqrt {1-c^2 x^2}}\\ &=-\frac {3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt {d-c^2 d x^2}-\frac {65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}-\frac {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {d^2 g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\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 \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{144 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 f g^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}{256 \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 f g^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 (15 b d^2 f g^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 f g^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}{96 \sqrt {1-c^2 x^2}}+\frac {\left (15 b^2 c^2 d^2 f g^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 {\left (2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{63 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{63 c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (63-90 c^2 x+35 c^4 x^2\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2835 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{441 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2}} \, dx}{105 \sqrt {1-c^2 x^2}}\\ &=-\frac {245 b^2 d^2 f^3 x \sqrt {d-c^2 d x^2}}{1152}-\frac {15}{512} b^2 d^2 f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {209 b^2 c^2 d^2 f g^2 x^5 \sqrt {d-c^2 d x^2}}{4608}-\frac {3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt {d-c^2 d x^2}-\frac {65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}+\frac {15 b d^2 f g^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 {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {2 d^2 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {d^2 g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\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 \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{288 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {16}{\sqrt {1-c^2 x}}+8 \sqrt {1-c^2 x}+6 \left (1-c^2 x\right )^{3/2}+5 \left (1-c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}}+\frac {\left (45 b^2 d^2 f g^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 (15 b^2 d^2 f g^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 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{1536 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{72 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {x^2}{c^2}\right )^2 \left (8+20 x^2+35 x^4\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{2835 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{189 \sqrt {1-c^2 x^2}}+\frac {\left (4 b d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (7-5 c^2 x\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{441 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{105 \sqrt {1-c^2 x^2}}\\ &=\frac {96 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {245 b^2 d^2 f^3 x \sqrt {d-c^2 d x^2}}{1152}+\frac {15 b^2 d^2 f g^2 x \sqrt {d-c^2 d x^2}}{1024 c^2}-\frac {1079 b^2 d^2 f g^2 x^3 \sqrt {d-c^2 d x^2}}{18432}+\frac {209 b^2 c^2 d^2 f g^2 x^5 \sqrt {d-c^2 d x^2}}{4608}-\frac {3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt {d-c^2 d x^2}+\frac {4 a b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {16 b^2 d^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {36 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {115 b^2 d^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}+\frac {15 b d^2 f g^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 {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {2 d^2 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {d^2 g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\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 \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (73 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2048 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{96 \sqrt {1-c^2 x^2}}+\frac {\left (45 b^2 d^2 f g^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 (15 b^2 d^2 f g^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 {\left (2 b^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{c^4}+\frac {4 x^2}{c^4}+\frac {3 x^4}{c^4}-\frac {50 x^6}{c^4}+\frac {35 x^8}{c^4}\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{2835 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{189 \sqrt {1-c^2 x^2}}+\frac {\left (4 b^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{c^4 \sqrt {1-c^2 x}}+\frac {\sqrt {1-c^2 x}}{c^4}-\frac {8 \left (1-c^2 x\right )^{3/2}}{c^4}+\frac {5 \left (1-c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right )}{441 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^4 \sqrt {1-c^2 x}}-\frac {2 \sqrt {1-c^2 x}}{c^4}+\frac {\left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{105 \sqrt {1-c^2 x^2}}\\ &=\frac {96 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {134 b^2 d^2 g^3 \sqrt {d-c^2 d x^2}}{3969 c^4}-\frac {245 b^2 d^2 f^3 x \sqrt {d-c^2 d x^2}}{1152}-\frac {359 b^2 d^2 f g^2 x \sqrt {d-c^2 d x^2}}{12288 c^2}-\frac {1079 b^2 d^2 f g^2 x^3 \sqrt {d-c^2 d x^2}}{18432}+\frac {209 b^2 c^2 d^2 f g^2 x^5 \sqrt {d-c^2 d x^2}}{4608}-\frac {3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt {d-c^2 d x^2}+\frac {4 a b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {16 b^2 d^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {122 b^2 d^2 g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11907 c^4}-\frac {65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {36 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {4 b^2 d^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1323 c^4}-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {50 b^2 d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{27783 c^4}-\frac {2 b^2 d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {115 b^2 d^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt {1-c^2 x^2}}-\frac {15 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1024 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 d^2 g^3 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}+\frac {15 b d^2 f g^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 {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {2 d^2 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {d^2 g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\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 \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (73 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4096 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{192 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{189 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{63 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {96 b^2 d^2 f^2 g \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {160 b^2 d^2 g^3 \sqrt {d-c^2 d x^2}}{3969 c^4}-\frac {245 b^2 d^2 f^3 x \sqrt {d-c^2 d x^2}}{1152}-\frac {359 b^2 d^2 f g^2 x \sqrt {d-c^2 d x^2}}{12288 c^2}-\frac {1079 b^2 d^2 f g^2 x^3 \sqrt {d-c^2 d x^2}}{18432}+\frac {209 b^2 c^2 d^2 f g^2 x^5 \sqrt {d-c^2 d x^2}}{4608}-\frac {3}{256} b^2 c^4 d^2 f g^2 x^7 \sqrt {d-c^2 d x^2}+\frac {4 a b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {16 b^2 d^2 f^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {80 b^2 d^2 g^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11907 c^4}-\frac {65 b^2 d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {36 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {4 b^2 d^2 g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1323 c^4}-\frac {1}{108} b^2 d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {6 b^2 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {50 b^2 d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{27783 c^4}-\frac {2 b^2 d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {115 b^2 d^2 f^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {359 b^2 d^2 f g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{12288 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 d^2 g^3 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {6 b d^2 f^2 g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt {1-c^2 x^2}}+\frac {15 b d^2 f g^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 {6 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g^3 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{189 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 \sqrt {1-c^2 x^2}}+\frac {18 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}-\frac {6 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{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} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{81 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^3 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 c}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac {2 d^2 g^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^4}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}-\frac {d^2 g^3 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{63 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{21} d^2 g^3 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {5}{63} d^2 g^3 x^4 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 \sin ^{-1}(c x)\right )^2+\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 \sin ^{-1}(c x)\right )^2+\frac {1}{9} d^2 g^3 x^4 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\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 \sin ^{-1}(c x)\right )^2}{7 c^2}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{128 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.99, size = 1114, normalized size = 0.49 \[ \frac {d^2 \sqrt {d-c^2 d x^2} \left (333396000 \left (8 c^3 f^3+3 c g^2 f\right ) a^3+3175200 b \sqrt {1-c^2 x^2} \left (16 x^5 \left (84 f^3+216 g x f^2+189 g^2 x^2 f+56 g^3 x^3\right ) c^8-8 x^3 \left (546 f^3+1296 g x f^2+1071 g^2 x^2 f+304 g^3 x^3\right ) c^6+6 x \left (924 f^3+1728 g x f^2+1239 g^2 x^2 f+320 g^3 x^3\right ) c^4-g \left (3456 f^2+945 g x f+128 g^2 x^2\right ) c^2-256 g^3\right ) a^2-10080 b^2 c x \left (20 x^5 \left (7056 f^3+15552 g x f^2+11907 g^2 x^2 f+3136 g^3 x^3\right ) c^8-72 x^3 \left (9555 f^3+18144 g x f^2+12495 g^2 x^2 f+3040 g^3 x^3\right ) c^6+945 x \left (1848 f^3+2304 g x f^2+1239 g^2 x^2 f+256 g^3 x^3\right ) c^4-105 g \left (20736 f^2+2835 g x f+256 g^2 x^2\right ) c^2-161280 g^3\right ) a+333396000 b^3 c f \left (8 c^2 f^2+3 g^2\right ) \sin ^{-1}(c x)^3+3175200 b^2 \left (315 a \left (8 c^3 f^3+3 c g^2 f\right )+b \sqrt {1-c^2 x^2} \left (16 x^5 \left (84 f^3+216 g x f^2+189 g^2 x^2 f+56 g^3 x^3\right ) c^8-8 x^3 \left (546 f^3+1296 g x f^2+1071 g^2 x^2 f+304 g^3 x^3\right ) c^6+6 x \left (924 f^3+1728 g x f^2+1239 g^2 x^2 f+320 g^3 x^3\right ) c^4-g \left (3456 f^2+945 g x f+128 g^2 x^2\right ) c^2-256 g^3\right )\right ) \sin ^{-1}(c x)^2-b^3 \sqrt {1-c^2 x^2} \left (400 x^5 \left (592704 f^3+1119744 g x f^2+750141 g^2 x^2 f+175616 g^3 x^3\right ) c^8-8 x^3 \left (179663400 f^3+262020096 g x f^2+145166175 g^2 x^2 f+29363200 g^3 x^3\right ) c^6+6 x \left (1107615600 f^3+753463296 g x f^2+249815475 g^2 x^2 f+34304000 g^3 x^3\right ) c^4+g \left (-12905422848 f^2+748057275 g x f+184115200 g^2 x^2\right ) c^2-1257472000 g^3\right )+315 b \left (3175200 \left (8 c^3 f^3+3 c g^2 f\right ) a^2+20160 b \sqrt {1-c^2 x^2} \left (16 x^5 \left (84 f^3+216 g x f^2+189 g^2 x^2 f+56 g^3 x^3\right ) c^8-8 x^3 \left (546 f^3+1296 g x f^2+1071 g^2 x^2 f+304 g^3 x^3\right ) c^6+6 x \left (924 f^3+1728 g x f^2+1239 g^2 x^2 f+320 g^3 x^3\right ) c^4-g \left (3456 f^2+945 g x f+128 g^2 x^2\right ) c^2-256 g^3\right ) a+b^2 c \left (-640 x^6 \left (7056 f^3+15552 g x f^2+11907 g^2 x^2 f+3136 g^3 x^3\right ) c^8+2304 x^4 \left (9555 f^3+18144 g x f^2+12495 g^2 x^2 f+3040 g^3 x^3\right ) c^6-30240 x^2 \left (1848 f^3+2304 g x f^2+1239 g^2 x^2 f+256 g^3 x^3\right ) c^4+3360 \left (6279 f^3+20736 g x f^2+2835 g^2 x^2 f+256 g^3 x^3\right ) c^2+315 g^2 (7539 f+16384 g x)\right )\right ) \sin ^{-1}(c x)\right )}{25604812800 b c^4 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c^{4} d^{2} g^{3} x^{7} + 3 \, a^{2} c^{4} d^{2} f g^{2} x^{6} + 3 \, a^{2} d^{2} f^{2} g x + a^{2} d^{2} f^{3} + {\left (3 \, a^{2} c^{4} d^{2} f^{2} g - 2 \, a^{2} c^{2} d^{2} g^{3}\right )} x^{5} + {\left (a^{2} c^{4} d^{2} f^{3} - 6 \, a^{2} c^{2} d^{2} f g^{2}\right )} x^{4} - {\left (6 \, a^{2} c^{2} d^{2} f^{2} g - a^{2} d^{2} g^{3}\right )} x^{3} - {\left (2 \, a^{2} c^{2} d^{2} f^{3} - 3 \, a^{2} d^{2} f g^{2}\right )} x^{2} + {\left (b^{2} c^{4} d^{2} g^{3} x^{7} + 3 \, b^{2} c^{4} d^{2} f g^{2} x^{6} + 3 \, b^{2} d^{2} f^{2} g x + b^{2} d^{2} f^{3} + {\left (3 \, b^{2} c^{4} d^{2} f^{2} g - 2 \, b^{2} c^{2} d^{2} g^{3}\right )} x^{5} + {\left (b^{2} c^{4} d^{2} f^{3} - 6 \, b^{2} c^{2} d^{2} f g^{2}\right )} x^{4} - {\left (6 \, b^{2} c^{2} d^{2} f^{2} g - b^{2} d^{2} g^{3}\right )} x^{3} - {\left (2 \, b^{2} c^{2} d^{2} f^{3} - 3 \, b^{2} d^{2} f g^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} g^{3} x^{7} + 3 \, a b c^{4} d^{2} f g^{2} x^{6} + 3 \, a b d^{2} f^{2} g x + a b d^{2} f^{3} + {\left (3 \, a b c^{4} d^{2} f^{2} g - 2 \, a b c^{2} d^{2} g^{3}\right )} x^{5} + {\left (a b c^{4} d^{2} f^{3} - 6 \, a b c^{2} d^{2} f g^{2}\right )} x^{4} - {\left (6 \, a b c^{2} d^{2} f^{2} g - a b d^{2} g^{3}\right )} x^{3} - {\left (2 \, a b c^{2} d^{2} f^{3} - 3 \, a b d^{2} f g^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
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 = 1.97, size = 11465, normalized size = 5.01 \[ \text {output too large to display} \]
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^{2} 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^{2} 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^{2} g^{3} - \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a^{2} f^{2} g}{7 \, c^{2} d} + \sqrt {d} \int {\left ({\left (b^{2} c^{4} d^{2} g^{3} x^{7} + 3 \, b^{2} c^{4} d^{2} f g^{2} x^{6} + 3 \, b^{2} d^{2} f^{2} g x + b^{2} d^{2} f^{3} + {\left (3 \, b^{2} c^{4} d^{2} f^{2} g - 2 \, b^{2} c^{2} d^{2} g^{3}\right )} x^{5} + {\left (b^{2} c^{4} d^{2} f^{3} - 6 \, b^{2} c^{2} d^{2} f g^{2}\right )} x^{4} - {\left (6 \, b^{2} c^{2} d^{2} f^{2} g - b^{2} d^{2} g^{3}\right )} x^{3} - {\left (2 \, b^{2} c^{2} d^{2} f^{3} - 3 \, b^{2} d^{2} f g^{2}\right )} 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} g^{3} x^{7} + 3 \, a b c^{4} d^{2} f g^{2} x^{6} + 3 \, a b d^{2} f^{2} g x + a b d^{2} f^{3} + {\left (3 \, a b c^{4} d^{2} f^{2} g - 2 \, a b c^{2} d^{2} g^{3}\right )} x^{5} + {\left (a b c^{4} d^{2} f^{3} - 6 \, a b c^{2} d^{2} f g^{2}\right )} x^{4} - {\left (6 \, a b c^{2} d^{2} f^{2} g - a b d^{2} g^{3}\right )} x^{3} - {\left (2 \, a b c^{2} d^{2} f^{3} - 3 \, a b d^{2} f g^{2}\right )} 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 \[ \int {\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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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