Optimal. Leaf size=1685 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.46911, antiderivative size = 1685, normalized size of antiderivative = 1., number of steps used = 56, number of rules used = 27, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.818, Rules used = {4777, 4763, 4649, 4647, 4641, 4627, 321, 216, 4677, 195, 194, 4645, 12, 1247, 698, 4699, 4697, 4707, 14, 4687, 459, 4619, 261, 266, 43, 446, 77} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4777
Rule 4763
Rule 4649
Rule 4647
Rule 4641
Rule 4627
Rule 321
Rule 216
Rule 4677
Rule 195
Rule 194
Rule 4645
Rule 12
Rule 1247
Rule 698
Rule 4699
Rule 4697
Rule 4707
Rule 14
Rule 4687
Rule 459
Rule 4619
Rule 261
Rule 266
Rule 43
Rule 446
Rule 77
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 )^2 \, 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 )^2 \, 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 )^2+3 f^2 g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+3 f g^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+g^3 x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2\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 )^2 \, 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 )^2 \, 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 )^2 \, 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 )^2 \, 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 )^2+\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 )^2+\frac{1}{7} d 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{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 )^2}{5 c^2}+\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 )^2 \, 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 ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (6 b d f^2 g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\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 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, 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 ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 d 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}{7 \sqrt{1-c^2 x^2}}-\frac{\left (2 b c d 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}{7 \sqrt{1-c^2 x^2}}\\ &=\frac{6 b d f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{4 b c d f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt{1-c^2 x^2}}-\frac{b c d f g^2 x^4 \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^3 d f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{2 b c d g^3 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{b c^3 d f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g^3 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{b d 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 )}{8 c}+\frac{3}{8} d f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 )^2+\frac{3}{35} d g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 )^2+\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 )^2+\frac{1}{7} d 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{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 )^2}{5 c^2}+\frac{\left (3 d 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}{8 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d f^3 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (3 b c d f^3 \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (6 b^2 d f^2 g \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt{1-c^2 x^2}} \, dx}{5 \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 )^2}{\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 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d 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}{\sqrt{1-c^2 x^2}}+\frac{\left (3 d 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}{35 \sqrt{1-c^2 x^2}}-\frac{\left (6 b c d g^3 \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d 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}{7 \sqrt{1-c^2 x^2}}\\ &=-\frac{1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{6 b d f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{3 b c d f^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}-\frac{4 b c d f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt{1-c^2 x^2}}-\frac{7 b c d 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{6 b c^3 d f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{16 b c d g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{b c^3 d f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g^3 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{b d 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 )}{8 c}+\frac{3}{8} d f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac{d g^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 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 )^2+\frac{3}{35} d g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 )^2+\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 )^2+\frac{1}{7} d 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{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 )^2}{5 c^2}+\frac{d f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 d f^3 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{32 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^2 d f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 d f^2 g \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx}{25 \sqrt{1-c^2 x^2}}+\frac{\left (3 d 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}{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 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 c \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d 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}{12 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^2 d f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{1-c^2 x^2}}+\frac{\left (2 d 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}{35 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (2 b d g^3 \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 c \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d 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}{245 \sqrt{1-c^2 x^2}}+\frac{\left (6 b^2 c^2 d g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5}{\sqrt{1-c^2 x^2}} \, dx}{175 \sqrt{1-c^2 x^2}}\\ &=-\frac{15}{64} b^2 d f^3 x \sqrt{d-c^2 d x^2}-\frac{3}{64} b^2 d f g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{1}{36} b^2 c^2 d f g^2 x^5 \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{6 b d f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{3 b c d f^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3 b d f g^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt{1-c^2 x^2}}-\frac{4 b c d f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt{1-c^2 x^2}}+\frac{2 b d g^3 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt{1-c^2 x^2}}-\frac{7 b c d 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{6 b c^3 d f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{16 b c d g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{b c^3 d f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g^3 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{b d 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 )}{8 c}-\frac{2 d g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}+\frac{3}{8} d f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac{d g^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 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 )^2+\frac{3}{35} d g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 )^2+\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 )^2+\frac{1}{7} d 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{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 )^2}{5 c^2}+\frac{d f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}+\frac{d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{16 b c^3 \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 d f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{64 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 d f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d f^2 g \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{15-10 c^2 x+3 c^4 x^2}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{25 \sqrt{1-c^2 x^2}}+\frac{\left (9 b^2 d f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{64 \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 d f g^2 \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 (b^2 c^2 d f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{9 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 d g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{1-c^2 x^2}} \, dx}{105 \sqrt{1-c^2 x^2}}+\frac{\left (4 b d g^3 \sqrt{d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d 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 )}{245 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^2 d 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 )}{175 \sqrt{1-c^2 x^2}}\\ &=-\frac{15}{64} b^2 d f^3 x \sqrt{d-c^2 d x^2}+\frac{3 b^2 d f g^2 x \sqrt{d-c^2 d x^2}}{128 c^2}-\frac{43}{576} b^2 d f g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{1}{36} b^2 c^2 d f g^2 x^5 \sqrt{d-c^2 d x^2}+\frac{4 a b d g^3 x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}-\frac{1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{9 b^2 d f^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{6 b d f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{3 b c d f^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3 b d f g^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt{1-c^2 x^2}}-\frac{4 b c d f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt{1-c^2 x^2}}+\frac{2 b d g^3 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt{1-c^2 x^2}}-\frac{7 b c d 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{6 b c^3 d f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{16 b c d g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{b c^3 d f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g^3 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{b d 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 )}{8 c}-\frac{2 d g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}+\frac{3}{8} d f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac{d g^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 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 )^2+\frac{3}{35} d g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 )^2+\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 )^2+\frac{1}{7} d 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{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 )^2}{5 c^2}+\frac{d f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}+\frac{d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{16 b c^3 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d f^2 g \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{1-c^2 x}}+4 \sqrt{1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{25 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 d f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{12 \sqrt{1-c^2 x^2}}+\frac{\left (9 b^2 d f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 d f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{32 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d 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 )}{105 \sqrt{1-c^2 x^2}}+\frac{\left (4 b^2 d g^3 \sqrt{d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d 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 )}{245 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^2 d 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 )}{175 \sqrt{1-c^2 x^2}}\\ &=\frac{16 b^2 d f^2 g \sqrt{d-c^2 d x^2}}{25 c^2}-\frac{62 b^2 d g^3 \sqrt{d-c^2 d x^2}}{1225 c^4}-\frac{15}{64} b^2 d f^3 x \sqrt{d-c^2 d x^2}-\frac{7 b^2 d f g^2 x \sqrt{d-c^2 d x^2}}{384 c^2}-\frac{43}{576} b^2 d f g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{1}{36} b^2 c^2 d f g^2 x^5 \sqrt{d-c^2 d x^2}+\frac{4 a b d g^3 x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{8 b^2 d f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{74 b^2 d g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3675 c^4}-\frac{1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{6 b^2 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{38 b^2 d g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{6125 c^4}-\frac{2 b^2 d g^3 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^4}+\frac{9 b^2 d f^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}-\frac{3 b^2 d f g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{128 c^3 \sqrt{1-c^2 x^2}}+\frac{4 b^2 d g^3 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{6 b d f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{3 b c d f^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3 b d f g^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt{1-c^2 x^2}}-\frac{4 b c d f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt{1-c^2 x^2}}+\frac{2 b d g^3 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt{1-c^2 x^2}}-\frac{7 b c d 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{6 b c^3 d f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{16 b c d g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{b c^3 d f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g^3 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{b d 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 )}{8 c}-\frac{2 d g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}+\frac{3}{8} d f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac{d g^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 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 )^2+\frac{3}{35} d g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 )^2+\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 )^2+\frac{1}{7} d 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{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 )^2}{5 c^2}+\frac{d f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}+\frac{d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{16 b c^3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 d f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{24 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d 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 )}{105 \sqrt{1-c^2 x^2}}-\frac{\left (4 b^2 d g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{35 c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{16 b^2 d f^2 g \sqrt{d-c^2 d x^2}}{25 c^2}+\frac{304 b^2 d g^3 \sqrt{d-c^2 d x^2}}{3675 c^4}-\frac{15}{64} b^2 d f^3 x \sqrt{d-c^2 d x^2}-\frac{7 b^2 d f g^2 x \sqrt{d-c^2 d x^2}}{384 c^2}-\frac{43}{576} b^2 d f g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{1}{36} b^2 c^2 d f g^2 x^5 \sqrt{d-c^2 d x^2}+\frac{4 a b d g^3 x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{8 b^2 d f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{152 b^2 d g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{11025 c^4}-\frac{1}{32} b^2 d f^3 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{6 b^2 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{38 b^2 d g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{6125 c^4}-\frac{2 b^2 d g^3 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^4}+\frac{9 b^2 d f^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{7 b^2 d f g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{384 c^3 \sqrt{1-c^2 x^2}}+\frac{4 b^2 d g^3 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{6 b d f^2 g x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt{1-c^2 x^2}}-\frac{3 b c d f^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3 b d f g^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt{1-c^2 x^2}}-\frac{4 b c d f^2 g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 \sqrt{1-c^2 x^2}}+\frac{2 b d g^3 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt{1-c^2 x^2}}-\frac{7 b c d 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{6 b c^3 d f^2 g x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt{1-c^2 x^2}}-\frac{16 b c d g^3 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{b c^3 d f g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{6 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g^3 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{b d 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 )}{8 c}-\frac{2 d g^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}+\frac{3}{8} d f^3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{3 d f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac{d g^3 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 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 )^2+\frac{3}{35} d g^3 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^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 )^2+\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 )^2+\frac{1}{7} d 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{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 )^2}{5 c^2}+\frac{d f^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt{1-c^2 x^2}}+\frac{d f g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{16 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 2.48067, size = 872, normalized size = 0.52 \[ \frac{d \sqrt{d-c^2 d x^2} \left (3087000 c f \left (2 c^2 f^2+g^2\right ) a^3-88200 b \sqrt{1-c^2 x^2} \left (4 x^3 \left (35 f^3+84 g x f^2+70 g^2 x^2 f+20 g^3 x^3\right ) c^6-2 x \left (175 f^3+336 g x f^2+245 g^2 x^2 f+64 g^3 x^3\right ) c^4+g \left (336 f^2+105 g x f+16 g^2 x^2\right ) c^2+32 g^3\right ) a^2+840 b^2 c x \left (2 x^3 \left (3675 f^3+7056 g x f^2+4900 g^2 x^2 f+1200 g^3 x^3\right ) c^6-21 x \left (1750 f^3+2240 g x f^2+1225 g^2 x^2 f+256 g^3 x^3\right ) c^4+35 g \left (2016 f^2+315 g x f+32 g^2 x^2\right ) c^2+6720 g^3\right ) a+3087000 b^3 c f \left (2 c^2 f^2+g^2\right ) \sin ^{-1}(c x)^3-88200 b^2 \left (b \sqrt{1-c^2 x^2} \left (4 x^3 \left (35 f^3+84 g x f^2+70 g^2 x^2 f+20 g^3 x^3\right ) c^6-2 x \left (175 f^3+336 g x f^2+245 g^2 x^2 f+64 g^3 x^3\right ) c^4+g \left (336 f^2+105 g x f+16 g^2 x^2\right ) c^2+32 g^3\right )-105 a c f \left (2 c^2 f^2+g^2\right )\right ) \sin ^{-1}(c x)^2+b^3 \sqrt{1-c^2 x^2} \left (4 x^3 \left (385875 f^3+592704 g x f^2+343000 g^2 x^2 f+72000 g^3 x^3\right ) c^6-2 x \left (6559875 f^3+5005056 g x f^2+1843625 g^2 x^2 f+278784 g^3 x^3\right ) c^4+g \left (39250176 f^2-900375 g x f-429824 g^2 x^2\right ) c^2+4785152 g^3\right )+105 b \left (88200 c f \left (2 c^2 f^2+g^2\right ) a^2-1680 b \sqrt{1-c^2 x^2} \left (4 x^3 \left (35 f^3+84 g x f^2+70 g^2 x^2 f+20 g^3 x^3\right ) c^6-2 x \left (175 f^3+336 g x f^2+245 g^2 x^2 f+64 g^3 x^3\right ) c^4+g \left (336 f^2+105 g x f+16 g^2 x^2\right ) c^2+32 g^3\right ) a+b^2 c \left (16 x^4 \left (3675 f^3+7056 g x f^2+4900 g^2 x^2 f+1200 g^3 x^3\right ) c^6-168 x^2 \left (1750 f^3+2240 g x f^2+1225 g^2 x^2 f+256 g^3 x^3\right ) c^4+70 \left (1785 f^3+8064 g x f^2+1260 g^2 x^2 f+128 g^3 x^3\right ) c^2+35 g^2 (245 f+1536 g x)\right )\right ) \sin ^{-1}(c x)\right )}{49392000 b c^4 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
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Maple [B] time = 0.806, size = 4018, normalized size = 2.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a^{2} c^{2} d g^{3} x^{5} + 3 \, a^{2} c^{2} d f g^{2} x^{4} - 3 \, a^{2} d f^{2} g x - a^{2} d f^{3} +{\left (3 \, a^{2} c^{2} d f^{2} g - a^{2} d g^{3}\right )} x^{3} +{\left (a^{2} c^{2} d f^{3} - 3 \, a^{2} d f g^{2}\right )} x^{2} +{\left (b^{2} c^{2} d g^{3} x^{5} + 3 \, b^{2} c^{2} d f g^{2} x^{4} - 3 \, b^{2} d f^{2} g x - b^{2} d f^{3} +{\left (3 \, b^{2} c^{2} d f^{2} g - b^{2} d g^{3}\right )} x^{3} +{\left (b^{2} c^{2} d f^{3} - 3 \, b^{2} d f g^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d g^{3} x^{5} + 3 \, a b c^{2} d f g^{2} x^{4} - 3 \, a b d f^{2} g x - a b d f^{3} +{\left (3 \, a b c^{2} d f^{2} g - a b d g^{3}\right )} x^{3} +{\left (a b c^{2} d f^{3} - 3 \, a b d f g^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{3}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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