Optimal. Leaf size=1108 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.52367, antiderivative size = 1108, normalized size of antiderivative = 1., number of steps used = 36, number of rules used = 21, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {4777, 4763, 4649, 4647, 4641, 4627, 321, 216, 4677, 195, 194, 4645, 12, 1247, 698, 4699, 4697, 4707, 14, 4687, 459} \[ \frac{b c^3 d g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^6}{18 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 d f g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^5}{25 \sqrt{1-c^2 x^2}}+\frac{1}{108} b^2 c^2 d g^2 \sqrt{d-c^2 d x^2} x^5-\frac{7 b c d g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^4}{48 \sqrt{1-c^2 x^2}}+\frac{1}{8} d g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3+\frac{1}{6} d g^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3-\frac{8 b c d f g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{15 \sqrt{1-c^2 x^2}}-\frac{43 b^2 d g^2 \sqrt{d-c^2 d x^2} x^3}{1728}-\frac{3 b c d f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{8 \sqrt{1-c^2 x^2}}+\frac{b d g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{16 c \sqrt{1-c^2 x^2}}+\frac{3}{8} d f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x-\frac{d g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x}{16 c^2}+\frac{1}{4} d f^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x+\frac{4 b d f g \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{5 c \sqrt{1-c^2 x^2}}-\frac{15}{64} b^2 d f^2 \sqrt{d-c^2 d x^2} x-\frac{7 b^2 d g^2 \sqrt{d-c^2 d x^2} x}{1152 c^2}-\frac{1}{32} b^2 d f^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} x+\frac{d f^2 \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 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{1-c^2 x^2}}-\frac{2 d f 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{9 b^2 d f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{7 b^2 d g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt{1-c^2 x^2}}+\frac{b d f^2 \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{4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{32 b^2 d f g \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{225 c^2} \]
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
Rubi steps
\begin{align*} \int (f+g x)^2 \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)^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 \sqrt{d-c^2 d x^2}\right ) \int \left (f^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+2 f g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+g^2 x^2 \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^2 \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 (2 d f 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 (d 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{1}{4} d f^2 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}{6} d 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{2 d f 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^2 \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^2 \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 (4 b d f 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 (d 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 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}{3 \sqrt{1-c^2 x^2}}\\ &=\frac{4 b d f 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{8 b c d f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}-\frac{b c d g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 d f 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{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt{1-c^2 x^2}}+\frac{b d f^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d f^2 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}{6} d 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{2 d f 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^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}{8 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d f^2 \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^2 \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 (4 b^2 d f 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 (d 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 (b c d 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 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}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{4 b d f 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^2 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{8 b c d f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}-\frac{7 b c d g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 d f 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{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt{1-c^2 x^2}}+\frac{b d f^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d f^2 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}{6} d 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{2 d f 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^2 \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^2 \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^2 \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 (4 b^2 d f 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}{75 \sqrt{1-c^2 x^2}}+\frac{\left (d 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 (b d 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 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}{36 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d 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{15}{64} b^2 d f^2 x \sqrt{d-c^2 d x^2}-\frac{1}{64} b^2 d g^2 x^3 \sqrt{d-c^2 d x^2}+\frac{1}{108} b^2 c^2 d g^2 x^5 \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{4 b d f 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^2 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{b d 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{8 b c d f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}-\frac{7 b c d g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 d f 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{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt{1-c^2 x^2}}+\frac{b d f^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d f^2 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}{6} d 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{2 d f 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^2 \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 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 d f^2 \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^2 \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 (2 b^2 d f 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 )}{75 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 d 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 (b^2 d 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 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{27 \sqrt{1-c^2 x^2}}\\ &=-\frac{15}{64} b^2 d f^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 d g^2 x \sqrt{d-c^2 d x^2}}{128 c^2}-\frac{43 b^2 d g^2 x^3 \sqrt{d-c^2 d x^2}}{1728}+\frac{1}{108} b^2 c^2 d g^2 x^5 \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{9 b^2 d f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{4 b d f 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^2 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{b d 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{8 b c d f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}-\frac{7 b c d g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 d f 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{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt{1-c^2 x^2}}+\frac{b d f^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d f^2 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}{6} d 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{2 d f 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^2 \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 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 d f 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 )}{75 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 d g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{36 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 d 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 (b^2 d 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{32 b^2 d f g \sqrt{d-c^2 d x^2}}{75 c^2}-\frac{15}{64} b^2 d f^2 x \sqrt{d-c^2 d x^2}-\frac{7 b^2 d g^2 x \sqrt{d-c^2 d x^2}}{1152 c^2}-\frac{43 b^2 d g^2 x^3 \sqrt{d-c^2 d x^2}}{1728}+\frac{1}{108} b^2 c^2 d g^2 x^5 \sqrt{d-c^2 d x^2}+\frac{16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{225 c^2}-\frac{1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{9 b^2 d f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}-\frac{b^2 d 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 d f 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^2 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{b d 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{8 b c d f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}-\frac{7 b c d g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 d f 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{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt{1-c^2 x^2}}+\frac{b d f^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d f^2 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}{6} d 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{2 d f 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^2 \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 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 d g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{72 c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{32 b^2 d f g \sqrt{d-c^2 d x^2}}{75 c^2}-\frac{15}{64} b^2 d f^2 x \sqrt{d-c^2 d x^2}-\frac{7 b^2 d g^2 x \sqrt{d-c^2 d x^2}}{1152 c^2}-\frac{43 b^2 d g^2 x^3 \sqrt{d-c^2 d x^2}}{1728}+\frac{1}{108} b^2 c^2 d g^2 x^5 \sqrt{d-c^2 d x^2}+\frac{16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{225 c^2}-\frac{1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}+\frac{4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{9 b^2 d f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{7 b^2 d g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt{1-c^2 x^2}}+\frac{4 b d f 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^2 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{b d 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{8 b c d f g x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt{1-c^2 x^2}}-\frac{7 b c d g^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 d f 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{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt{1-c^2 x^2}}+\frac{b d f^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d f^2 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}{6} d 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{2 d f 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^2 \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 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.03594, size = 616, normalized size = 0.56 \[ \frac{d \sqrt{d-c^2 d x^2} \left (15 b \sin ^{-1}(c x) \left (1800 a^2 \left (6 c^2 f^2+g^2\right )-240 a b c \sqrt{1-c^2 x^2} \left (30 c^2 f^2 x \left (2 c^2 x^2-5\right )+96 f g \left (c^2 x^2-1\right )^2+5 g^2 x \left (8 c^4 x^4-14 c^2 x^2+3\right )\right )+b^2 \left (16 c^6 x^4 \left (225 f^2+288 f g x+100 g^2 x^2\right )-120 c^4 x^2 \left (150 f^2+128 f g x+35 g^2 x^2\right )+90 c^2 \left (85 f^2+256 f g x+20 g^2 x^2\right )+175 g^2\right )\right )-1800 a^2 b c \sqrt{1-c^2 x^2} \left (30 c^2 f^2 x \left (2 c^2 x^2-5\right )+96 f g \left (c^2 x^2-1\right )^2+5 g^2 x \left (8 c^4 x^4-14 c^2 x^2+3\right )\right )+9000 a^3 \left (6 c^2 f^2+g^2\right )+120 a b^2 c^2 x \left (450 c^2 f^2 x \left (c^2 x^2-5\right )+192 f g \left (3 c^4 x^4-10 c^2 x^2+15\right )+25 g^2 x \left (8 c^4 x^4-21 c^2 x^2+9\right )\right )+1800 b^2 \sin ^{-1}(c x)^2 \left (15 a \left (6 c^2 f^2+g^2\right )-b c \sqrt{1-c^2 x^2} \left (30 c^2 f^2 x \left (2 c^2 x^2-5\right )+96 f g \left (c^2 x^2-1\right )^2+5 g^2 x \left (8 c^4 x^4-14 c^2 x^2+3\right )\right )\right )+b^3 c \sqrt{1-c^2 x^2} \left (6750 c^2 f^2 x \left (2 c^2 x^2-17\right )+1536 f g \left (9 c^4 x^4-38 c^2 x^2+149\right )+125 g^2 x \left (32 c^4 x^4-86 c^2 x^2-21\right )\right )+9000 b^3 \left (6 c^2 f^2+g^2\right ) \sin ^{-1}(c x)^3\right )}{432000 b c^3 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.855, size = 2850, normalized size = 2.6 \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^{2} x^{4} + 2 \, a^{2} c^{2} d f g x^{3} - 2 \, a^{2} d f g x - a^{2} d f^{2} +{\left (a^{2} c^{2} d f^{2} - a^{2} d g^{2}\right )} x^{2} +{\left (b^{2} c^{2} d g^{2} x^{4} + 2 \, b^{2} c^{2} d f g x^{3} - 2 \, b^{2} d f g x - b^{2} d f^{2} +{\left (b^{2} c^{2} d f^{2} - b^{2} d g^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d g^{2} x^{4} + 2 \, a b c^{2} d f g x^{3} - 2 \, a b d f g x - a b d f^{2} +{\left (a b c^{2} d f^{2} - a b d 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 )}^{2}{\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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