Optimal. Leaf size=940 \[ -\frac{b c^5 d^2 g^2 \sqrt{d-c^2 d x^2} x^8}{64 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 f g \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 g^2 \sqrt{d-c^2 d x^2} x^6}{288 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 f g \sqrt{d-c^2 d x^2} x^5}{35 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d^2 f^2 \sqrt{d-c^2 d x^2} x^4}{96 \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 g^2 \sqrt{d-c^2 d x^2} x^4}{768 \sqrt{1-c^2 x^2}}+\frac{5}{64} d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3+\frac{1}{8} d^2 g^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3+\frac{5}{48} d^2 g^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3-\frac{2 b c d^2 f g \sqrt{d-c^2 d x^2} x^3}{7 \sqrt{1-c^2 x^2}}-\frac{25 b c d^2 f^2 \sqrt{d-c^2 d x^2} x^2}{96 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 g^2 \sqrt{d-c^2 d x^2} x^2}{256 c \sqrt{1-c^2 x^2}}+\frac{5}{16} d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{128 c^2}+\frac{1}{6} d^2 f^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x+\frac{5}{24} d^2 f^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x+\frac{2 b d^2 f g \sqrt{d-c^2 d x^2} x}{7 c \sqrt{1-c^2 x^2}}+\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}+\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1-c^2 x^2}}-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac{b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c} \]
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Rubi [A] time = 0.919956, antiderivative size = 940, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 15, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.484, Rules used = {4777, 4763, 4649, 4647, 4641, 30, 14, 261, 4677, 194, 4699, 4697, 4707, 266, 43} \[ -\frac{b c^5 d^2 g^2 \sqrt{d-c^2 d x^2} x^8}{64 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 f g \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 g^2 \sqrt{d-c^2 d x^2} x^6}{288 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 f g \sqrt{d-c^2 d x^2} x^5}{35 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d^2 f^2 \sqrt{d-c^2 d x^2} x^4}{96 \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 g^2 \sqrt{d-c^2 d x^2} x^4}{768 \sqrt{1-c^2 x^2}}+\frac{5}{64} d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3+\frac{1}{8} d^2 g^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3+\frac{5}{48} d^2 g^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3-\frac{2 b c d^2 f g \sqrt{d-c^2 d x^2} x^3}{7 \sqrt{1-c^2 x^2}}-\frac{25 b c d^2 f^2 \sqrt{d-c^2 d x^2} x^2}{96 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 g^2 \sqrt{d-c^2 d x^2} x^2}{256 c \sqrt{1-c^2 x^2}}+\frac{5}{16} d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{128 c^2}+\frac{1}{6} d^2 f^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x+\frac{5}{24} d^2 f^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x+\frac{2 b d^2 f g \sqrt{d-c^2 d x^2} x}{7 c \sqrt{1-c^2 x^2}}+\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}+\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1-c^2 x^2}}-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac{b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c} \]
Antiderivative was successfully verified.
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Rule 4777
Rule 4763
Rule 4649
Rule 4647
Rule 4641
Rule 30
Rule 14
Rule 261
Rule 4677
Rule 194
Rule 4699
Rule 4697
Rule 4707
Rule 266
Rule 43
Rubi steps
\begin{align*} \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int (f+g x)^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (f^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+2 f g x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+g^2 x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (2 d^2 f g \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac{\left (5 d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{6 \sqrt{1-c^2 x^2}}-\frac{\left (b c d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 \, dx}{6 \sqrt{1-c^2 x^2}}+\frac{\left (2 b d^2 f g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \, dx}{7 c \sqrt{1-c^2 x^2}}+\frac{\left (5 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (b c d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 \, dx}{8 \sqrt{1-c^2 x^2}}\\ &=\frac{b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac{\left (5 d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{24 \sqrt{1-c^2 x^2}}+\frac{\left (2 b d^2 f g \sqrt{d-c^2 d x^2}\right ) \int \left (1-3 c^2 x^2+3 c^4 x^4-c^6 x^6\right ) \, dx}{7 c \sqrt{1-c^2 x^2}}+\frac{\left (5 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{16 \sqrt{1-c^2 x^2}}-\frac{\left (b c d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int x \left (1-c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{48 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d^2 f g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 f g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 f g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 f g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}+\frac{b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac{\left (5 d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{24 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 f^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt{1-c^2 x^2}}+\frac{\left (5 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{64 \sqrt{1-c^2 x^2}}-\frac{\left (b c d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (x-2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{48 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d^2 f g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}-\frac{25 b c d^2 f^2 x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 f g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d^2 f^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 g^2 x^4 \sqrt{d-c^2 d x^2}}{768 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 f g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2}}{288 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 f g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 g^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}+\frac{b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}+\frac{\left (5 d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{128 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (5 b d^2 g^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d^2 f g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}-\frac{25 b c d^2 f^2 x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 g^2 x^2 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 f g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d^2 f^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 g^2 x^4 \sqrt{d-c^2 d x^2}}{768 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 f g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 g^2 x^6 \sqrt{d-c^2 d x^2}}{288 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 f g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 g^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}+\frac{b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{5 d^2 g^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac{5}{64} d^2 g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2}+\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c \sqrt{1-c^2 x^2}}+\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.729484, size = 390, normalized size = 0.41 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (11025 a^2 \left (8 c^2 f^2+g^2\right )+210 a b c \sqrt{1-c^2 x^2} \left (56 c^2 f^2 x \left (8 c^4 x^4-26 c^2 x^2+33\right )+768 f g \left (c^2 x^2-1\right )^3+7 g^2 x \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )\right )+210 b \sin ^{-1}(c x) \left (105 a \left (8 c^2 f^2+g^2\right )+b c \sqrt{1-c^2 x^2} \left (56 c^2 f^2 x \left (8 c^4 x^4-26 c^2 x^2+33\right )+768 f g \left (c^2 x^2-1\right )^3+7 g^2 x \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )\right )\right )+b^2 c^2 x \left (-1960 c^2 f^2 x \left (8 c^4 x^4-39 c^2 x^2+99\right )-4608 f g \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )-245 g^2 x \left (36 c^6 x^6-136 c^4 x^4+177 c^2 x^2-45\right )\right )+11025 b^2 \left (8 c^2 f^2+g^2\right ) \sin ^{-1}(c x)^2\right )}{564480 b c^3 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.664, size = 1633, normalized size = 1.7 \begin{align*} \text{result 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 c^{4} d^{2} g^{2} x^{6} + 2 \, a c^{4} d^{2} f g x^{5} - 4 \, a c^{2} d^{2} f g x^{3} + 2 \, a d^{2} f g x + a d^{2} f^{2} +{\left (a c^{4} d^{2} f^{2} - 2 \, a c^{2} d^{2} g^{2}\right )} x^{4} -{\left (2 \, a c^{2} d^{2} f^{2} - a d^{2} g^{2}\right )} x^{2} +{\left (b c^{4} d^{2} g^{2} x^{6} + 2 \, b c^{4} d^{2} f g x^{5} - 4 \, b c^{2} d^{2} f g x^{3} + 2 \, b d^{2} f g x + b d^{2} f^{2} +{\left (b c^{4} d^{2} f^{2} - 2 \, b c^{2} d^{2} g^{2}\right )} x^{4} -{\left (2 \, b c^{2} d^{2} f^{2} - b d^{2} g^{2}\right )} x^{2}\right )} \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{5}{2}}{\left (g x + f\right )}^{2}{\left (b \arcsin \left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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