Optimal. Leaf size=1648 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.68744, antiderivative size = 1648, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 28, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.903, Rules used = {4777, 4767, 4647, 4641, 30, 4677, 4697, 4707, 266, 43, 4689, 12, 4765, 683, 4757, 6742, 725, 204, 1654, 4799, 4797, 8, 4773, 3323, 2264, 2190, 2279, 2391} \[ -\frac{b d^2 x^5 \sqrt{d-c^2 d x^2} c^5}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 f x^4 \sqrt{d-c^2 d x^2} c^5}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) c^4}{4 g^2}-\frac{b d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2} c^3}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b d^2 x^3 \sqrt{d-c^2 d x^2} c^3}{45 g \sqrt{1-c^2 x^2}}+\frac{b d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2} c^3}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b d^2 f x^2 \sqrt{d-c^2 d x^2} c^3}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) c^2}{2 g^4}+\frac{d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) c^2}{8 g^2}-\frac{d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c}{2 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 c}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{b d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} c}{g^5 \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} c}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 x \sqrt{d-c^2 d x^2} c}{15 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{a d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) c}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^6 (f+g x) \sqrt{1-c^2 x^2} c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4777
Rule 4767
Rule 4647
Rule 4641
Rule 30
Rule 4677
Rule 4697
Rule 4707
Rule 266
Rule 43
Rule 4689
Rule 12
Rule 4765
Rule 683
Rule 4757
Rule 6742
Rule 725
Rule 204
Rule 1654
Rule 4799
Rule 4797
Rule 8
Rule 4773
Rule 3323
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{f+g x} \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{c^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g^4}+\frac{c^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g^3}-\frac{c^4 f x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g^2}+\frac{c^4 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g}+\frac{\left (-c^2 f^2+g^2\right )^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g^4 (f+g x)}\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\left (c^4 d^2 f \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}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (c^4 d^2 \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 ) \, dx}{g \sqrt{1-c^2 x^2}}-\frac{\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{g^4 \sqrt{1-c^2 x^2}}+\frac{\left (c^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{g^3 \sqrt{1-c^2 x^2}}+\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{f+g x} \, dx}{g^4 \sqrt{1-c^2 x^2}}\\ &=-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (c^4 d^2 f \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}{4 g^2 \sqrt{1-c^2 x^2}}+\frac{\left (b c^5 d^2 f \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{\left (b c^5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{g \sqrt{1-c^2 x^2}}-\frac{\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 g^4 \sqrt{1-c^2 x^2}}+\frac{\left (b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 g^4 \sqrt{1-c^2 x^2}}+\frac{\left (b c d^2 \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-g-2 c^2 f x-c^2 g x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c g^4 \sqrt{1-c^2 x^2}}\\ &=\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (c^2 d^2 f \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 g^2 \sqrt{1-c^2 x^2}}-\frac{\left (b c^3 d^2 f \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 g^2 \sqrt{1-c^2 x^2}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{15 g \sqrt{1-c^2 x^2}}+\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (\frac{1}{f+g x}-\frac{c^2 \left (g x+\frac{f^2}{f+g x}\right )}{g^2}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{g^4 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}+\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt{1-c^2 x^2}}-\frac{b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{g^2 (f+g x) \sqrt{1-c^2 x^2}}\right ) \, dx}{g^4 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^6 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 g^2 \left (c^2 f^2-g^2\right )}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{c^2 g^8 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^2 g x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{\left (c^2 f^2-g^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt{1-c^2 x^2}}\right ) \, dx}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (b c^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{g^5 \sqrt{1-c^2 x^2}}-\frac{\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^6 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\sin ^{-1}(c x)}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{\left (b c d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{g^5 \sqrt{1-c^2 x^2}}+\frac{\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-c^2 f^2+g^2-x^2} \, dx,x,\frac{g+c^2 f x}{\sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{\left (2 i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f-2 i e^{i x} g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g^5 \sqrt{1-c^2 x^2}}-\frac{\left (2 i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f-2 i e^{i x} g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g^5 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{\left (i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e^{i x} g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{\left (i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e^{i x} g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i g x}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i g x}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^6 \sqrt{1-c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{1-c^2 x^2}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1-c^2 x^2}}+\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1-c^2 x^2}}+\frac{d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}+\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}-\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 2.68865, size = 787, normalized size = 0.48 \[ -\frac{d^2 \sqrt{d-c^2 d x^2} \left (-\frac{1800 \left (g^2-c^2 f^2\right )^2 \left (-2 b c \left (-i \sqrt{c^2 f^2-g^2} \left (-i b \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )+i b \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )+\left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right )-\log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )\right )\right )-g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+b c g x\right )+\frac{\left (c^2 f^2-g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x}+c^2 g x \left (a+b \sin ^{-1}(c x)\right )^2\right )}{b c g^2}+1800 c^2 f x \sqrt{1-c^2 x^2} \left (c^2 f^2-2 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )+1200 g \left (1-c^2 x^2\right )^{3/2} \left (c^2 f^2-2 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1800 \left (c^2 x^2-1\right ) \left (g^2-c^2 f^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{b c (f+g x)}+\frac{900 c f \left (c^2 f^2-2 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{b}+900 c^4 f g^2 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+225 c f g^2 \left (-2 c x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b}+b c^2 x^2\right )-720 c^4 g^3 x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-80 g^3 \left (-3 c^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-6 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+b c^3 x^3+6 b c x\right )-900 b c^3 f x^2 \left (c^2 f^2-2 g^2\right )+400 b c g x \left (c^2 x^2-3\right ) \left (c^2 f^2-2 g^2\right )-225 b c^5 f g^2 x^4+144 b c^5 g^3 x^5\right )}{3600 g^4 \sqrt{1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.419, size = 4685, normalized size = 2.8 \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 (\frac{{\left (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{g x + f}, 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 \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}}{g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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