Integrand size = 31, antiderivative size = 1073 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}+\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g^3 \sqrt {1-c^2 x^2}}-\frac {d \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c g^4 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d (c f-g) (c f+g) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c g^2 (f+g x)}+\frac {a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}} \]
[Out]
Time = 1.47 (sec) , antiderivative size = 1073, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.742, Rules used = {4861, 4851, 4741, 4737, 30, 4767, 4849, 697, 4841, 6874, 739, 210, 1668, 12, 4883, 4881, 8, 4857, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\frac {b d x^3 \sqrt {d-c^2 d x^2} c^3}{9 g \sqrt {1-c^2 x^2}}-\frac {b d f x^2 \sqrt {d-c^2 d x^2} c^3}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) c^2}{2 g^2}-\frac {d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 c}{2 b g^3 \sqrt {1-c^2 x^2}}+\frac {d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 c}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {b d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} c}{g^3 \sqrt {1-c^2 x^2}}-\frac {b d x \sqrt {d-c^2 d x^2} c}{3 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^3}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {d (c f-g) (c f+g) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g^2 (f+g x) c}-\frac {d \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g^4 (f+g x) \sqrt {1-c^2 x^2} c} \]
[In]
[Out]
Rule 8
Rule 12
Rule 30
Rule 210
Rule 697
Rule 739
Rule 1668
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3404
Rule 4737
Rule 4741
Rule 4767
Rule 4841
Rule 4849
Rule 4851
Rule 4857
Rule 4861
Rule 4881
Rule 4883
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{g^2}-\frac {c^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{g}+\frac {\left (-c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{g^2 (f+g x)}\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{f+g x} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 d f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (c^2 d \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{g \sqrt {1-c^2 x^2}} \\ & = \frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}-\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-g-2 c^2 f x-c^2 g x^2\right ) (a+b \arcsin (c x))^2}{(f+g x)^2} \, dx}{2 b c \sqrt {1-c^2 x^2}}+\frac {\left (c^2 d f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 d f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 g \sqrt {1-c^2 x^2}} \\ & = -\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}+\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \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 ) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}+\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \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 ) \arcsin (c x)}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}-\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) \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^2 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) \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 ) \arcsin (c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}} \\ & = -\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}-\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) \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^4 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \arcsin (c x)}{(f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^2 \sqrt {1-c^2 x^2}} \\ & = -\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}-\frac {\left (b c^2 d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {x \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {\arcsin (c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}} \\ & = -\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}-\frac {\left (b c d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt {1-c^2 x^2}}+\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {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^2 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \sin (x)} \, dx,x,\arcsin (c x)\right )}{g^2 \sqrt {1-c^2 x^2}} \\ & = -\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\arcsin (c x)\right )}{g^2 \sqrt {1-c^2 x^2}} \\ & = -\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {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,\arcsin (c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {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,\arcsin (c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \\ & = -\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {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,\arcsin (c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {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,\arcsin (c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \\ & = -\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {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 \arcsin (c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {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 \arcsin (c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \\ & = -\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 1.19 (sec) , antiderivative size = 507, normalized size of antiderivative = 0.47 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (-9 b c^3 f x^2+4 b c g x \left (-3+c^2 x^2\right )+18 c^2 f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+12 g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {9 c f (a+b \arcsin (c x))^2}{b}+\frac {18 \left (c^2 f^2-g^2\right ) \left (-1+c^2 x^2\right ) (a+b \arcsin (c x))^2}{b c (f+g x)}-\frac {18 \left (c^2 f^2-g^2\right ) \left (\left (c^2 f^2-g^2\right ) (a+b \arcsin (c x))^2+c^2 g x (f+g x) (a+b \arcsin (c x))^2-2 b c (f+g x) \left (b c g x-g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-i \sqrt {c^2 f^2-g^2} \left ((a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-i b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+i b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )\right )}{b c g^2 (f+g x)}\right )}{36 g^2 \sqrt {1-c^2 x^2}} \]
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Time = 0.64 (sec) , antiderivative size = 1546, normalized size of antiderivative = 1.44
method | result | size |
default | \(\text {Expression too large to display}\) | \(1546\) |
parts | \(\text {Expression too large to display}\) | \(1546\) |
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{g x + f} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{f + g x}\, dx \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{f+g\,x} \,d x \]
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