Integrand size = 31, antiderivative size = 1281 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=-\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2}}{256 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{9 c^4}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \]
[Out]
Time = 0.77 (sec) , antiderivative size = 1281, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {4862, 4848, 4744, 4742, 4738, 30, 14, 267, 4768, 200, 4788, 4784, 4796, 272, 45, 4780, 12, 380} \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {b c^5 d^2 g^3 \sqrt {d-c^2 d x^2} x^9}{81 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f g^2 \sqrt {d-c^2 d x^2} x^8}{64 \sqrt {1-c^2 x^2}}-\frac {19 b c^3 d^2 g^3 \sqrt {d-c^2 d x^2} x^7}{441 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 \sqrt {d-c^2 d x^2} x^6}{96 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g^3 \sqrt {d-c^2 d x^2} x^5}{21 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 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^3 \sqrt {d-c^2 d x^2} x^4}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 f g^2 \sqrt {d-c^2 d x^2} x^4}{256 \sqrt {1-c^2 x^2}}+\frac {15}{64} d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x^3+\frac {3}{8} d^2 f g^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x^3+\frac {5}{16} d^2 f g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x^3-\frac {b d^2 g^3 \sqrt {d-c^2 d x^2} x^3}{189 c \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g \sqrt {d-c^2 d x^2} x^3}{7 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^3 \sqrt {d-c^2 d x^2} x^2}{96 \sqrt {1-c^2 x^2}}-\frac {15 b d^2 f 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^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x}{128 c^2}+\frac {1}{6} d^2 f^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x+\frac {5}{24} d^2 f^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) x-\frac {2 b d^2 g^3 \sqrt {d-c^2 d x^2} x}{63 c^3 \sqrt {1-c^2 x^2}}-\frac {3 b d^2 f^2 g \sqrt {d-c^2 d x^2} x}{7 c \sqrt {1-c^2 x^2}}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{9 c^4}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^4}-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c} \]
[In]
[Out]
Rule 12
Rule 14
Rule 30
Rule 45
Rule 200
Rule 267
Rule 272
Rule 380
Rule 4738
Rule 4742
Rule 4744
Rule 4768
Rule 4780
Rule 4784
Rule 4788
Rule 4796
Rule 4848
Rule 4862
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (f+g x)^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (f^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+3 f^2 g x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+3 f g^2 x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+g^3 x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) \, dx}{6 \sqrt {1-c^2 x^2}}+\frac {\left (b c d^2 f^3 \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 (3 b d^2 f^2 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 (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (3 b c d^2 f 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 {\left (b c d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 c^4} \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 f^3 \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 (3 b d^2 f^2 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 (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (3 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \text {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 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (b d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3 \, dx}{63 c^3 \sqrt {1-c^2 x^2}} \\ & = -\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c d^2 f^3 \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^3 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (3 b c d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \text {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 (15 b c d^2 f 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 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (b d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+15 c^4 x^4-19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt {1-c^2 x^2}} \\ & = -\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2}}{256 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{9 c^4}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (15 b d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {1-c^2 x^2}} \\ & = -\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {25 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}+\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2}}{256 \sqrt {1-c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}+\frac {b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{9 c^4}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c \sqrt {1-c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 9.41 (sec) , antiderivative size = 1582, normalized size of antiderivative = 1.23 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (-\frac {a d^2 g \left (27 c^2 f^2+2 g^2\right )}{63 c^4}+\frac {a d^2 f \left (88 c^2 f^2-15 g^2\right ) x}{128 c^2}-\frac {a d^2 g \left (-81 c^2 f^2+g^2\right ) x^2}{63 c^2}-\frac {1}{192} a d^2 f \left (104 c^2 f^2-177 g^2\right ) x^3+\frac {1}{21} a d^2 g \left (-27 c^2 f^2+5 g^2\right ) x^4+\frac {1}{48} a c^2 d^2 f \left (8 c^2 f^2-51 g^2\right ) x^5-\frac {1}{63} a c^2 d^2 g \left (-27 c^2 f^2+19 g^2\right ) x^6+\frac {3}{8} a c^4 d^2 f g^2 x^7+\frac {1}{9} a c^4 d^2 g^3 x^8\right )-\frac {5 a d^{5/2} f \left (8 c^2 f^2+3 g^2\right ) \arctan \left (\frac {c x \sqrt {-d \left (-1+c^2 x^2\right )}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{128 c^3}-\frac {b d^2 f^2 g \sqrt {d \left (1-c^2 x^2\right )} \left (9 c x+12 \left (1-c^2 x^2\right )^{3/2} \arccos (c x)-\cos (3 \arccos (c x))\right )}{12 c^2 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^2 g \sqrt {d \left (1-c^2 x^2\right )} \left (55125 c x-1225 \cos (3 \arccos (c x))+840 \left (1-c^2 x^2\right )^{3/2} \arccos (c x) (157+108 \cos (2 \arccos (c x))+15 \cos (4 \arccos (c x)))-1323 \cos (5 \arccos (c x))-225 \cos (7 \arccos (c x))\right )}{235200 c^2 \sqrt {1-c^2 x^2}}+\frac {b d^2 g^3 \sqrt {d \left (1-c^2 x^2\right )} \left (55125 c x-1225 \cos (3 \arccos (c x))+840 \left (1-c^2 x^2\right )^{3/2} \arccos (c x) (157+108 \cos (2 \arccos (c x))+15 \cos (4 \arccos (c x)))-1323 \cos (5 \arccos (c x))-225 \cos (7 \arccos (c x))\right )}{352800 c^4 \sqrt {1-c^2 x^2}}+\frac {b d^2 f^3 \sqrt {d \left (1-c^2 x^2\right )} (\cos (2 \arccos (c x))+2 \arccos (c x) (-\arccos (c x)+\sin (2 \arccos (c x))))}{8 c \sqrt {1-c^2 x^2}}+\frac {b d^2 f^3 \sqrt {d \left (1-c^2 x^2\right )} \left (8 \arccos (c x)^2-\cos (4 \arccos (c x))-4 \arccos (c x) \sin (4 \arccos (c x))\right )}{64 c \sqrt {1-c^2 x^2}}-\frac {3 b d^2 f g^2 \sqrt {d \left (1-c^2 x^2\right )} \left (8 \arccos (c x)^2-\cos (4 \arccos (c x))-4 \arccos (c x) \sin (4 \arccos (c x))\right )}{128 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 f^2 g \sqrt {d \left (1-c^2 x^2\right )} \left (450 c x+450 \sqrt {1-c^2 x^2} \arccos (c x)-25 \cos (3 \arccos (c x))-9 \cos (5 \arccos (c x))-75 \arccos (c x) \sin (3 \arccos (c x))-45 \arccos (c x) \sin (5 \arccos (c x))\right )}{600 c^2 \sqrt {1-c^2 x^2}}-\frac {b d^2 g^3 \sqrt {d \left (1-c^2 x^2\right )} \left (450 c x+450 \sqrt {1-c^2 x^2} \arccos (c x)-25 \cos (3 \arccos (c x))-9 \cos (5 \arccos (c x))-75 \arccos (c x) \sin (3 \arccos (c x))-45 \arccos (c x) \sin (5 \arccos (c x))\right )}{3600 c^4 \sqrt {1-c^2 x^2}}-\frac {b d^2 f^3 \sqrt {d \left (1-c^2 x^2\right )} \left (18 \cos (2 \arccos (c x))-9 \cos (4 \arccos (c x))-2 \left (-36 \arccos (c x)^2+\cos (6 \arccos (c x))-18 \arccos (c x) \sin (2 \arccos (c x))+18 \arccos (c x) \sin (4 \arccos (c x))+6 \arccos (c x) \sin (6 \arccos (c x))\right )\right )}{2304 c \sqrt {1-c^2 x^2}}+\frac {b d^2 f g^2 \sqrt {d \left (1-c^2 x^2\right )} \left (18 \cos (2 \arccos (c x))-9 \cos (4 \arccos (c x))-2 \left (-36 \arccos (c x)^2+\cos (6 \arccos (c x))-18 \arccos (c x) \sin (2 \arccos (c x))+18 \arccos (c x) \sin (4 \arccos (c x))+6 \arccos (c x) \sin (6 \arccos (c x))\right )\right )}{384 c^3 \sqrt {1-c^2 x^2}}-\frac {b d^2 f g^2 \sqrt {d \left (1-c^2 x^2\right )} \left (1440 \arccos (c x)^2+576 \cos (2 \arccos (c x))-144 \cos (4 \arccos (c x))-64 \cos (6 \arccos (c x))-9 \cos (8 \arccos (c x))+1152 \arccos (c x) \sin (2 \arccos (c x))-576 \arccos (c x) \sin (4 \arccos (c x))-384 \arccos (c x) \sin (6 \arccos (c x))-72 \arccos (c x) \sin (8 \arccos (c x))\right )}{24576 c^3 \sqrt {1-c^2 x^2}}-\frac {b d^2 g^3 \sqrt {d \left (1-c^2 x^2\right )} \left (1389150 c x-31752 \cos (5 \arccos (c x))-5 \left (2025 \cos (7 \arccos (c x))+245 \cos (9 \arccos (c x))+63 \arccos (c x) \left (-4410 \sqrt {1-c^2 x^2}+504 \sin (5 \arccos (c x))+225 \sin (7 \arccos (c x))+35 \sin (9 \arccos (c x))\right )\right )\right )}{25401600 c^4 \sqrt {1-c^2 x^2}} \]
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Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 3019, normalized size of antiderivative = 2.36
method | result | size |
default | \(\text {Expression too large to display}\) | \(3019\) |
parts | \(\text {Expression too large to display}\) | \(3019\) |
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\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Timed out} \]
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\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]
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Exception generated. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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