Integrand size = 30, antiderivative size = 315 \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=-\frac {25 b c x^2 (d+c d x)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b c^3 x^4 (d+c d x)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}}+\frac {b (d+c d x)^{5/2} (f-c f x)^{5/2} \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))^2}{32 b c \left (1-c^2 x^2\right )^{5/2}} \]
[Out]
Time = 0.19 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {4763, 4743, 4741, 4737, 30, 14, 267} \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\frac {5 x (c d x+d)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))}{24 \left (1-c^2 x^2\right )}+\frac {5 x (c d x+d)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))}{16 \left (1-c^2 x^2\right )^2}+\frac {5 (c d x+d)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))^2}{32 b c \left (1-c^2 x^2\right )^{5/2}}+\frac {1}{6} x (c d x+d)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))-\frac {25 b c x^2 (c d x+d)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}}+\frac {b \sqrt {1-c^2 x^2} (c d x+d)^{5/2} (f-c f x)^{5/2}}{36 c}+\frac {5 b c^3 x^4 (c d x+d)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}} \]
[In]
[Out]
Rule 14
Rule 30
Rule 267
Rule 4737
Rule 4741
Rule 4743
Rule 4763
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx}{\left (1-c^2 x^2\right )^{5/2}} \\ & = \frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))+\frac {\left (5 (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx}{6 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (b c (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int x \left (1-c^2 x^2\right )^2 \, dx}{6 \left (1-c^2 x^2\right )^{5/2}} \\ & = \frac {b (d+c d x)^{5/2} (f-c f x)^{5/2} \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))}{24 \left (1-c^2 x^2\right )}+\frac {\left (5 (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{8 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{24 \left (1-c^2 x^2\right )^{5/2}} \\ & = \frac {b (d+c d x)^{5/2} (f-c f x)^{5/2} \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))}{24 \left (1-c^2 x^2\right )}+\frac {\left (5 (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{24 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int x \, dx}{16 \left (1-c^2 x^2\right )^{5/2}} \\ & = -\frac {25 b c x^2 (d+c d x)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b c^3 x^4 (d+c d x)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}}+\frac {b (d+c d x)^{5/2} (f-c f x)^{5/2} \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x))^2}{32 b c \left (1-c^2 x^2\right )^{5/2}} \\ \end{align*}
Time = 3.06 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.96 \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\frac {d^2 f^2 \left (360 b \sqrt {d+c d x} \sqrt {f-c f x} \arcsin (c x)^2-720 a \sqrt {d} \sqrt {f} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )+\sqrt {d+c d x} \sqrt {f-c f x} \left (1584 a c x \sqrt {1-c^2 x^2}-1248 a c^3 x^3 \sqrt {1-c^2 x^2}+384 a c^5 x^5 \sqrt {1-c^2 x^2}+270 b \cos (2 \arcsin (c x))+27 b \cos (4 \arcsin (c x))+2 b \cos (6 \arcsin (c x))\right )+12 b \sqrt {d+c d x} \sqrt {f-c f x} \arcsin (c x) (45 \sin (2 \arcsin (c x))+9 \sin (4 \arcsin (c x))+\sin (6 \arcsin (c x)))\right )}{2304 c \sqrt {1-c^2 x^2}} \]
[In]
[Out]
\[\int \left (c d x +d \right )^{\frac {5}{2}} \left (-c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )d x\]
[In]
[Out]
\[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
[In]
[Out]
\[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^{5/2}\,{\left (f-c\,f\,x\right )}^{5/2} \,d x \]
[In]
[Out]