Integrand size = 32, antiderivative size = 559 \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\frac {68 b^2 d^3 \left (1-c^2 x^2\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b^2 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 d^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 b^2 d^3 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {22 b d^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b c^2 d^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}} \]
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Time = 0.46 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4763, 4857, 3398, 3377, 2718, 3392, 32, 2715, 8, 2713} \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\frac {5 d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {22 b d^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b c^2 d^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {3 b^2 d^3 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b^2 d^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {3 b^2 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {68 b^2 d^3 \left (1-c^2 x^2\right )}{9 c \sqrt {c d x+d} \sqrt {e-c e x}} \]
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Rule 8
Rule 32
Rule 2713
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 3398
Rule 4763
Rule 4857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {(d+c d x)^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x)^2 (c d+c d \sin (x))^3 \, dx,x,\arcsin (c x)\right )}{c^4 \sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \left (c^3 d^3 (a+b x)^2+3 c^3 d^3 (a+b x)^2 \sin (x)+3 c^3 d^3 (a+b x)^2 \sin ^2(x)+c^3 d^3 (a+b x)^2 \sin ^3(x)\right ) \, dx,x,\arcsin (c x)\right )}{c^4 \sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin ^3(x) \, dx,x,\arcsin (c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (3 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\arcsin (c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (3 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\arcsin (c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b c^2 d^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\arcsin (c x)\right )}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (3 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \, dx,x,\arcsin (c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (6 b d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \cos (x) \, dx,x,\arcsin (c x))}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b^2 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \sin ^3(x) \, dx,x,\arcsin (c x)\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (3 b^2 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \sin ^2(x) \, dx,x,\arcsin (c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {3 b^2 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {6 b d^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b c^2 d^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (4 b d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \cos (x) \, dx,x,\arcsin (c x))}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b^2 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (3 b^2 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int 1 \, dx,x,\arcsin (c x))}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (6 b^2 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int \sin (x) \, dx,x,\arcsin (c x))}{c \sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {56 b^2 d^3 \left (1-c^2 x^2\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b^2 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 d^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 b^2 d^3 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {22 b d^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b c^2 d^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (4 b^2 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int \sin (x) \, dx,x,\arcsin (c x))}{3 c \sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {68 b^2 d^3 \left (1-c^2 x^2\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b^2 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 d^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 b^2 d^3 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {22 b d^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b c^2 d^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}} \\ \end{align*}
Time = 9.93 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.78 \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=-\frac {d^2 \left (-180 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3+540 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-6 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) \left (-18 b+264 b c x+36 b c^2 x^2+8 b c^3 x^3-270 a \sqrt {1-c^2 x^2}-108 a c x \sqrt {1-c^2 x^2}-9 b \cos (2 \arcsin (c x))+6 a \cos (3 \arcsin (c x))\right )+18 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 \left (-30 a+9 b (5+2 c x) \sqrt {1-c^2 x^2}-b \cos (3 \arcsin (c x))\right )+\sqrt {d+c d x} \sqrt {e-c e x} \left (6 \left (-27 b^2 (10+c x) \sqrt {1-c^2 x^2}-8 a b c x \left (33+c^2 x^2\right )+6 a^2 \sqrt {1-c^2 x^2} \left (22+9 c x+2 c^2 x^2\right )\right )+162 a b \cos (2 \arcsin (c x))+4 b^2 \cos (3 \arcsin (c x))\right )\right )}{216 c e \sqrt {1-c^2 x^2}} \]
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\[\int \frac {\left (c d x +d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {-c e x +e}}d x\]
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\[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c e x + e}} \,d x } \]
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Timed out. \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c e x + e}} \,d x } \]
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Timed out. \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{5/2}}{\sqrt {e-c\,e\,x}} \,d x \]
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