Integrand size = 10, antiderivative size = 211 \[ \int x \arcsin (a+b x)^3 \, dx=\frac {6 a \sqrt {1-(a+b x)^2}}{b^2}-\frac {3 (a+b x) \sqrt {1-(a+b x)^2}}{8 b^2}+\frac {3 \arcsin (a+b x)}{8 b^2}+\frac {6 a (a+b x) \arcsin (a+b x)}{b^2}-\frac {3 (a+b x)^2 \arcsin (a+b x)}{4 b^2}-\frac {3 a \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b^2}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{4 b^2}-\frac {\arcsin (a+b x)^3}{4 b^2}-\frac {a^2 \arcsin (a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \arcsin (a+b x)^3 \]
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Time = 0.22 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4889, 4827, 4857, 3398, 3377, 2718, 3392, 30, 2715, 8} \[ \int x \arcsin (a+b x)^3 \, dx=-\frac {a^2 \arcsin (a+b x)^3}{2 b^2}-\frac {\arcsin (a+b x)^3}{4 b^2}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{4 b^2}-\frac {3 a \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b^2}-\frac {3 (a+b x)^2 \arcsin (a+b x)}{4 b^2}+\frac {6 a (a+b x) \arcsin (a+b x)}{b^2}+\frac {3 \arcsin (a+b x)}{8 b^2}+\frac {1}{2} x^2 \arcsin (a+b x)^3-\frac {3 (a+b x) \sqrt {1-(a+b x)^2}}{8 b^2}+\frac {6 a \sqrt {1-(a+b x)^2}}{b^2} \]
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Rule 8
Rule 30
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 3398
Rule 4827
Rule 4857
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \arcsin (x)^3 \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{2} x^2 \arcsin (a+b x)^3-\frac {3}{2} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \arcsin (x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right ) \\ & = \frac {1}{2} x^2 \arcsin (a+b x)^3-\frac {3}{2} \text {Subst}\left (\int x^2 \left (-\frac {a}{b}+\frac {\sin (x)}{b}\right )^2 \, dx,x,\arcsin (a+b x)\right ) \\ & = \frac {1}{2} x^2 \arcsin (a+b x)^3-\frac {3}{2} \text {Subst}\left (\int \left (\frac {a^2 x^2}{b^2}-\frac {2 a x^2 \sin (x)}{b^2}+\frac {x^2 \sin ^2(x)}{b^2}\right ) \, dx,x,\arcsin (a+b x)\right ) \\ & = -\frac {a^2 \arcsin (a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \arcsin (a+b x)^3-\frac {3 \text {Subst}\left (\int x^2 \sin ^2(x) \, dx,x,\arcsin (a+b x)\right )}{2 b^2}+\frac {(3 a) \text {Subst}\left (\int x^2 \sin (x) \, dx,x,\arcsin (a+b x)\right )}{b^2} \\ & = -\frac {3 (a+b x)^2 \arcsin (a+b x)}{4 b^2}-\frac {3 a \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b^2}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{4 b^2}-\frac {a^2 \arcsin (a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \arcsin (a+b x)^3-\frac {3 \text {Subst}\left (\int x^2 \, dx,x,\arcsin (a+b x)\right )}{4 b^2}+\frac {3 \text {Subst}\left (\int \sin ^2(x) \, dx,x,\arcsin (a+b x)\right )}{4 b^2}+\frac {(6 a) \text {Subst}(\int x \cos (x) \, dx,x,\arcsin (a+b x))}{b^2} \\ & = -\frac {3 (a+b x) \sqrt {1-(a+b x)^2}}{8 b^2}+\frac {6 a (a+b x) \arcsin (a+b x)}{b^2}-\frac {3 (a+b x)^2 \arcsin (a+b x)}{4 b^2}-\frac {3 a \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b^2}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{4 b^2}-\frac {\arcsin (a+b x)^3}{4 b^2}-\frac {a^2 \arcsin (a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \arcsin (a+b x)^3+\frac {3 \text {Subst}(\int 1 \, dx,x,\arcsin (a+b x))}{8 b^2}-\frac {(6 a) \text {Subst}(\int \sin (x) \, dx,x,\arcsin (a+b x))}{b^2} \\ & = \frac {6 a \sqrt {1-(a+b x)^2}}{b^2}-\frac {3 (a+b x) \sqrt {1-(a+b x)^2}}{8 b^2}+\frac {3 \arcsin (a+b x)}{8 b^2}+\frac {6 a (a+b x) \arcsin (a+b x)}{b^2}-\frac {3 (a+b x)^2 \arcsin (a+b x)}{4 b^2}-\frac {3 a \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b^2}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{4 b^2}-\frac {\arcsin (a+b x)^3}{4 b^2}-\frac {a^2 \arcsin (a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \arcsin (a+b x)^3 \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.64 \[ \int x \arcsin (a+b x)^3 \, dx=\frac {3 (15 a-b x) \sqrt {1-a^2-2 a b x-b^2 x^2}+\left (3+42 a^2+36 a b x-6 b^2 x^2\right ) \arcsin (a+b x)-6 (3 a-b x) \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2+\left (-2-4 a^2+4 b^2 x^2\right ) \arcsin (a+b x)^3}{8 b^2} \]
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Time = 0.44 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {\left (-1+\left (b x +a \right )^{2}\right ) \arcsin \left (b x +a \right )^{3}}{2}+\frac {3 \arcsin \left (b x +a \right )^{2} \left (\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{4}-\frac {3 \left (-1+\left (b x +a \right )^{2}\right ) \arcsin \left (b x +a \right )}{4}-\frac {3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{8}-\frac {3 \arcsin \left (b x +a \right )}{8}-\frac {\arcsin \left (b x +a \right )^{3}}{2}-a \left (\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )\right )}{b^{2}}\) | \(185\) |
default | \(\frac {\frac {\left (-1+\left (b x +a \right )^{2}\right ) \arcsin \left (b x +a \right )^{3}}{2}+\frac {3 \arcsin \left (b x +a \right )^{2} \left (\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{4}-\frac {3 \left (-1+\left (b x +a \right )^{2}\right ) \arcsin \left (b x +a \right )}{4}-\frac {3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{8}-\frac {3 \arcsin \left (b x +a \right )}{8}-\frac {\arcsin \left (b x +a \right )^{3}}{2}-a \left (\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )\right )}{b^{2}}\) | \(185\) |
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Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.51 \[ \int x \arcsin (a+b x)^3 \, dx=\frac {2 \, {\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \arcsin \left (b x + a\right )^{3} - 3 \, {\left (2 \, b^{2} x^{2} - 12 \, a b x - 14 \, a^{2} - 1\right )} \arcsin \left (b x + a\right ) + 3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (2 \, {\left (b x - 3 \, a\right )} \arcsin \left (b x + a\right )^{2} - b x + 15 \, a\right )}}{8 \, b^{2}} \]
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Time = 0.33 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.18 \[ \int x \arcsin (a+b x)^3 \, dx=\begin {cases} - \frac {a^{2} \operatorname {asin}^{3}{\left (a + b x \right )}}{2 b^{2}} + \frac {21 a^{2} \operatorname {asin}{\left (a + b x \right )}}{4 b^{2}} + \frac {9 a x \operatorname {asin}{\left (a + b x \right )}}{2 b} - \frac {9 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac {45 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{8 b^{2}} + \frac {x^{2} \operatorname {asin}^{3}{\left (a + b x \right )}}{2} - \frac {3 x^{2} \operatorname {asin}{\left (a + b x \right )}}{4} + \frac {3 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4 b} - \frac {3 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{8 b} - \frac {\operatorname {asin}^{3}{\left (a + b x \right )}}{4 b^{2}} + \frac {3 \operatorname {asin}{\left (a + b x \right )}}{8 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asin}^{3}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \]
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\[ \int x \arcsin (a+b x)^3 \, dx=\int { x \arcsin \left (b x + a\right )^{3} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.96 \[ \int x \arcsin (a+b x)^3 \, dx=-\frac {{\left (b x + a\right )} a \arcsin \left (b x + a\right )^{3}}{b^{2}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )^{3}}{2 \, b^{2}} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{4 \, b^{2}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a \arcsin \left (b x + a\right )^{2}}{b^{2}} + \frac {6 \, {\left (b x + a\right )} a \arcsin \left (b x + a\right )}{b^{2}} + \frac {\arcsin \left (b x + a\right )^{3}}{4 \, b^{2}} - \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )}{4 \, b^{2}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}}{8 \, b^{2}} + \frac {6 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a}{b^{2}} - \frac {3 \, \arcsin \left (b x + a\right )}{8 \, b^{2}} \]
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Timed out. \[ \int x \arcsin (a+b x)^3 \, dx=\int x\,{\mathrm {asin}\left (a+b\,x\right )}^3 \,d x \]
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