Integrand size = 10, antiderivative size = 201 \[ \int x^4 \arcsin (a x)^3 \, dx=-\frac {298 \sqrt {1-a^2 x^2}}{375 a^5}+\frac {76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {16 x \arcsin (a x)}{25 a^4}-\frac {8 x^3 \arcsin (a x)}{75 a^2}-\frac {6}{125} x^5 \arcsin (a x)+\frac {8 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a}+\frac {1}{5} x^5 \arcsin (a x)^3 \]
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Time = 0.24 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4723, 4795, 4767, 4715, 267, 272, 45} \[ \int x^4 \arcsin (a x)^3 \, dx=-\frac {16 x \arcsin (a x)}{25 a^4}-\frac {8 x^3 \arcsin (a x)}{75 a^2}+\frac {3 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a}+\frac {8 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^5}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}+\frac {76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}-\frac {298 \sqrt {1-a^2 x^2}}{375 a^5}+\frac {4 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^3}+\frac {1}{5} x^5 \arcsin (a x)^3-\frac {6}{125} x^5 \arcsin (a x) \]
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Rule 45
Rule 267
Rule 272
Rule 4715
Rule 4723
Rule 4767
Rule 4795
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \arcsin (a x)^3-\frac {1}{5} (3 a) \int \frac {x^5 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {3 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a}+\frac {1}{5} x^5 \arcsin (a x)^3-\frac {6}{25} \int x^4 \arcsin (a x) \, dx-\frac {12 \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{25 a} \\ & = -\frac {6}{125} x^5 \arcsin (a x)+\frac {4 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a}+\frac {1}{5} x^5 \arcsin (a x)^3-\frac {8 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{25 a^3}-\frac {8 \int x^2 \arcsin (a x) \, dx}{25 a^2}+\frac {1}{125} (6 a) \int \frac {x^5}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {8 x^3 \arcsin (a x)}{75 a^2}-\frac {6}{125} x^5 \arcsin (a x)+\frac {8 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a}+\frac {1}{5} x^5 \arcsin (a x)^3-\frac {16 \int \arcsin (a x) \, dx}{25 a^4}+\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2}} \, dx}{75 a}+\frac {1}{125} (3 a) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-a^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {16 x \arcsin (a x)}{25 a^4}-\frac {8 x^3 \arcsin (a x)}{75 a^2}-\frac {6}{125} x^5 \arcsin (a x)+\frac {8 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a}+\frac {1}{5} x^5 \arcsin (a x)^3+\frac {16 \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{25 a^3}+\frac {4 \text {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )}{75 a}+\frac {1}{125} (3 a) \text {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {1-a^2 x}}-\frac {2 \sqrt {1-a^2 x}}{a^4}+\frac {\left (1-a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right ) \\ & = -\frac {86 \sqrt {1-a^2 x^2}}{125 a^5}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{125 a^5}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {16 x \arcsin (a x)}{25 a^4}-\frac {8 x^3 \arcsin (a x)}{75 a^2}-\frac {6}{125} x^5 \arcsin (a x)+\frac {8 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a}+\frac {1}{5} x^5 \arcsin (a x)^3+\frac {4 \text {Subst}\left (\int \left (\frac {1}{a^2 \sqrt {1-a^2 x}}-\frac {\sqrt {1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )}{75 a} \\ & = -\frac {298 \sqrt {1-a^2 x^2}}{375 a^5}+\frac {76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {16 x \arcsin (a x)}{25 a^4}-\frac {8 x^3 \arcsin (a x)}{75 a^2}-\frac {6}{125} x^5 \arcsin (a x)+\frac {8 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a}+\frac {1}{5} x^5 \arcsin (a x)^3 \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.61 \[ \int x^4 \arcsin (a x)^3 \, dx=\frac {-2 \sqrt {1-a^2 x^2} \left (2072+136 a^2 x^2+27 a^4 x^4\right )-30 a x \left (120+20 a^2 x^2+9 a^4 x^4\right ) \arcsin (a x)+225 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \arcsin (a x)^2+1125 a^5 x^5 \arcsin (a x)^3}{5625 a^5} \]
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Time = 0.05 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \arcsin \left (a x \right )^{3}}{5}+\frac {\arcsin \left (a x \right )^{2} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{25}-\frac {6 a^{5} x^{5} \arcsin \left (a x \right )}{125}-\frac {2 \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}-\frac {8 a^{3} x^{3} \arcsin \left (a x \right )}{75}-\frac {8 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {16 \sqrt {-a^{2} x^{2}+1}}{25}-\frac {16 a x \arcsin \left (a x \right )}{25}}{a^{5}}\) | \(159\) |
default | \(\frac {\frac {a^{5} x^{5} \arcsin \left (a x \right )^{3}}{5}+\frac {\arcsin \left (a x \right )^{2} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{25}-\frac {6 a^{5} x^{5} \arcsin \left (a x \right )}{125}-\frac {2 \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}-\frac {8 a^{3} x^{3} \arcsin \left (a x \right )}{75}-\frac {8 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {16 \sqrt {-a^{2} x^{2}+1}}{25}-\frac {16 a x \arcsin \left (a x \right )}{25}}{a^{5}}\) | \(159\) |
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Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.52 \[ \int x^4 \arcsin (a x)^3 \, dx=\frac {1125 \, a^{5} x^{5} \arcsin \left (a x\right )^{3} - 30 \, {\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \arcsin \left (a x\right ) - {\left (54 \, a^{4} x^{4} + 272 \, a^{2} x^{2} - 225 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \arcsin \left (a x\right )^{2} + 4144\right )} \sqrt {-a^{2} x^{2} + 1}}{5625 \, a^{5}} \]
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Time = 0.54 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.98 \[ \int x^4 \arcsin (a x)^3 \, dx=\begin {cases} \frac {x^{5} \operatorname {asin}^{3}{\left (a x \right )}}{5} - \frac {6 x^{5} \operatorname {asin}{\left (a x \right )}}{125} + \frac {3 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{25 a} - \frac {6 x^{4} \sqrt {- a^{2} x^{2} + 1}}{625 a} - \frac {8 x^{3} \operatorname {asin}{\left (a x \right )}}{75 a^{2}} + \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{25 a^{3}} - \frac {272 x^{2} \sqrt {- a^{2} x^{2} + 1}}{5625 a^{3}} - \frac {16 x \operatorname {asin}{\left (a x \right )}}{25 a^{4}} + \frac {8 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{25 a^{5}} - \frac {4144 \sqrt {- a^{2} x^{2} + 1}}{5625 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.85 \[ \int x^4 \arcsin (a x)^3 \, dx=\frac {1}{5} \, x^{5} \arcsin \left (a x\right )^{3} + \frac {1}{25} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arcsin \left (a x\right )^{2} - \frac {2}{5625} \, a {\left (\frac {27 \, \sqrt {-a^{2} x^{2} + 1} a^{2} x^{4} + 136 \, \sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}}{a^{4}} + \frac {15 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \arcsin \left (a x\right )}{a^{5}}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.24 \[ \int x^4 \arcsin (a x)^3 \, dx=\frac {{\left (a^{2} x^{2} - 1\right )}^{2} x \arcsin \left (a x\right )^{3}}{5 \, a^{4}} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{3}}{5 \, a^{4}} - \frac {6 \, {\left (a^{2} x^{2} - 1\right )}^{2} x \arcsin \left (a x\right )}{125 \, a^{4}} + \frac {x \arcsin \left (a x\right )^{3}}{5 \, a^{4}} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{25 \, a^{5}} - \frac {76 \, {\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )}{375 \, a^{4}} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (a x\right )^{2}}{5 \, a^{5}} - \frac {298 \, x \arcsin \left (a x\right )}{375 \, a^{4}} - \frac {6 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{625 \, a^{5}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{5 \, a^{5}} + \frac {76 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{1125 \, a^{5}} - \frac {298 \, \sqrt {-a^{2} x^{2} + 1}}{375 \, a^{5}} \]
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Timed out. \[ \int x^4 \arcsin (a x)^3 \, dx=\int x^4\,{\mathrm {asin}\left (a\,x\right )}^3 \,d x \]
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