Integrand size = 23, antiderivative size = 140 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^2 \, dx=-\frac {4}{9} b^2 e^2 x-\frac {2 b^2 e^2 (c+d x)^3}{27 d}+\frac {4 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{9 d}+\frac {2 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{9 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^2}{3 d} \]
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Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4889, 12, 4723, 4795, 4767, 8, 30} \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^2 \, dx=\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^2}{3 d}+\frac {2 b e^2 \sqrt {1-(c+d x)^2} (c+d x)^2 (a+b \arcsin (c+d x))}{9 d}+\frac {4 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{9 d}-\frac {2 b^2 e^2 (c+d x)^3}{27 d}-\frac {4}{9} b^2 e^2 x \]
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Rule 8
Rule 12
Rule 30
Rule 4723
Rule 4767
Rule 4795
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^2 x^2 (a+b \arcsin (x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 (a+b \arcsin (x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^2}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x^3 (a+b \arcsin (x))}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d} \\ & = \frac {2 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{9 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^2}{3 d}-\frac {\left (4 b e^2\right ) \text {Subst}\left (\int \frac {x (a+b \arcsin (x))}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d}-\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int x^2 \, dx,x,c+d x\right )}{9 d} \\ & = -\frac {2 b^2 e^2 (c+d x)^3}{27 d}+\frac {4 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{9 d}+\frac {2 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{9 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^2}{3 d}-\frac {\left (4 b^2 e^2\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{9 d} \\ & = -\frac {4}{9} b^2 e^2 x-\frac {2 b^2 e^2 (c+d x)^3}{27 d}+\frac {4 b e^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{9 d}+\frac {2 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{9 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^2}{3 d} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.80 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^2 \, dx=\frac {e^2 \left ((c+d x)^3 (a+b \arcsin (c+d x))^2-\frac {2}{9} b \left (6 b d x+b (c+d x)^3-6 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))-3 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )}{3 d} \]
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Time = 1.00 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(152\) |
default | \(\frac {\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(152\) |
parts | \(\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )^{2}}{3}+\frac {2 \arcsin \left (d x +c \right ) \left (\left (d x +c \right )^{2}+2\right ) \sqrt {1-\left (d x +c \right )^{2}}}{9}-\frac {2 \left (d x +c \right )^{3}}{27}-\frac {4 d x}{9}-\frac {4 c}{9}\right )}{d}+\frac {2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \arcsin \left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1-\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(157\) |
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Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (126) = 252\).
Time = 0.26 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.19 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^2 \, dx=\frac {{\left (9 \, a^{2} - 2 \, b^{2}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c d^{2} e^{2} x^{2} + 3 \, {\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{2} - 4 \, b^{2}\right )} d e^{2} x + 9 \, {\left (b^{2} d^{3} e^{2} x^{3} + 3 \, b^{2} c d^{2} e^{2} x^{2} + 3 \, b^{2} c^{2} d e^{2} x + b^{2} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + 18 \, {\left (a b d^{3} e^{2} x^{3} + 3 \, a b c d^{2} e^{2} x^{2} + 3 \, a b c^{2} d e^{2} x + a b c^{3} e^{2}\right )} \arcsin \left (d x + c\right ) + 6 \, {\left (a b d^{2} e^{2} x^{2} + 2 \, a b c d e^{2} x + {\left (a b c^{2} + 2 \, a b\right )} e^{2} + {\left (b^{2} d^{2} e^{2} x^{2} + 2 \, b^{2} c d e^{2} x + {\left (b^{2} c^{2} + 2 \, b^{2}\right )} e^{2}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{27 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (126) = 252\).
Time = 0.31 (sec) , antiderivative size = 610, normalized size of antiderivative = 4.36 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^2 \, dx=\begin {cases} a^{2} c^{2} e^{2} x + a^{2} c d e^{2} x^{2} + \frac {a^{2} d^{2} e^{2} x^{3}}{3} + \frac {2 a b c^{3} e^{2} \operatorname {asin}{\left (c + d x \right )}}{3 d} + 2 a b c^{2} e^{2} x \operatorname {asin}{\left (c + d x \right )} + \frac {2 a b c^{2} e^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{9 d} + 2 a b c d e^{2} x^{2} \operatorname {asin}{\left (c + d x \right )} + \frac {4 a b c e^{2} x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{9} + \frac {2 a b d^{2} e^{2} x^{3} \operatorname {asin}{\left (c + d x \right )}}{3} + \frac {2 a b d e^{2} x^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{9} + \frac {4 a b e^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{9 d} + \frac {b^{2} c^{3} e^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{3 d} + b^{2} c^{2} e^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - \frac {2 b^{2} c^{2} e^{2} x}{9} + \frac {2 b^{2} c^{2} e^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{9 d} + b^{2} c d e^{2} x^{2} \operatorname {asin}^{2}{\left (c + d x \right )} - \frac {2 b^{2} c d e^{2} x^{2}}{9} + \frac {4 b^{2} c e^{2} x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{9} + \frac {b^{2} d^{2} e^{2} x^{3} \operatorname {asin}^{2}{\left (c + d x \right )}}{3} - \frac {2 b^{2} d^{2} e^{2} x^{3}}{27} + \frac {2 b^{2} d e^{2} x^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{9} - \frac {4 b^{2} e^{2} x}{9} + \frac {4 b^{2} e^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{9 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {asin}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \]
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\[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (126) = 252\).
Time = 0.31 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.96 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^2 \, dx=\frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{2} e^{2} \arcsin \left (d x + c\right )^{2}}{3 \, d} + \frac {{\left (d x + c\right )}^{3} a^{2} e^{2}}{3 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b e^{2} \arcsin \left (d x + c\right )}{3 \, d} + \frac {{\left (d x + c\right )} b^{2} e^{2} \arcsin \left (d x + c\right )^{2}}{3 \, d} - \frac {2 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{2} e^{2} \arcsin \left (d x + c\right )}{9 \, d} - \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{2} e^{2}}{27 \, d} + \frac {2 \, {\left (d x + c\right )} a b e^{2} \arcsin \left (d x + c\right )}{3 \, d} - \frac {2 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a b e^{2}}{9 \, d} + \frac {2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{2} e^{2} \arcsin \left (d x + c\right )}{3 \, d} - \frac {14 \, {\left (d x + c\right )} b^{2} e^{2}}{27 \, d} + \frac {2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b e^{2}}{3 \, d} \]
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Timed out. \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^2 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2 \,d x \]
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