Integrand size = 23, antiderivative size = 203 \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^2 \, dx=-\frac {16}{75} b^2 e^4 x-\frac {8 b^2 e^4 (c+d x)^3}{225 d}-\frac {2 b^2 e^4 (c+d x)^5}{125 d}+\frac {16 b e^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{75 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{75 d}+\frac {2 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \arcsin (c+d x))^2}{5 d} \]
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Time = 0.24 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4889, 12, 4723, 4795, 4767, 8, 30} \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^2 \, dx=\frac {e^4 (c+d x)^5 (a+b \arcsin (c+d x))^2}{5 d}+\frac {2 b e^4 \sqrt {1-(c+d x)^2} (c+d x)^4 (a+b \arcsin (c+d x))}{25 d}+\frac {8 b e^4 \sqrt {1-(c+d x)^2} (c+d x)^2 (a+b \arcsin (c+d x))}{75 d}+\frac {16 b e^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{75 d}-\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {8 b^2 e^4 (c+d x)^3}{225 d}-\frac {16}{75} b^2 e^4 x \]
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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^4 x^4 (a+b \arcsin (x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int x^4 (a+b \arcsin (x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 (c+d x)^5 (a+b \arcsin (c+d x))^2}{5 d}-\frac {\left (2 b e^4\right ) \text {Subst}\left (\int \frac {x^5 (a+b \arcsin (x))}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{5 d} \\ & = \frac {2 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \arcsin (c+d x))^2}{5 d}-\frac {\left (8 b e^4\right ) \text {Subst}\left (\int \frac {x^3 (a+b \arcsin (x))}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{25 d}-\frac {\left (2 b^2 e^4\right ) \text {Subst}\left (\int x^4 \, dx,x,c+d x\right )}{25 d} \\ & = -\frac {2 b^2 e^4 (c+d x)^5}{125 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{75 d}+\frac {2 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \arcsin (c+d x))^2}{5 d}-\frac {\left (16 b e^4\right ) \text {Subst}\left (\int \frac {x (a+b \arcsin (x))}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{75 d}-\frac {\left (8 b^2 e^4\right ) \text {Subst}\left (\int x^2 \, dx,x,c+d x\right )}{75 d} \\ & = -\frac {8 b^2 e^4 (c+d x)^3}{225 d}-\frac {2 b^2 e^4 (c+d x)^5}{125 d}+\frac {16 b e^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{75 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{75 d}+\frac {2 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \arcsin (c+d x))^2}{5 d}-\frac {\left (16 b^2 e^4\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{75 d} \\ & = -\frac {16}{75} b^2 e^4 x-\frac {8 b^2 e^4 (c+d x)^3}{225 d}-\frac {2 b^2 e^4 (c+d x)^5}{125 d}+\frac {16 b e^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{75 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{75 d}+\frac {2 b e^4 (c+d x)^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{25 d}+\frac {e^4 (c+d x)^5 (a+b \arcsin (c+d x))^2}{5 d} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.81 \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^2 \, dx=\frac {e^4 \left ((c+d x)^5 (a+b \arcsin (c+d x))^2-\frac {2}{25} b \left (\frac {20}{9} b (c+d x)^3+b (c+d x)^5-\frac {20}{3} (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))-5 (c+d x)^4 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))+\frac {40}{3} \left (b d x-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))\right )\right )\right )}{5 d} \]
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Time = 1.04 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {e^{4} a^{2} \left (d x +c \right )^{5}}{5}+e^{4} b^{2} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{2}}{5}+\frac {2 \arcsin \left (d x +c \right ) \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}-\frac {16 d x}{75}-\frac {16 c}{75}\right )+2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{5}+\frac {\left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{75}+\frac {8 \sqrt {1-\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(194\) |
default | \(\frac {\frac {e^{4} a^{2} \left (d x +c \right )^{5}}{5}+e^{4} b^{2} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{2}}{5}+\frac {2 \arcsin \left (d x +c \right ) \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}-\frac {16 d x}{75}-\frac {16 c}{75}\right )+2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{5}+\frac {\left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{75}+\frac {8 \sqrt {1-\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(194\) |
parts | \(\frac {e^{4} a^{2} \left (d x +c \right )^{5}}{5 d}+\frac {e^{4} b^{2} \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )^{2}}{5}+\frac {2 \arcsin \left (d x +c \right ) \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right ) \sqrt {1-\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}-\frac {16 d x}{75}-\frac {16 c}{75}\right )}{d}+\frac {2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{5}+\frac {\left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{75}+\frac {8 \sqrt {1-\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(199\) |
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Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (183) = 366\).
Time = 0.28 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.79 \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^2 \, dx=\frac {9 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} d^{5} e^{4} x^{5} + 45 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c d^{4} e^{4} x^{4} + 10 \, {\left (9 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{2} - 4 \, b^{2}\right )} d^{3} e^{4} x^{3} + 30 \, {\left (3 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{3} - 4 \, b^{2} c\right )} d^{2} e^{4} x^{2} + 15 \, {\left (3 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{4} - 8 \, b^{2} c^{2} - 16 \, b^{2}\right )} d e^{4} x + 225 \, {\left (b^{2} d^{5} e^{4} x^{5} + 5 \, b^{2} c d^{4} e^{4} x^{4} + 10 \, b^{2} c^{2} d^{3} e^{4} x^{3} + 10 \, b^{2} c^{3} d^{2} e^{4} x^{2} + 5 \, b^{2} c^{4} d e^{4} x + b^{2} c^{5} e^{4}\right )} \arcsin \left (d x + c\right )^{2} + 450 \, {\left (a b d^{5} e^{4} x^{5} + 5 \, a b c d^{4} e^{4} x^{4} + 10 \, a b c^{2} d^{3} e^{4} x^{3} + 10 \, a b c^{3} d^{2} e^{4} x^{2} + 5 \, a b c^{4} d e^{4} x + a b c^{5} e^{4}\right )} \arcsin \left (d x + c\right ) + 30 \, {\left (3 \, a b d^{4} e^{4} x^{4} + 12 \, a b c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, a b c^{2} + 2 \, a b\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, a b c^{3} + 2 \, a b c\right )} d e^{4} x + {\left (3 \, a b c^{4} + 4 \, a b c^{2} + 8 \, a b\right )} e^{4} + {\left (3 \, b^{2} d^{4} e^{4} x^{4} + 12 \, b^{2} c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, b^{2} c^{2} + 2 \, b^{2}\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, b^{2} c^{3} + 2 \, b^{2} c\right )} d e^{4} x + {\left (3 \, b^{2} c^{4} + 4 \, b^{2} c^{2} + 8 \, b^{2}\right )} e^{4}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{1125 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1268 vs. \(2 (184) = 368\).
Time = 0.64 (sec) , antiderivative size = 1268, normalized size of antiderivative = 6.25 \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^2 \, dx=\text {Too large to display} \]
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\[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (183) = 366\).
Time = 0.32 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.18 \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^2 \, dx=\frac {{\left (d x + c\right )}^{5} a^{2} e^{4}}{5 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} {\left (d x + c\right )} b^{2} e^{4} \arcsin \left (d x + c\right )^{2}}{5 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} {\left (d x + c\right )} a b e^{4} \arcsin \left (d x + c\right )}{5 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{2} e^{4} \arcsin \left (d x + c\right )^{2}}{5 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{2} e^{4} \arcsin \left (d x + c\right )}{25 \, d} - \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} {\left (d x + c\right )} b^{2} e^{4}}{125 \, d} + \frac {4 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} a b e^{4} \arcsin \left (d x + c\right )}{5 \, d} + \frac {{\left (d x + c\right )} b^{2} e^{4} \arcsin \left (d x + c\right )^{2}}{5 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} \sqrt {-{\left (d x + c\right )}^{2} + 1} a b e^{4}}{25 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b^{2} e^{4} \arcsin \left (d x + c\right )}{15 \, d} - \frac {76 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b^{2} e^{4}}{1125 \, d} + \frac {2 \, {\left (d x + c\right )} a b e^{4} \arcsin \left (d x + c\right )}{5 \, d} - \frac {4 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} a b e^{4}}{15 \, d} + \frac {2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{2} e^{4} \arcsin \left (d x + c\right )}{5 \, d} - \frac {298 \, {\left (d x + c\right )} b^{2} e^{4}}{1125 \, d} + \frac {2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b e^{4}}{5 \, d} \]
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Timed out. \[ \int (c e+d e x)^4 (a+b \arcsin (c+d x))^2 \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2 \,d x \]
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