Integrand size = 16, antiderivative size = 142 \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=-2 b^2 d x-\frac {1}{4} b^2 e x^2+\frac {2 b d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {b e x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}-\frac {d^2 (a+b \arcsin (c x))^2}{2 e}-\frac {e (a+b \arcsin (c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e} \]
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Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4827, 4847, 4737, 4767, 8, 4795, 30} \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\frac {2 b d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {b e x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}-\frac {e (a+b \arcsin (c x))^2}{4 c^2}-\frac {d^2 (a+b \arcsin (c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e}-2 b^2 d x-\frac {1}{4} b^2 e x^2 \]
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Rule 8
Rule 30
Rule 4737
Rule 4767
Rule 4795
Rule 4827
Rule 4847
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e}-\frac {(b c) \int \frac {(d+e x)^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{e} \\ & = \frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e}-\frac {(b c) \int \left (\frac {d^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {2 d e x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {e^2 x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}\right ) \, dx}{e} \\ & = \frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e}-(2 b c d) \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx-\frac {\left (b c d^2\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{e}-(b c e) \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {2 b d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {b e x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}-\frac {d^2 (a+b \arcsin (c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e}-\left (2 b^2 d\right ) \int 1 \, dx-\frac {1}{2} \left (b^2 e\right ) \int x \, dx-\frac {(b e) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c} \\ & = -2 b^2 d x-\frac {1}{4} b^2 e x^2+\frac {2 b d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {b e x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}-\frac {d^2 (a+b \arcsin (c x))^2}{2 e}-\frac {e (a+b \arcsin (c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.08 \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\frac {(d+e x)^2 (a+b \arcsin (c x))^2}{2 e}-\frac {b \left (2 b d e x+\frac {1}{4} b e^2 x^2-\frac {2 d e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}-\frac {e^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}+\frac {d^2 (a+b \arcsin (c x))^2}{2 b}+\frac {e^2 (a+b \arcsin (c x))^2}{4 b c^2}\right )}{e} \]
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Time = 0.20 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.33
method | result | size |
parts | \(a^{2} \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b^{2} \left (\frac {e \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4 c}+d \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c}+\frac {2 a b \left (\frac {c \arcsin \left (c x \right ) x^{2} e}{2}+\arcsin \left (c x \right ) d c x -\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-2 d c \sqrt {-c^{2} x^{2}+1}}{2 c}\right )}{c}\) | \(189\) |
derivativedivides | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}\right )}{c}+\frac {2 a b \left (\arcsin \left (c x \right ) c^{2} x d +\frac {\arcsin \left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}+d c \sqrt {-c^{2} x^{2}+1}\right )}{c}}{c}\) | \(198\) |
default | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}\right )}{c}+\frac {2 a b \left (\arcsin \left (c x \right ) c^{2} x d +\frac {\arcsin \left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}+d c \sqrt {-c^{2} x^{2}+1}\right )}{c}}{c}\) | \(198\) |
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Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.10 \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\frac {{\left (2 \, a^{2} - b^{2}\right )} c^{2} e x^{2} + 4 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{2} d x + {\left (2 \, b^{2} c^{2} e x^{2} + 4 \, b^{2} c^{2} d x - b^{2} e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (2 \, a b c^{2} e x^{2} + 4 \, a b c^{2} d x - a b e\right )} \arcsin \left (c x\right ) + 2 \, {\left (a b c e x + 4 \, a b c d + {\left (b^{2} c e x + 4 \, b^{2} c d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, c^{2}} \]
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Time = 0.25 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.64 \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\begin {cases} a^{2} d x + \frac {a^{2} e x^{2}}{2} + 2 a b d x \operatorname {asin}{\left (c x \right )} + a b e x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 a b d \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {a b e x \sqrt {- c^{2} x^{2} + 1}}{2 c} - \frac {a b e \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + b^{2} d x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d x + \frac {b^{2} e x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} e x^{2}}{4} + \frac {2 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {b^{2} e x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} - \frac {b^{2} e \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
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\[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\int { {\left (e x + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.72 \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=b^{2} d x \arcsin \left (c x\right )^{2} + 2 \, a b d x \arcsin \left (c x\right ) + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} e x \arcsin \left (c x\right )}{2 \, c} + a^{2} d x - 2 \, b^{2} d x + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b e x}{2 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a b e \arcsin \left (c x\right )}{c^{2}} + \frac {b^{2} e \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} e}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e}{4 \, c^{2}} + \frac {a b e \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {b^{2} e}{8 \, c^{2}} \]
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Timed out. \[ \int (d+e x) (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \]
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