Integrand size = 19, antiderivative size = 148 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (9 c^2 d f+2 e g\right )+9 c^2 (e f+d g) x\right ) \sqrt {1-c^2 x^2}}{36 c^3}-\frac {b (e f+d g) \arcsin (c x)}{4 c^2}+d f x (a+b \arcsin (c x))+\frac {1}{2} (e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x)) \]
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Time = 0.14 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4833, 12, 1823, 794, 222} \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))-\frac {b \arcsin (c x) (d g+e f)}{4 c^2}+\frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \sqrt {1-c^2 x^2} \left (9 c^2 x (d g+e f)+4 \left (9 c^2 d f+2 e g\right )\right )}{36 c^3} \]
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Rule 12
Rule 222
Rule 794
Rule 1823
Rule 4833
Rubi steps \begin{align*} \text {integral}& = d f x (a+b \arcsin (c x))+\frac {1}{2} (e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))-(b c) \int \frac {x \left (6 d f+3 (e f+d g) x+2 e g x^2\right )}{6 \sqrt {1-c^2 x^2}} \, dx \\ & = d f x (a+b \arcsin (c x))+\frac {1}{2} (e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))-\frac {1}{6} (b c) \int \frac {x \left (6 d f+3 (e f+d g) x+2 e g x^2\right )}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}+d f x (a+b \arcsin (c x))+\frac {1}{2} (e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))+\frac {b \int \frac {x \left (-2 \left (9 c^2 d f+2 e g\right )-9 c^2 (e f+d g) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{18 c} \\ & = \frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (9 c^2 d f+2 e g\right )+9 c^2 (e f+d g) x\right ) \sqrt {1-c^2 x^2}}{36 c^3}+d f x (a+b \arcsin (c x))+\frac {1}{2} (e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))-\frac {(b (e f+d g)) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c} \\ & = \frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (9 c^2 d f+2 e g\right )+9 c^2 (e f+d g) x\right ) \sqrt {1-c^2 x^2}}{36 c^3}-\frac {b (e f+d g) \arcsin (c x)}{4 c^2}+d f x (a+b \arcsin (c x))+\frac {1}{2} (e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x)) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.93 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {6 a c^3 x (3 d (2 f+g x)+e x (3 f+2 g x))+b \sqrt {1-c^2 x^2} \left (8 e g+c^2 (9 d (4 f+g x)+e x (9 f+4 g x))\right )+3 b c \left (12 c^2 d f x+4 c^2 e g x^3+3 d g \left (-1+2 c^2 x^2\right )+e f \left (-3+6 c^2 x^2\right )\right ) \arcsin (c x)}{36 c^3} \]
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Time = 0.08 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.20
method | result | size |
parts | \(a \left (\frac {e g \,x^{3}}{3}+\frac {\left (d g +e f \right ) x^{2}}{2}+d f x \right )+\frac {b \left (\frac {c \arcsin \left (c x \right ) e g \,x^{3}}{3}+\frac {c \arcsin \left (c x \right ) x^{2} d g}{2}+\frac {c \arcsin \left (c x \right ) x^{2} e f}{2}+\arcsin \left (c x \right ) d f c x -\frac {2 e g \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-6 d \,c^{2} f \sqrt {-c^{2} x^{2}+1}+\left (3 d c g +3 e c f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{6 c^{2}}\right )}{c}\) | \(178\) |
derivativedivides | \(\frac {\frac {a \left (\frac {e g \,c^{3} x^{3}}{3}+\frac {\left (d c g +e c f \right ) c^{2} x^{2}}{2}+d \,c^{3} f x \right )}{c^{2}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e g \,c^{3} x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{3} d g \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{3} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d \,c^{3} f x -\frac {e g \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+d \,c^{2} f \sqrt {-c^{2} x^{2}+1}-\frac {\left (3 d c g +3 e c f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{6}\right )}{c^{2}}}{c}\) | \(198\) |
default | \(\frac {\frac {a \left (\frac {e g \,c^{3} x^{3}}{3}+\frac {\left (d c g +e c f \right ) c^{2} x^{2}}{2}+d \,c^{3} f x \right )}{c^{2}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e g \,c^{3} x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{3} d g \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{3} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d \,c^{3} f x -\frac {e g \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+d \,c^{2} f \sqrt {-c^{2} x^{2}+1}-\frac {\left (3 d c g +3 e c f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{6}\right )}{c^{2}}}{c}\) | \(198\) |
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Time = 0.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.09 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {12 \, a c^{3} e g x^{3} + 36 \, a c^{3} d f x + 18 \, {\left (a c^{3} e f + a c^{3} d g\right )} x^{2} + 3 \, {\left (4 \, b c^{3} e g x^{3} + 12 \, b c^{3} d f x - 3 \, b c e f - 3 \, b c d g + 6 \, {\left (b c^{3} e f + b c^{3} d g\right )} x^{2}\right )} \arcsin \left (c x\right ) + {\left (4 \, b c^{2} e g x^{2} + 36 \, b c^{2} d f + 8 \, b e g + 9 \, {\left (b c^{2} e f + b c^{2} d g\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{36 \, c^{3}} \]
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Time = 0.25 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.80 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\begin {cases} a d f x + \frac {a d g x^{2}}{2} + \frac {a e f x^{2}}{2} + \frac {a e g x^{3}}{3} + b d f x \operatorname {asin}{\left (c x \right )} + \frac {b d g x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e f x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e g x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d f \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d g x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b e f x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b e g x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {b d g \operatorname {asin}{\left (c x \right )}}{4 c^{2}} - \frac {b e f \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {2 b e g \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d f x + \frac {d g x^{2}}{2} + \frac {e f x^{2}}{2} + \frac {e g x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.34 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {1}{3} \, a e g x^{3} + \frac {1}{2} \, a e f x^{2} + \frac {1}{2} \, a d g x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b e f + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d g + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e g + a d f x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d f}{c} \]
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Time = 0.31 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.75 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {1}{3} \, a e g x^{3} + b d f x \arcsin \left (c x\right ) + a d f x + \frac {{\left (c^{2} x^{2} - 1\right )} b e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e f x}{4 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} b d g x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b e f \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {b e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d f}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a e f}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d g}{2 \, c^{2}} + \frac {b e f \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {b d g \arcsin \left (c x\right )}{4 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e g}{9 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e g}{3 \, c^{3}} \]
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Timed out. \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\left \{\begin {array}{cl} \frac {a\,x^2\,\left (d\,g+e\,f\right )}{2}+a\,d\,f\,x+b\,e\,g\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}+\frac {x^3\,\mathrm {asin}\left (c\,x\right )}{3}\right )+\frac {a\,e\,g\,x^3}{3}+\frac {b\,d\,f\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c}+\frac {b\,d\,g\,\left (\frac {\mathrm {asin}\left (c\,x\right )\,\left (2\,c^2\,x^2-1\right )}{4}+\frac {c\,x\,\sqrt {1-c^2\,x^2}}{4}\right )}{c^2}+\frac {b\,e\,f\,\left (\frac {\mathrm {asin}\left (c\,x\right )\,\left (2\,c^2\,x^2-1\right )}{4}+\frac {c\,x\,\sqrt {1-c^2\,x^2}}{4}\right )}{c^2} & \text {\ if\ \ }0<c\\ \int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d+e\,x\right ) \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \]
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