Integrand size = 26, antiderivative size = 512 \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\frac {b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b \left (32 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+75 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) \arcsin (c x)}{96 c^6}+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^2 (e g+3 d h) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x)) \]
[Out]
Time = 1.28 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4833, 12, 1823, 794, 222} \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))-\frac {b \arcsin (c x) \left (24 c^4 d^2 (d g+3 e f)+9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^3 h\right )}{96 c^6}+\frac {b e x^3 \sqrt {1-c^2 x^2} \left (e^2 \left (\frac {5 h}{c^2}+9 f\right )+27 d^2 h+27 d e g\right )}{144 c}+\frac {b e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )}{225 c^3}+\frac {b \sqrt {1-c^2 x^2} \left (75 x \left (24 c^4 d^2 (d g+3 e f)+9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^3 h\right )+32 \left (225 c^4 d^3 f+50 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+24 e^2 (3 d h+e g)\right )\right )}{7200 c^5} \]
[In]
[Out]
Rule 12
Rule 222
Rule 794
Rule 1823
Rule 4833
Rubi steps \begin{align*} \text {integral}& = d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^2 (e g+3 d h) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))-(b c) \int \frac {x \left (10 d^3 (6 f+x (3 g+2 h x))+15 d^2 e x (6 f+x (4 g+3 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right )}{60 \sqrt {1-c^2 x^2}} \, dx \\ & = d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^2 (e g+3 d h) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))-\frac {1}{60} (b c) \int \frac {x \left (10 d^3 (6 f+x (3 g+2 h x))+15 d^2 e x (6 f+x (4 g+3 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right )}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^2 (e g+3 d h) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))+\frac {b \int \frac {x \left (-360 c^2 d^3 f-180 c^2 d^2 (3 e f+d g) x-120 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right ) x^2-10 e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3-72 c^2 e^2 (e g+3 d h) x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{360 c} \\ & = \frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^2 (e g+3 d h) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))-\frac {b \int \frac {x \left (1800 c^4 d^3 f+900 c^4 d^2 (3 e f+d g) x+24 c^2 \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2+50 c^2 e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3\right )}{\sqrt {1-c^2 x^2}} \, dx}{1800 c^3} \\ & = \frac {b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^2 (e g+3 d h) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))+\frac {b \int \frac {x \left (-7200 c^6 d^3 f-150 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x-96 c^4 \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{7200 c^5} \\ & = \frac {b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^2 (e g+3 d h) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))-\frac {b \int \frac {x \left (96 c^4 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+450 c^4 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{21600 c^7} \\ & = \frac {b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b \left (32 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+75 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^2 (e g+3 d h) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))-\frac {\left (b \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{96 c^5} \\ & = \frac {b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b \left (32 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+75 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) \arcsin (c x)}{96 c^6}+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^2 (e g+3 d h) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x)) \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 463, normalized size of antiderivative = 0.90 \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=a d^3 f x+\frac {1}{2} a d^2 (3 e f+d g) x^2+\frac {1}{3} a d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3+\frac {1}{4} a e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4+\frac {1}{5} a e^2 (e g+3 d h) x^5+\frac {1}{6} a e^3 h x^6+\frac {b \sqrt {1-c^2 x^2} \left (3 e^2 (256 e g+768 d h+125 e h x)+c^2 \left (1600 d^3 h+75 d^2 e (64 g+27 h x)+e^3 x \left (675 f+384 g x+250 h x^2\right )+3 d e^2 \left (1600 f+675 g x+384 h x^2\right )\right )+2 c^4 \left (100 d^3 (36 f+x (9 g+4 h x))+75 d^2 e x (36 f+x (16 g+9 h x))+3 d e^2 x^2 (400 f+9 x (25 g+16 h x))+e^3 x^3 (225 f+4 x (36 g+25 h x))\right )\right )}{7200 c^5}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) \arcsin (c x)}{96 c^6}+\frac {1}{60} b x \left (10 d^3 (6 f+x (3 g+2 h x))+15 d^2 e x (6 f+x (4 g+3 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right ) \arcsin (c x) \]
[In]
[Out]
Time = 0.37 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.24
method | result | size |
parts | \(a \left (\frac {e^{3} h \,x^{6}}{6}+\frac {\left (3 d \,e^{2} h +e^{3} g \right ) x^{5}}{5}+\frac {\left (3 d^{2} e h +3 d \,e^{2} g +e^{3} f \right ) x^{4}}{4}+\frac {\left (d^{3} h +3 d^{2} e g +3 d \,e^{2} f \right ) x^{3}}{3}+\frac {\left (d^{3} g +3 d^{2} e f \right ) x^{2}}{2}+d^{3} f x \right )+\frac {b \left (\frac {c \arcsin \left (c x \right ) e^{3} h \,x^{6}}{6}+\frac {3 c \arcsin \left (c x \right ) x^{5} d \,e^{2} h}{5}+\frac {c \arcsin \left (c x \right ) e^{3} g \,x^{5}}{5}+\frac {3 c \arcsin \left (c x \right ) x^{4} d^{2} e h}{4}+\frac {3 c \arcsin \left (c x \right ) x^{4} d \,e^{2} g}{4}+\frac {c \arcsin \left (c x \right ) x^{4} e^{3} f}{4}+\frac {c \arcsin \left (c x \right ) x^{3} d^{3} h}{3}+c \arcsin \left (c x \right ) x^{3} d^{2} e g +c \arcsin \left (c x \right ) x^{3} d \,e^{2} f +\frac {c \arcsin \left (c x \right ) x^{2} d^{3} g}{2}+\frac {3 c \arcsin \left (c x \right ) x^{2} d^{2} e f}{2}+\arcsin \left (c x \right ) d^{3} f c x -\frac {10 e^{3} h \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )-60 d^{3} c^{5} f \sqrt {-c^{2} x^{2}+1}+\left (36 d c \,e^{2} h +12 e^{3} c g \right ) \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )+\left (30 c^{4} d^{3} g +90 d^{2} c^{4} e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+\left (45 d^{2} c^{2} e h +45 d \,c^{2} e^{2} g +15 e^{3} c^{2} f \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+\left (20 c^{3} d^{3} h +60 d^{2} c^{3} e g +60 d \,c^{3} e^{2} f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{60 c^{5}}\right )}{c}\) | \(633\) |
derivativedivides | \(\frac {\frac {a \left (\frac {e^{3} h \,c^{6} x^{6}}{6}+\frac {\left (3 d c \,e^{2} h +e^{3} c g \right ) c^{5} x^{5}}{5}+\frac {\left (3 d^{2} c^{2} e h +3 d \,c^{2} e^{2} g +e^{3} c^{2} f \right ) c^{4} x^{4}}{4}+\frac {\left (c^{3} d^{3} h +3 d^{2} c^{3} e g +3 d \,c^{3} e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{4} d^{3} g +3 d^{2} c^{4} e f \right ) c^{2} x^{2}}{2}+d^{3} c^{6} f x \right )}{c^{5}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} h \,c^{6} x^{6}}{6}+\frac {3 \arcsin \left (c x \right ) c^{6} d \,e^{2} h \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{6} e^{3} g \,x^{5}}{5}+\frac {3 \arcsin \left (c x \right ) c^{6} d^{2} e h \,x^{4}}{4}+\frac {3 \arcsin \left (c x \right ) c^{6} d \,e^{2} g \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{6} e^{3} f \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{6} d^{3} h \,x^{3}}{3}+\arcsin \left (c x \right ) c^{6} d^{2} e g \,x^{3}+\arcsin \left (c x \right ) c^{6} d \,e^{2} f \,x^{3}+\frac {\arcsin \left (c x \right ) c^{6} d^{3} g \,x^{2}}{2}+\frac {3 \arcsin \left (c x \right ) c^{6} d^{2} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d^{3} c^{6} f x -\frac {e^{3} h \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{6}+d^{3} c^{5} f \sqrt {-c^{2} x^{2}+1}-\frac {\left (36 d c \,e^{2} h +12 e^{3} c g \right ) \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{60}-\frac {\left (30 c^{4} d^{3} g +90 d^{2} c^{4} e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{60}-\frac {\left (45 d^{2} c^{2} e h +45 d \,c^{2} e^{2} g +15 e^{3} c^{2} f \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{60}-\frac {\left (20 c^{3} d^{3} h +60 d^{2} c^{3} e g +60 d \,c^{3} e^{2} f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{60}\right )}{c^{5}}}{c}\) | \(705\) |
default | \(\frac {\frac {a \left (\frac {e^{3} h \,c^{6} x^{6}}{6}+\frac {\left (3 d c \,e^{2} h +e^{3} c g \right ) c^{5} x^{5}}{5}+\frac {\left (3 d^{2} c^{2} e h +3 d \,c^{2} e^{2} g +e^{3} c^{2} f \right ) c^{4} x^{4}}{4}+\frac {\left (c^{3} d^{3} h +3 d^{2} c^{3} e g +3 d \,c^{3} e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{4} d^{3} g +3 d^{2} c^{4} e f \right ) c^{2} x^{2}}{2}+d^{3} c^{6} f x \right )}{c^{5}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} h \,c^{6} x^{6}}{6}+\frac {3 \arcsin \left (c x \right ) c^{6} d \,e^{2} h \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{6} e^{3} g \,x^{5}}{5}+\frac {3 \arcsin \left (c x \right ) c^{6} d^{2} e h \,x^{4}}{4}+\frac {3 \arcsin \left (c x \right ) c^{6} d \,e^{2} g \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{6} e^{3} f \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{6} d^{3} h \,x^{3}}{3}+\arcsin \left (c x \right ) c^{6} d^{2} e g \,x^{3}+\arcsin \left (c x \right ) c^{6} d \,e^{2} f \,x^{3}+\frac {\arcsin \left (c x \right ) c^{6} d^{3} g \,x^{2}}{2}+\frac {3 \arcsin \left (c x \right ) c^{6} d^{2} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d^{3} c^{6} f x -\frac {e^{3} h \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{6}+d^{3} c^{5} f \sqrt {-c^{2} x^{2}+1}-\frac {\left (36 d c \,e^{2} h +12 e^{3} c g \right ) \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{60}-\frac {\left (30 c^{4} d^{3} g +90 d^{2} c^{4} e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{60}-\frac {\left (45 d^{2} c^{2} e h +45 d \,c^{2} e^{2} g +15 e^{3} c^{2} f \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{60}-\frac {\left (20 c^{3} d^{3} h +60 d^{2} c^{3} e g +60 d \,c^{3} e^{2} f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{60}\right )}{c^{5}}}{c}\) | \(705\) |
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Time = 0.29 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.32 \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\frac {1200 \, a c^{6} e^{3} h x^{6} + 7200 \, a c^{6} d^{3} f x + 1440 \, {\left (a c^{6} e^{3} g + 3 \, a c^{6} d e^{2} h\right )} x^{5} + 1800 \, {\left (a c^{6} e^{3} f + 3 \, a c^{6} d e^{2} g + 3 \, a c^{6} d^{2} e h\right )} x^{4} + 2400 \, {\left (3 \, a c^{6} d e^{2} f + 3 \, a c^{6} d^{2} e g + a c^{6} d^{3} h\right )} x^{3} + 3600 \, {\left (3 \, a c^{6} d^{2} e f + a c^{6} d^{3} g\right )} x^{2} + 15 \, {\left (80 \, b c^{6} e^{3} h x^{6} + 480 \, b c^{6} d^{3} f x + 96 \, {\left (b c^{6} e^{3} g + 3 \, b c^{6} d e^{2} h\right )} x^{5} + 120 \, {\left (b c^{6} e^{3} f + 3 \, b c^{6} d e^{2} g + 3 \, b c^{6} d^{2} e h\right )} x^{4} + 160 \, {\left (3 \, b c^{6} d e^{2} f + 3 \, b c^{6} d^{2} e g + b c^{6} d^{3} h\right )} x^{3} + 240 \, {\left (3 \, b c^{6} d^{2} e f + b c^{6} d^{3} g\right )} x^{2} - 45 \, {\left (8 \, b c^{4} d^{2} e + b c^{2} e^{3}\right )} f - 15 \, {\left (8 \, b c^{4} d^{3} + 9 \, b c^{2} d e^{2}\right )} g - 5 \, {\left (27 \, b c^{2} d^{2} e + 5 \, b e^{3}\right )} h\right )} \arcsin \left (c x\right ) + {\left (200 \, b c^{5} e^{3} h x^{5} + 288 \, {\left (b c^{5} e^{3} g + 3 \, b c^{5} d e^{2} h\right )} x^{4} + 50 \, {\left (9 \, b c^{5} e^{3} f + 27 \, b c^{5} d e^{2} g + {\left (27 \, b c^{5} d^{2} e + 5 \, b c^{3} e^{3}\right )} h\right )} x^{3} + 32 \, {\left (75 \, b c^{5} d e^{2} f + 3 \, {\left (25 \, b c^{5} d^{2} e + 4 \, b c^{3} e^{3}\right )} g + {\left (25 \, b c^{5} d^{3} + 36 \, b c^{3} d e^{2}\right )} h\right )} x^{2} + 2400 \, {\left (3 \, b c^{5} d^{3} + 2 \, b c^{3} d e^{2}\right )} f + 192 \, {\left (25 \, b c^{3} d^{2} e + 4 \, b c e^{3}\right )} g + 64 \, {\left (25 \, b c^{3} d^{3} + 36 \, b c d e^{2}\right )} h + 75 \, {\left (9 \, {\left (8 \, b c^{5} d^{2} e + b c^{3} e^{3}\right )} f + 3 \, {\left (8 \, b c^{5} d^{3} + 9 \, b c^{3} d e^{2}\right )} g + {\left (27 \, b c^{3} d^{2} e + 5 \, b c e^{3}\right )} h\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{7200 \, c^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1263 vs. \(2 (505) = 1010\).
Time = 0.67 (sec) , antiderivative size = 1263, normalized size of antiderivative = 2.47 \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\text {Too large to display} \]
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Time = 0.29 (sec) , antiderivative size = 859, normalized size of antiderivative = 1.68 \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\frac {1}{6} \, a e^{3} h x^{6} + \frac {1}{5} \, a e^{3} g x^{5} + \frac {3}{5} \, a d e^{2} h x^{5} + \frac {1}{4} \, a e^{3} f x^{4} + \frac {3}{4} \, a d e^{2} g x^{4} + \frac {3}{4} \, a d^{2} e h x^{4} + a d e^{2} f x^{3} + a d^{2} e g x^{3} + \frac {1}{3} \, a d^{3} h x^{3} + \frac {3}{2} \, a d^{2} e f x^{2} + \frac {1}{2} \, a d^{3} g x^{2} + \frac {3}{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^{2} e f + \frac {1}{3} \, {\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 d e^{2} f + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b e^{3} 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^{3} g + \frac {1}{3} \, {\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 d^{2} e g + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d e^{2} g + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b e^{3} 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 d^{3} h + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} e h + \frac {1}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d e^{2} h + \frac {1}{288} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b e^{3} h + a d^{3} f x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{3} f}{c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1337 vs. \(2 (477) = 954\).
Time = 0.34 (sec) , antiderivative size = 1337, normalized size of antiderivative = 2.61 \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3\,\left (h\,x^2+g\,x+f\right ) \,d x \]
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