Integrand size = 26, antiderivative size = 512 \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\frac {b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b \left (32 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+75 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) \arcsin (c x)}{96 c^6}+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^2 (e g+3 d h) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x)) \]
-1/96*b*(24*c^4*d^2*(d*g+3*e*f)+5*e^3*h+9*c^2*e*(3*d^2*h+3*d*e*g+e^2*f))*a rcsin(c*x)/c^6+d^3*f*x*(a+b*arcsin(c*x))+1/2*d^2*(d*g+3*e*f)*x^2*(a+b*arcs in(c*x))+1/3*d*(d^2*h+3*d*e*g+3*e^2*f)*x^3*(a+b*arcsin(c*x))+1/4*e*(3*d^2* h+3*d*e*g+e^2*f)*x^4*(a+b*arcsin(c*x))+1/5*e^2*(3*d*h+e*g)*x^5*(a+b*arcsin (c*x))+1/6*e^3*h*x^6*(a+b*arcsin(c*x))+1/225*b*(12*e^2*(3*d*h+e*g)+25*c^2* d*(d^2*h+3*d*e*g+3*e^2*f))*x^2*(-c^2*x^2+1)^(1/2)/c^3+1/144*b*e*(5*e^2*h+9 *c^2*(3*d^2*h+3*d*e*g+e^2*f))*x^3*(-c^2*x^2+1)^(1/2)/c^3+1/25*b*e^2*(3*d*h +e*g)*x^4*(-c^2*x^2+1)^(1/2)/c+1/36*b*e^3*h*x^5*(-c^2*x^2+1)^(1/2)/c+1/720 0*b*(7200*c^4*d^3*f+768*e^2*(3*d*h+e*g)+1600*c^2*d*(d^2*h+3*d*e*g+3*e^2*f) +75*(24*c^4*d^2*(d*g+3*e*f)+5*e^3*h+9*c^2*e*(3*d^2*h+3*d*e*g+e^2*f))*x)*(- c^2*x^2+1)^(1/2)/c^5
Time = 0.53 (sec) , antiderivative size = 463, normalized size of antiderivative = 0.90 \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=a d^3 f x+\frac {1}{2} a d^2 (3 e f+d g) x^2+\frac {1}{3} a d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3+\frac {1}{4} a e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4+\frac {1}{5} a e^2 (e g+3 d h) x^5+\frac {1}{6} a e^3 h x^6+\frac {b \sqrt {1-c^2 x^2} \left (3 e^2 (256 e g+768 d h+125 e h x)+c^2 \left (1600 d^3 h+75 d^2 e (64 g+27 h x)+e^3 x \left (675 f+384 g x+250 h x^2\right )+3 d e^2 \left (1600 f+675 g x+384 h x^2\right )\right )+2 c^4 \left (100 d^3 (36 f+x (9 g+4 h x))+75 d^2 e x (36 f+x (16 g+9 h x))+3 d e^2 x^2 (400 f+9 x (25 g+16 h x))+e^3 x^3 (225 f+4 x (36 g+25 h x))\right )\right )}{7200 c^5}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) \arcsin (c x)}{96 c^6}+\frac {1}{60} b x \left (10 d^3 (6 f+x (3 g+2 h x))+15 d^2 e x (6 f+x (4 g+3 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right ) \arcsin (c x) \]
a*d^3*f*x + (a*d^2*(3*e*f + d*g)*x^2)/2 + (a*d*(3*e^2*f + 3*d*e*g + d^2*h) *x^3)/3 + (a*e*(e^2*f + 3*d*e*g + 3*d^2*h)*x^4)/4 + (a*e^2*(e*g + 3*d*h)*x ^5)/5 + (a*e^3*h*x^6)/6 + (b*Sqrt[1 - c^2*x^2]*(3*e^2*(256*e*g + 768*d*h + 125*e*h*x) + c^2*(1600*d^3*h + 75*d^2*e*(64*g + 27*h*x) + e^3*x*(675*f + 384*g*x + 250*h*x^2) + 3*d*e^2*(1600*f + 675*g*x + 384*h*x^2)) + 2*c^4*(10 0*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)))))/(7200*c^5) - (b*(24*c^4*d^2*(3*e*f + d*g) + 5*e^3*h + 9*c^2*e*(e^ 2*f + 3*d*e*g + 3*d^2*h))*ArcSin[c*x])/(96*c^6) + (b*x*(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)))*ArcSin[c*x])/60
Time = 2.24 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {5248, 27, 2340, 27, 2340, 25, 2340, 27, 2340, 25, 27, 533, 455, 223}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5248 |
| \(\displaystyle -b c \int \frac {x \left (10 (6 f+x (3 g+2 h x)) d^3+15 e x (6 f+x (4 g+3 h x)) d^2+3 e^2 x^2 (20 f+3 x (5 g+4 h x)) d+e^3 x^3 (15 f+2 x (6 g+5 h x))\right )}{60 \sqrt {1-c^2 x^2}}dx+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{60} b c \int \frac {x \left (10 (6 f+x (3 g+2 h x)) d^3+15 e x (6 f+x (4 g+3 h x)) d^2+3 e^2 x^2 (20 f+3 x (5 g+4 h x)) d+e^3 x^3 (15 f+2 x (6 g+5 h x))\right )}{\sqrt {1-c^2 x^2}}dx+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{60} b c \left (-\frac {\int -\frac {2 x \left (36 c^2 e^2 (e g+3 d h) x^4+5 e \left (9 \left (3 h d^2+3 e g d+e^2 f\right ) c^2+5 e^2 h\right ) x^3+60 c^2 d \left (h d^2+3 e g d+3 e^2 f\right ) x^2+90 c^2 d^2 (3 e f+d g) x+180 c^2 d^3 f\right )}{\sqrt {1-c^2 x^2}}dx}{6 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{60} b c \left (\frac {\int \frac {x \left (36 c^2 e^2 (e g+3 d h) x^4+5 e \left (9 \left (3 h d^2+3 e g d+e^2 f\right ) c^2+5 e^2 h\right ) x^3+60 c^2 d \left (h d^2+3 e g d+3 e^2 f\right ) x^2+90 c^2 d^2 (3 e f+d g) x+180 c^2 d^3 f\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{60} b c \left (\frac {-\frac {\int -\frac {x \left (900 d^3 f c^4+450 d^2 (3 e f+d g) x c^4+25 e \left (9 \left (3 h d^2+3 e g d+e^2 f\right ) c^2+5 e^2 h\right ) x^3 c^2+12 \left (25 d \left (h d^2+3 e g d+3 e^2 f\right ) c^2+12 e^2 (e g+3 d h)\right ) x^2 c^2\right )}{\sqrt {1-c^2 x^2}}dx}{5 c^2}-\frac {36}{5} e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{60} b c \left (\frac {\frac {\int \frac {x \left (900 d^3 f c^4+450 d^2 (3 e f+d g) x c^4+25 e \left (9 \left (3 h d^2+3 e g d+e^2 f\right ) c^2+5 e^2 h\right ) x^3 c^2+12 \left (25 d \left (h d^2+3 e g d+3 e^2 f\right ) c^2+12 e^2 (e g+3 d h)\right ) x^2 c^2\right )}{\sqrt {1-c^2 x^2}}dx}{5 c^2}-\frac {36}{5} e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{60} b c \left (\frac {\frac {-\frac {\int -\frac {3 x \left (1200 d^3 f c^6+16 \left (25 d \left (h d^2+3 e g d+3 e^2 f\right ) c^2+12 e^2 (e g+3 d h)\right ) x^2 c^4+25 \left (24 d^2 (3 e f+d g) c^4+9 e \left (3 h d^2+3 e g d+e^2 f\right ) c^2+5 e^3 h\right ) x c^2\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {25}{4} e x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^2 h\right )}{5 c^2}-\frac {36}{5} e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{60} b c \left (\frac {\frac {\frac {3 \int \frac {x \left (1200 d^3 f c^6+16 \left (25 d \left (h d^2+3 e g d+3 e^2 f\right ) c^2+12 e^2 (e g+3 d h)\right ) x^2 c^4+25 \left (24 d^2 (3 e f+d g) c^4+9 e \left (3 h d^2+3 e g d+e^2 f\right ) c^2+5 e^3 h\right ) x c^2\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {25}{4} e x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^2 h\right )}{5 c^2}-\frac {36}{5} e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{60} b c \left (\frac {\frac {\frac {3 \left (-\frac {\int -\frac {c^4 x \left (16 \left (225 d^3 f c^4+50 d \left (h d^2+3 e g d+3 e^2 f\right ) c^2+24 e^2 (e g+3 d h)\right )+75 \left (24 d^2 (3 e f+d g) c^4+9 e \left (3 h d^2+3 e g d+e^2 f\right ) c^2+5 e^3 h\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )\right )}{4 c^2}-\frac {25}{4} e x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^2 h\right )}{5 c^2}-\frac {36}{5} e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{60} b c \left (\frac {\frac {\frac {3 \left (\frac {\int \frac {c^4 x \left (16 \left (225 d^3 f c^4+50 d \left (h d^2+3 e g d+3 e^2 f\right ) c^2+24 e^2 (e g+3 d h)\right )+75 \left (24 d^2 (3 e f+d g) c^4+9 e \left (3 h d^2+3 e g d+e^2 f\right ) c^2+5 e^3 h\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )\right )}{4 c^2}-\frac {25}{4} e x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^2 h\right )}{5 c^2}-\frac {36}{5} e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{60} b c \left (\frac {\frac {\frac {3 \left (\frac {1}{3} c^2 \int \frac {x \left (16 \left (225 d^3 f c^4+50 d \left (h d^2+3 e g d+3 e^2 f\right ) c^2+24 e^2 (e g+3 d h)\right )+75 \left (24 d^2 (3 e f+d g) c^4+9 e \left (3 h d^2+3 e g d+e^2 f\right ) c^2+5 e^3 h\right ) x\right )}{\sqrt {1-c^2 x^2}}dx-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )\right )}{4 c^2}-\frac {25}{4} e x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^2 h\right )}{5 c^2}-\frac {36}{5} e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle -\frac {1}{60} b c \left (\frac {\frac {\frac {3 \left (\frac {1}{3} c^2 \left (\frac {\int \frac {32 \left (225 d^3 f c^4+50 d \left (h d^2+3 e g d+3 e^2 f\right ) c^2+24 e^2 (e g+3 d h)\right ) x c^2+75 \left (24 d^2 (3 e f+d g) c^4+9 e \left (3 h d^2+3 e g d+e^2 f\right ) c^2+5 e^3 h\right )}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {75 x \sqrt {1-c^2 x^2} \left (24 c^4 d^2 (d g+3 e f)+9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^3 h\right )}{2 c^2}\right )-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )\right )}{4 c^2}-\frac {25}{4} e x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^2 h\right )}{5 c^2}-\frac {36}{5} e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {1}{60} b c \left (\frac {\frac {\frac {3 \left (\frac {1}{3} c^2 \left (\frac {75 \left (24 c^4 d^2 (d g+3 e f)+9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^3 h\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx-32 \sqrt {1-c^2 x^2} \left (225 c^4 d^3 f+50 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+24 e^2 (3 d h+e g)\right )}{2 c^2}-\frac {75 x \sqrt {1-c^2 x^2} \left (24 c^4 d^2 (d g+3 e f)+9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^3 h\right )}{2 c^2}\right )-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )\right )}{4 c^2}-\frac {25}{4} e x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^2 h\right )}{5 c^2}-\frac {36}{5} e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle d^3 f x (a+b \arcsin (c x))+\frac {1}{4} e x^4 (a+b \arcsin (c x)) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{5} e^2 x^5 (3 d h+e g) (a+b \arcsin (c x))+\frac {1}{6} e^3 h x^6 (a+b \arcsin (c x))-\frac {1}{60} b c \left (\frac {\frac {\frac {3 \left (\frac {1}{3} c^2 \left (\frac {\frac {75 \arcsin (c x) \left (24 c^4 d^2 (d g+3 e f)+9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^3 h\right )}{c}-32 \sqrt {1-c^2 x^2} \left (225 c^4 d^3 f+50 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+24 e^2 (3 d h+e g)\right )}{2 c^2}-\frac {75 x \sqrt {1-c^2 x^2} \left (24 c^4 d^2 (d g+3 e f)+9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^3 h\right )}{2 c^2}\right )-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )\right )}{4 c^2}-\frac {25}{4} e x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^2 h\right )}{5 c^2}-\frac {36}{5} e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{3 c^2}-\frac {5 e^3 h x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )\) |
d^3*f*x*(a + b*ArcSin[c*x]) + (d^2*(3*e*f + d*g)*x^2*(a + b*ArcSin[c*x]))/ 2 + (d*(3*e^2*f + 3*d*e*g + d^2*h)*x^3*(a + b*ArcSin[c*x]))/3 + (e*(e^2*f + 3*d*e*g + 3*d^2*h)*x^4*(a + b*ArcSin[c*x]))/4 + (e^2*(e*g + 3*d*h)*x^5*( a + b*ArcSin[c*x]))/5 + (e^3*h*x^6*(a + b*ArcSin[c*x]))/6 - (b*c*((-5*e^3* h*x^5*Sqrt[1 - c^2*x^2])/(3*c^2) + ((-36*e^2*(e*g + 3*d*h)*x^4*Sqrt[1 - c^ 2*x^2])/5 + ((-25*e*(5*e^2*h + 9*c^2*(e^2*f + 3*d*e*g + 3*d^2*h))*x^3*Sqrt [1 - c^2*x^2])/4 + (3*((-16*c^2*(12*e^2*(e*g + 3*d*h) + 25*c^2*d*(3*e^2*f + 3*d*e*g + d^2*h))*x^2*Sqrt[1 - c^2*x^2])/3 + (c^2*((-75*(24*c^4*d^2*(3*e *f + d*g) + 5*e^3*h + 9*c^2*e*(e^2*f + 3*d*e*g + 3*d^2*h))*x*Sqrt[1 - c^2* x^2])/(2*c^2) + (-32*(225*c^4*d^3*f + 24*e^2*(e*g + 3*d*h) + 50*c^2*d*(3*e ^2*f + 3*d*e*g + d^2*h))*Sqrt[1 - c^2*x^2] + (75*(24*c^4*d^2*(3*e*f + d*g) + 5*e^3*h + 9*c^2*e*(e^2*f + 3*d*e*g + 3*d^2*h))*ArcSin[c*x])/c)/(2*c^2)) )/3))/(4*c^2))/(5*c^2))/(3*c^2)))/60
3.1.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_), x_Symbol] :> With[{u = IntHid e[ExpandExpression[Px, x], x]}, Simp[(a + b*ArcSin[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c }, x] && PolynomialQ[Px, x]
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}\) | \(705\) |
a*(1/6*e^3*h*x^6+1/5*(3*d*e^2*h+e^3*g)*x^5+1/4*(3*d^2*e*h+3*d*e^2*g+e^3*f) *x^4+1/3*(d^3*h+3*d^2*e*g+3*d*e^2*f)*x^3+1/2*(d^3*g+3*d^2*e*f)*x^2+d^3*f*x )+b/c*(1/6*c*arcsin(c*x)*e^3*h*x^6+3/5*c*arcsin(c*x)*x^5*d*e^2*h+1/5*c*arc sin(c*x)*e^3*g*x^5+3/4*c*arcsin(c*x)*x^4*d^2*e*h+3/4*c*arcsin(c*x)*x^4*d*e ^2*g+1/4*c*arcsin(c*x)*x^4*e^3*f+1/3*c*arcsin(c*x)*x^3*d^3*h+c*arcsin(c*x) *x^3*d^2*e*g+c*arcsin(c*x)*x^3*d*e^2*f+1/2*c*arcsin(c*x)*x^2*d^3*g+3/2*c*a rcsin(c*x)*x^2*d^2*e*f+arcsin(c*x)*d^3*f*c*x-1/60/c^5*(10*e^3*h*(-1/6*c^5* x^5*(-c^2*x^2+1)^(1/2)-5/24*c^3*x^3*(-c^2*x^2+1)^(1/2)-5/16*c*x*(-c^2*x^2+ 1)^(1/2)+5/16*arcsin(c*x))-60*d^3*c^5*f*(-c^2*x^2+1)^(1/2)+(36*c*d*e^2*h+1 2*c*e^3*g)*(-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2 )-8/15*(-c^2*x^2+1)^(1/2))+(30*c^4*d^3*g+90*c^4*d^2*e*f)*(-1/2*c*x*(-c^2*x ^2+1)^(1/2)+1/2*arcsin(c*x))+(45*c^2*d^2*e*h+45*c^2*d*e^2*g+15*c^2*e^3*f)* (-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2+1)^(1/2)+3/8*arcsin(c*x ))+(20*c^3*d^3*h+60*c^3*d^2*e*g+60*c^3*d*e^2*f)*(-1/3*c^2*x^2*(-c^2*x^2+1) ^(1/2)-2/3*(-c^2*x^2+1)^(1/2))))
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}} \]
1/7200*(1200*a*c^6*e^3*h*x^6 + 7200*a*c^6*d^3*f*x + 1440*(a*c^6*e^3*g + 3* a*c^6*d*e^2*h)*x^5 + 1800*(a*c^6*e^3*f + 3*a*c^6*d*e^2*g + 3*a*c^6*d^2*e*h )*x^4 + 2400*(3*a*c^6*d*e^2*f + 3*a*c^6*d^2*e*g + a*c^6*d^3*h)*x^3 + 3600* (3*a*c^6*d^2*e*f + a*c^6*d^3*g)*x^2 + 15*(80*b*c^6*e^3*h*x^6 + 480*b*c^6*d ^3*f*x + 96*(b*c^6*e^3*g + 3*b*c^6*d*e^2*h)*x^5 + 120*(b*c^6*e^3*f + 3*b*c ^6*d*e^2*g + 3*b*c^6*d^2*e*h)*x^4 + 160*(3*b*c^6*d*e^2*f + 3*b*c^6*d^2*e*g + b*c^6*d^3*h)*x^3 + 240*(3*b*c^6*d^2*e*f + b*c^6*d^3*g)*x^2 - 45*(8*b*c^ 4*d^2*e + b*c^2*e^3)*f - 15*(8*b*c^4*d^3 + 9*b*c^2*d*e^2)*g - 5*(27*b*c^2* d^2*e + 5*b*e^3)*h)*arcsin(c*x) + (200*b*c^5*e^3*h*x^5 + 288*(b*c^5*e^3*g + 3*b*c^5*d*e^2*h)*x^4 + 50*(9*b*c^5*e^3*f + 27*b*c^5*d*e^2*g + (27*b*c^5* d^2*e + 5*b*c^3*e^3)*h)*x^3 + 32*(75*b*c^5*d*e^2*f + 3*(25*b*c^5*d^2*e + 4 *b*c^3*e^3)*g + (25*b*c^5*d^3 + 36*b*c^3*d*e^2)*h)*x^2 + 2400*(3*b*c^5*d^3 + 2*b*c^3*d*e^2)*f + 192*(25*b*c^3*d^2*e + 4*b*c*e^3)*g + 64*(25*b*c^3*d^ 3 + 36*b*c*d*e^2)*h + 75*(9*(8*b*c^5*d^2*e + b*c^3*e^3)*f + 3*(8*b*c^5*d^3 + 9*b*c^3*d*e^2)*g + (27*b*c^3*d^2*e + 5*b*c*e^3)*h)*x)*sqrt(-c^2*x^2 + 1 ))/c^6
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 \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\text {Too large to display} \]
Piecewise((a*d**3*f*x + a*d**3*g*x**2/2 + a*d**3*h*x**3/3 + 3*a*d**2*e*f*x **2/2 + a*d**2*e*g*x**3 + 3*a*d**2*e*h*x**4/4 + a*d*e**2*f*x**3 + 3*a*d*e* *2*g*x**4/4 + 3*a*d*e**2*h*x**5/5 + a*e**3*f*x**4/4 + a*e**3*g*x**5/5 + a* e**3*h*x**6/6 + b*d**3*f*x*asin(c*x) + b*d**3*g*x**2*asin(c*x)/2 + b*d**3* h*x**3*asin(c*x)/3 + 3*b*d**2*e*f*x**2*asin(c*x)/2 + b*d**2*e*g*x**3*asin( c*x) + 3*b*d**2*e*h*x**4*asin(c*x)/4 + b*d*e**2*f*x**3*asin(c*x) + 3*b*d*e **2*g*x**4*asin(c*x)/4 + 3*b*d*e**2*h*x**5*asin(c*x)/5 + b*e**3*f*x**4*asi n(c*x)/4 + b*e**3*g*x**5*asin(c*x)/5 + b*e**3*h*x**6*asin(c*x)/6 + b*d**3* f*sqrt(-c**2*x**2 + 1)/c + b*d**3*g*x*sqrt(-c**2*x**2 + 1)/(4*c) + b*d**3* h*x**2*sqrt(-c**2*x**2 + 1)/(9*c) + 3*b*d**2*e*f*x*sqrt(-c**2*x**2 + 1)/(4 *c) + b*d**2*e*g*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 3*b*d**2*e*h*x**3*sqrt( -c**2*x**2 + 1)/(16*c) + b*d*e**2*f*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 3*b* d*e**2*g*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + 3*b*d*e**2*h*x**4*sqrt(-c**2*x **2 + 1)/(25*c) + b*e**3*f*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + b*e**3*g*x** 4*sqrt(-c**2*x**2 + 1)/(25*c) + b*e**3*h*x**5*sqrt(-c**2*x**2 + 1)/(36*c) - b*d**3*g*asin(c*x)/(4*c**2) - 3*b*d**2*e*f*asin(c*x)/(4*c**2) + 2*b*d**3 *h*sqrt(-c**2*x**2 + 1)/(9*c**3) + 2*b*d**2*e*g*sqrt(-c**2*x**2 + 1)/(3*c* *3) + 9*b*d**2*e*h*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 2*b*d*e**2*f*sqrt(-c **2*x**2 + 1)/(3*c**3) + 9*b*d*e**2*g*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 4 *b*d*e**2*h*x**2*sqrt(-c**2*x**2 + 1)/(25*c**3) + 3*b*e**3*f*x*sqrt(-c*...
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} \]
1/6*a*e^3*h*x^6 + 1/5*a*e^3*g*x^5 + 3/5*a*d*e^2*h*x^5 + 1/4*a*e^3*f*x^4 + 3/4*a*d*e^2*g*x^4 + 3/4*a*d^2*e*h*x^4 + a*d*e^2*f*x^3 + a*d^2*e*g*x^3 + 1/ 3*a*d^3*h*x^3 + 3/2*a*d^2*e*f*x^2 + 1/2*a*d^3*g*x^2 + 3/4*(2*x^2*arcsin(c* x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*b*d^2*e*f + 1/3*(3*x^ 3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)) *b*d*e^2*f + 1/32*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*s qrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*e^3*f + 1/4*(2*x^2*arcsi n(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*b*d^3*g + 1/3*(3* x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4 ))*b*d^2*e*g + 3/32*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3 *sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*d*e^2*g + 1/75*(15*x^5 *arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^ 4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*e^3*g + 1/9*(3*x^3*arcsin(c*x) + c*(sqr t(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*d^3*h + 3/32*(8*x^4 *arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*d^2*e*h + 1/25*(15*x^5*arcsin(c*x) + (3*sqrt(-c^ 2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c ^6)*c)*b*d*e^2*h + 1/288*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c ^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcs in(c*x)/c^7)*c)*b*e^3*h + a*d^3*f*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 ...
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 \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\text {Too large to display} \]
1/6*a*e^3*h*x^6 + 1/5*a*e^3*g*x^5 + 3/5*a*d*e^2*h*x^5 + 1/4*a*e^3*f*x^4 + 3/4*a*d*e^2*g*x^4 + 3/4*a*d^2*e*h*x^4 + a*d*e^2*f*x^3 + a*d^2*e*g*x^3 + 1/ 3*a*d^3*h*x^3 + b*d^3*f*x*arcsin(c*x) + a*d^3*f*x + (c^2*x^2 - 1)*b*d*e^2* f*x*arcsin(c*x)/c^2 + (c^2*x^2 - 1)*b*d^2*e*g*x*arcsin(c*x)/c^2 + 1/3*(c^2 *x^2 - 1)*b*d^3*h*x*arcsin(c*x)/c^2 + 3/4*sqrt(-c^2*x^2 + 1)*b*d^2*e*f*x/c + 1/4*sqrt(-c^2*x^2 + 1)*b*d^3*g*x/c + 3/2*(c^2*x^2 - 1)*b*d^2*e*f*arcsin (c*x)/c^2 + 1/2*(c^2*x^2 - 1)*b*d^3*g*arcsin(c*x)/c^2 + b*d*e^2*f*x*arcsin (c*x)/c^2 + b*d^2*e*g*x*arcsin(c*x)/c^2 + 1/5*(c^2*x^2 - 1)^2*b*e^3*g*x*ar csin(c*x)/c^4 + 1/3*b*d^3*h*x*arcsin(c*x)/c^2 + 3/5*(c^2*x^2 - 1)^2*b*d*e^ 2*h*x*arcsin(c*x)/c^4 + sqrt(-c^2*x^2 + 1)*b*d^3*f/c - 1/16*(-c^2*x^2 + 1) ^(3/2)*b*e^3*f*x/c^3 - 3/16*(-c^2*x^2 + 1)^(3/2)*b*d*e^2*g*x/c^3 - 3/16*(- c^2*x^2 + 1)^(3/2)*b*d^2*e*h*x/c^3 + 3/2*(c^2*x^2 - 1)*a*d^2*e*f/c^2 + 1/2 *(c^2*x^2 - 1)*a*d^3*g/c^2 + 3/4*b*d^2*e*f*arcsin(c*x)/c^2 + 1/4*(c^2*x^2 - 1)^2*b*e^3*f*arcsin(c*x)/c^4 + 1/4*b*d^3*g*arcsin(c*x)/c^2 + 3/4*(c^2*x^ 2 - 1)^2*b*d*e^2*g*arcsin(c*x)/c^4 + 3/4*(c^2*x^2 - 1)^2*b*d^2*e*h*arcsin( c*x)/c^4 + 2/5*(c^2*x^2 - 1)*b*e^3*g*x*arcsin(c*x)/c^4 + 6/5*(c^2*x^2 - 1) *b*d*e^2*h*x*arcsin(c*x)/c^4 - 1/3*(-c^2*x^2 + 1)^(3/2)*b*d*e^2*f/c^3 - 1/ 3*(-c^2*x^2 + 1)^(3/2)*b*d^2*e*g/c^3 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*d^3*h/c^ 3 + 5/32*sqrt(-c^2*x^2 + 1)*b*e^3*f*x/c^3 + 15/32*sqrt(-c^2*x^2 + 1)*b*d*e ^2*g*x/c^3 + 15/32*sqrt(-c^2*x^2 + 1)*b*d^2*e*h*x/c^3 + 1/36*(c^2*x^2 -...
Timed out. \[ \int (d+e x)^3 \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3\,\left (h\,x^2+g\,x+f\right ) \,d x \]