Integrand size = 31, antiderivative size = 689 \[ \int (d+e x)^3 \left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x)) \, dx=\frac {b \left (105 c^6 d^3 f+35 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+15 e^3 i+21 c^2 e \left (e^2 g+3 d e h+3 d^2 i\right )\right ) \sqrt {1-c^2 x^2}}{105 c^7}+\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {b \left (5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e h+3 d i) x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {b \left (35 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+45 e^3 i+42 c^2 e \left (e^2 g+3 d e h+3 d^2 i\right )\right ) \left (1-c^2 x^2\right )^{3/2}}{315 c^7}+\frac {b e \left (15 e^2 i+7 c^2 \left (e^2 g+3 d e h+3 d^2 i\right )\right ) \left (1-c^2 x^2\right )^{5/2}}{175 c^7}-\frac {b e^3 i \left (1-c^2 x^2\right )^{7/2}}{49 c^7}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) \arcsin (c x)}{96 c^6}+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \arcsin (c x))+\frac {1}{4} \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right ) x^4 (a+b \arcsin (c x))+\frac {1}{5} e \left (e^2 g+3 d e h+3 d^2 i\right ) x^5 (a+b \arcsin (c x))+\frac {1}{6} e^2 (e h+3 d i) x^6 (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x)) \] Output:
1/105*b*(105*c^6*d^3*f+35*c^4*d*(d^2*h+3*d*e*g+3*e^2*f)+15*e^3*i+21*c^2*e* (3*d^2*i+3*d*e*h+e^2*g))*(-c^2*x^2+1)^(1/2)/c^7+1/96*b*(24*c^4*d^2*(d*g+3* e*f)+5*e^2*(3*d*i+e*h)+9*c^2*(d^3*i+3*d^2*e*h+3*d*e^2*g+e^3*f))*x*(-c^2*x^ 2+1)^(1/2)/c^5+1/144*b*(5*e^2*(3*d*i+e*h)+9*c^2*(d^3*i+3*d^2*e*h+3*d*e^2*g +e^3*f))*x^3*(-c^2*x^2+1)^(1/2)/c^3+1/36*b*e^2*(3*d*i+e*h)*x^5*(-c^2*x^2+1 )^(1/2)/c-1/315*b*(35*c^4*d*(d^2*h+3*d*e*g+3*e^2*f)+45*e^3*i+42*c^2*e*(3*d ^2*i+3*d*e*h+e^2*g))*(-c^2*x^2+1)^(3/2)/c^7+1/175*b*e*(15*e^2*i+7*c^2*(3*d ^2*i+3*d*e*h+e^2*g))*(-c^2*x^2+1)^(5/2)/c^7-1/49*b*e^3*i*(-c^2*x^2+1)^(7/2 )/c^7-1/96*b*(24*c^4*d^2*(d*g+3*e*f)+5*e^2*(3*d*i+e*h)+9*c^2*(d^3*i+3*d^2* e*h+3*d*e^2*g+e^3*f))*arcsin(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*arcsin(c*x))+1/3*d*(d^2*h+3*d*e*g+3*e^2*f)*x^3*(a+b*arc sin(c*x))+1/4*(d^3*i+3*d^2*e*h+3*d*e^2*g+e^3*f)*x^4*(a+b*arcsin(c*x))+1/5* e*(3*d^2*i+3*d*e*h+e^2*g)*x^5*(a+b*arcsin(c*x))+1/6*e^2*(3*d*i+e*h)*x^6*(a +b*arcsin(c*x))+1/7*e^3*i*x^7*(a+b*arcsin(c*x))
Time = 0.55 (sec) , antiderivative size = 619, normalized size of antiderivative = 0.90 \[ \int (d+e x)^3 \left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x)) \, dx=a d^3 f x+\frac {1}{2} a d^2 (3 e f+d g) x^2+\frac {1}{3} a d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3+\frac {1}{4} a \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right ) x^4+\frac {1}{5} a e \left (e^2 g+3 d e h+3 d^2 i\right ) x^5+\frac {1}{6} a e^2 (e h+3 d i) x^6+\frac {1}{7} a e^3 i x^7+\frac {b \sqrt {1-c^2 x^2} \left (23040 e^3 i+3 c^2 e \left (37632 d^2 i+147 d e (256 h+125 i x)+e^2 (12544 g+5 x (1225 h+768 i x))\right )+c^4 \left (1225 d^3 (64 h+27 i x)+147 d^2 e \left (1600 g+675 h x+384 i x^2\right )+147 d e^2 \left (1600 f+x \left (675 g+384 h x+250 i x^2\right )\right )+e^3 x \left (33075 f+2 x \left (9408 g+6125 h x+4320 i x^2\right )\right )\right )+2 c^6 \left (1225 d^3 (144 f+x (36 g+x (16 h+9 i x)))+147 d^2 e x (900 f+x (400 g+9 x (25 h+16 i x)))+147 d e^2 x^2 (400 f+x (225 g+4 x (36 h+25 i x)))+e^3 x^3 (11025 f+4 x (1764 g+25 x (49 h+36 i x)))\right )\right )}{352800 c^7}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) \arcsin (c x)}{96 c^6}+\frac {1}{420} b x \left (35 d^3 (12 f+x (6 g+x (4 h+3 i x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+4 i x)))+21 d e^2 x^2 (20 f+x (15 g+2 x (6 h+5 i x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+6 i x)))\right ) \arcsin (c x) \] Input:
Integrate[(d + e*x)^3*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]
Output:
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^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i)*x^4)/4 + (a*e*(e^2*g + 3*d*e*h + 3*d^2*i)*x^5)/5 + (a*e^2*(e*h + 3*d*i)*x^6)/6 + (a*e^3*i*x^7)/ 7 + (b*Sqrt[1 - c^2*x^2]*(23040*e^3*i + 3*c^2*e*(37632*d^2*i + 147*d*e*(25 6*h + 125*i*x) + e^2*(12544*g + 5*x*(1225*h + 768*i*x))) + c^4*(1225*d^3*( 64*h + 27*i*x) + 147*d^2*e*(1600*g + 675*h*x + 384*i*x^2) + 147*d*e^2*(160 0*f + x*(675*g + 384*h*x + 250*i*x^2)) + e^3*x*(33075*f + 2*x*(9408*g + 61 25*h*x + 4320*i*x^2))) + 2*c^6*(1225*d^3*(144*f + x*(36*g + x*(16*h + 9*i* x))) + 147*d^2*e*x*(900*f + x*(400*g + 9*x*(25*h + 16*i*x))) + 147*d*e^2*x ^2*(400*f + x*(225*g + 4*x*(36*h + 25*i*x))) + e^3*x^3*(11025*f + 4*x*(176 4*g + 25*x*(49*h + 36*i*x))))))/(352800*c^7) - (b*(24*c^4*d^2*(3*e*f + d*g ) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i))*A rcSin[c*x])/(96*c^6) + (b*x*(35*d^3*(12*f + x*(6*g + x*(4*h + 3*i*x))) + 2 1*d^2*e*x*(30*f + x*(20*g + 3*x*(5*h + 4*i*x))) + 21*d*e^2*x^2*(20*f + x*( 15*g + 2*x*(6*h + 5*i*x))) + e^3*x^3*(105*f + 2*x*(42*g + 5*x*(7*h + 6*i*x ))))*ArcSin[c*x])/420
Time = 4.18 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.06, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {5248, 27, 2340, 25, 2340, 27, 2340, 25, 2340, 27, 2340, 25, 27, 533, 27, 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.
| \(\displaystyle \int (d+e x)^3 (a+b \arcsin (c x)) \left (f+g x+h x^2+i x^3\right ) \, dx\) |
| \(\Big \downarrow \) 5248 |
| \(\displaystyle -b c \int \frac {x \left (35 (12 f+x (6 g+x (4 h+3 i x))) d^3+21 e x (30 f+x (20 g+3 x (5 h+4 i x))) d^2+21 e^2 x^2 (20 f+x (15 g+2 x (6 h+5 i x))) d+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+6 i x)))\right )}{420 \sqrt {1-c^2 x^2}}dx+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{420} b c \int \frac {x \left (35 (12 f+x (6 g+x (4 h+3 i x))) d^3+21 e x (30 f+x (20 g+3 x (5 h+4 i x))) d^2+21 e^2 x^2 (20 f+x (15 g+2 x (6 h+5 i x))) d+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+6 i x)))\right )}{\sqrt {1-c^2 x^2}}dx+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{420} b c \left (-\frac {\int -\frac {x \left (490 c^2 e^2 (e h+3 d i) x^5+12 e \left (49 \left (3 i d^2+3 e h d+e^2 g\right ) c^2+30 e^2 i\right ) x^4+735 c^2 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) x^3+980 c^2 d \left (h d^2+3 e g d+3 e^2 f\right ) x^2+1470 c^2 d^2 (3 e f+d g) x+2940 c^2 d^3 f\right )}{\sqrt {1-c^2 x^2}}dx}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\int \frac {x \left (490 c^2 e^2 (e h+3 d i) x^5+12 e \left (49 \left (3 i d^2+3 e h d+e^2 g\right ) c^2+30 e^2 i\right ) x^4+735 c^2 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) x^3+980 c^2 d \left (h d^2+3 e g d+3 e^2 f\right ) x^2+1470 c^2 d^2 (3 e f+d g) x+2940 c^2 d^3 f\right )}{\sqrt {1-c^2 x^2}}dx}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {-\frac {\int -\frac {2 x \left (2940 d \left (h d^2+3 e g d+3 e^2 f\right ) x^2 c^4+8820 d^3 f c^4+4410 d^2 (3 e f+d g) x c^4+36 e \left (49 \left (3 i d^2+3 e h d+e^2 g\right ) c^2+30 e^2 i\right ) x^4 c^2+245 \left (9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right ) x^3 c^2\right )}{\sqrt {1-c^2 x^2}}dx}{6 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {\int \frac {x \left (2940 d \left (h d^2+3 e g d+3 e^2 f\right ) x^2 c^4+8820 d^3 f c^4+4410 d^2 (3 e f+d g) x c^4+36 e \left (49 \left (3 i d^2+3 e h d+e^2 g\right ) c^2+30 e^2 i\right ) x^4 c^2+245 \left (9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right ) x^3 c^2\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {-\frac {\int -\frac {x \left (44100 d^3 f c^6+22050 d^2 (3 e f+d g) x c^6+1225 \left (9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right ) x^3 c^4+12 \left (1225 d \left (h d^2+3 e g d+3 e^2 f\right ) c^4+588 e \left (3 i d^2+3 e h d+e^2 g\right ) c^2+360 e^3 i\right ) x^2 c^2\right )}{\sqrt {1-c^2 x^2}}dx}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {\frac {\int \frac {x \left (44100 d^3 f c^6+22050 d^2 (3 e f+d g) x c^6+1225 \left (9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right ) x^3 c^4+12 \left (1225 d \left (h d^2+3 e g d+3 e^2 f\right ) c^4+588 e \left (3 i d^2+3 e h d+e^2 g\right ) c^2+360 e^3 i\right ) x^2 c^2\right )}{\sqrt {1-c^2 x^2}}dx}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {\frac {-\frac {\int -\frac {3 x \left (58800 d^3 f c^8+16 \left (1225 d \left (h d^2+3 e g d+3 e^2 f\right ) c^4+588 e \left (3 i d^2+3 e h d+e^2 g\right ) c^2+360 e^3 i\right ) x^2 c^4+1225 \left (24 d^2 (3 e f+d g) c^4+9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right ) x c^4\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {1225}{4} c^2 x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {\frac {\frac {3 \int \frac {x \left (58800 d^3 f c^8+16 \left (1225 d \left (h d^2+3 e g d+3 e^2 f\right ) c^4+588 e \left (3 i d^2+3 e h d+e^2 g\right ) c^2+360 e^3 i\right ) x^2 c^4+1225 \left (24 d^2 (3 e f+d g) c^4+9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right ) x c^4\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {1225}{4} c^2 x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {\frac {\frac {3 \left (-\frac {\int -\frac {c^4 x \left (3675 \left (24 d^2 (3 e f+d g) c^4+9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right ) x c^2+16 \left (11025 d^3 f c^6+2450 d \left (h d^2+3 e g d+3 e^2 f\right ) c^4+1176 e \left (3 i d^2+3 e h d+e^2 g\right ) c^2+720 e^3 i\right )\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )\right )}{4 c^2}-\frac {1225}{4} c^2 x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {\frac {\frac {3 \left (\frac {\int \frac {c^4 x \left (3675 \left (24 d^2 (3 e f+d g) c^4+9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right ) x c^2+16 \left (11025 d^3 f c^6+2450 d \left (h d^2+3 e g d+3 e^2 f\right ) c^4+1176 e \left (3 i d^2+3 e h d+e^2 g\right ) c^2+720 e^3 i\right )\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )\right )}{4 c^2}-\frac {1225}{4} c^2 x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {\frac {\frac {3 \left (\frac {1}{3} c^2 \int \frac {x \left (3675 \left (24 d^2 (3 e f+d g) c^4+9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right ) x c^2+16 \left (11025 d^3 f c^6+2450 d \left (h d^2+3 e g d+3 e^2 f\right ) c^4+1176 e \left (3 i d^2+3 e h d+e^2 g\right ) c^2+720 e^3 i\right )\right )}{\sqrt {1-c^2 x^2}}dx-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )\right )}{4 c^2}-\frac {1225}{4} c^2 x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {\frac {\frac {3 \left (\frac {1}{3} c^2 \left (\frac {\int \frac {c^2 \left (3675 \left (24 d^2 (3 e f+d g) c^4+9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right )+32 \left (11025 d^3 f c^6+2450 d \left (h d^2+3 e g d+3 e^2 f\right ) c^4+1176 e \left (3 i d^2+3 e h d+e^2 g\right ) c^2+720 e^3 i\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {3675}{2} x \sqrt {1-c^2 x^2} \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )\right )-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )\right )}{4 c^2}-\frac {1225}{4} c^2 x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {\frac {\frac {3 \left (\frac {1}{3} c^2 \left (\frac {1}{2} \int \frac {3675 \left (24 d^2 (3 e f+d g) c^4+9 \left (i d^3+3 e h d^2+3 e^2 g d+e^3 f\right ) c^2+5 e^2 (e h+3 d i)\right )+32 \left (11025 d^3 f c^6+2450 d \left (h d^2+3 e g d+3 e^2 f\right ) c^4+1176 e \left (3 i d^2+3 e h d+e^2 g\right ) c^2+720 e^3 i\right ) x}{\sqrt {1-c^2 x^2}}dx-\frac {3675}{2} x \sqrt {1-c^2 x^2} \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )\right )-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )\right )}{4 c^2}-\frac {1225}{4} c^2 x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {1}{420} b c \left (\frac {\frac {\frac {\frac {3 \left (\frac {1}{3} c^2 \left (\frac {1}{2} \left (3675 \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx-\frac {32 \sqrt {1-c^2 x^2} \left (11025 c^6 d^3 f+2450 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+1176 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+720 e^3 i\right )}{c^2}\right )-\frac {3675}{2} x \sqrt {1-c^2 x^2} \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )\right )-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )\right )}{4 c^2}-\frac {1225}{4} c^2 x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle d^3 f x (a+b \arcsin (c x))+\frac {1}{3} d x^3 (a+b \arcsin (c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \arcsin (c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \arcsin (c x))+\frac {1}{7} e^3 i x^7 (a+b \arcsin (c x))-\frac {1}{420} b c \left (\frac {\frac {\frac {\frac {3 \left (\frac {1}{3} c^2 \left (\frac {1}{2} \left (\frac {3675 \arcsin (c x) \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{c}-\frac {32 \sqrt {1-c^2 x^2} \left (11025 c^6 d^3 f+2450 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+1176 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+720 e^3 i\right )}{c^2}\right )-\frac {3675}{2} x \sqrt {1-c^2 x^2} \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )\right )-\frac {16}{3} c^2 x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )\right )}{4 c^2}-\frac {1225}{4} c^2 x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{5 c^2}-\frac {36}{5} e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{3 c^2}-\frac {245}{3} e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{7 c^2}-\frac {60 e^3 i x^6 \sqrt {1-c^2 x^2}}{7 c^2}\right )\) |
Input:
Int[(d + e*x)^3*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]
Output:
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^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i)*x^4*(a + b*ArcSin[c*x]))/4 + (e*(e^2*g + 3* d*e*h + 3*d^2*i)*x^5*(a + b*ArcSin[c*x]))/5 + (e^2*(e*h + 3*d*i)*x^6*(a + b*ArcSin[c*x]))/6 + (e^3*i*x^7*(a + b*ArcSin[c*x]))/7 - (b*c*((-60*e^3*i*x ^6*Sqrt[1 - c^2*x^2])/(7*c^2) + ((-245*e^2*(e*h + 3*d*i)*x^5*Sqrt[1 - c^2* x^2])/3 + ((-36*e*(30*e^2*i + 49*c^2*(e^2*g + 3*d*e*h + 3*d^2*i))*x^4*Sqrt [1 - c^2*x^2])/5 + ((-1225*c^2*(5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e ^2*g + 3*d^2*e*h + d^3*i))*x^3*Sqrt[1 - c^2*x^2])/4 + (3*((-16*c^2*(1225*c ^4*d*(3*e^2*f + 3*d*e*g + d^2*h) + 360*e^3*i + 588*c^2*e*(e^2*g + 3*d*e*h + 3*d^2*i))*x^2*Sqrt[1 - c^2*x^2])/3 + (c^2*((-3675*(24*c^4*d^2*(3*e*f + d *g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i)) *x*Sqrt[1 - c^2*x^2])/2 + ((-32*(11025*c^6*d^3*f + 2450*c^4*d*(3*e^2*f + 3 *d*e*g + d^2*h) + 720*e^3*i + 1176*c^2*e*(e^2*g + 3*d*e*h + 3*d^2*i))*Sqrt [1 - c^2*x^2])/c^2 + (3675*(24*c^4*d^2*(3*e*f + d*g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i))*ArcSin[c*x])/c)/2))/3))/ (4*c^2))/(5*c^2))/(3*c^2))/(7*c^2)))/420
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 = 0.29 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.16
| method | result | size |
| parts | \(a \left (\frac {e^{3} i \,x^{7}}{7}+\frac {\left (3 d \,e^{2} i +e^{3} h \right ) x^{6}}{6}+\frac {\left (3 d^{2} e i +3 d \,e^{2} h +e^{3} g \right ) x^{5}}{5}+\frac {\left (d^{3} i +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} i \,x^{7}}{7}+\frac {c \arcsin \left (c x \right ) x^{6} d \,e^{2} i}{2}+\frac {c \arcsin \left (c x \right ) e^{3} h \,x^{6}}{6}+\frac {3 c \arcsin \left (c x \right ) x^{5} d^{2} e i}{5}+\frac {3 c \arcsin \left (c x \right ) x^{5} d \,e^{2} h}{5}+\frac {c \arcsin \left (c x \right ) x^{5} e^{3} g}{5}+\frac {c \arcsin \left (c x \right ) x^{4} d^{3} i}{4}+\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 {210 c^{5} d^{2} \left (d g +3 e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+70 c \,e^{2} \left (3 d i +e h \right ) \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 )+140 c^{4} d \left (d^{2} h +3 d e g +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 )+84 c^{2} e \left (3 d^{2} i +3 d e h +e^{2} 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 )+105 c^{3} \left (d^{3} i +3 d^{2} e h +3 d \,e^{2} g +e^{3} 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 e^{3} i \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )-420 d^{3} c^{6} f \sqrt {-c^{2} x^{2}+1}}{420 c^{6}}\right )}{c}\) | \(802\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {e^{3} i \,c^{7} x^{7}}{7}+\frac {\left (3 c d \,e^{2} i +e^{3} c h \right ) c^{6} x^{6}}{6}+\frac {\left (3 c^{2} d^{2} e i +3 c^{2} d \,e^{2} h +e^{3} c^{2} g \right ) c^{5} x^{5}}{5}+\frac {\left (c^{3} d^{3} i +3 c^{3} d^{2} e h +3 c^{3} d \,e^{2} g +e^{3} f \,c^{3}\right ) c^{4} x^{4}}{4}+\frac {\left (c^{4} d^{3} h +3 c^{4} d^{2} e g +3 c^{4} d \,e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{5} d^{3} g +3 c^{5} d^{2} e f \right ) c^{2} x^{2}}{2}+d^{3} c^{7} f x \right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} i \,c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{7} d \,e^{2} i \,x^{6}}{2}+\frac {\arcsin \left (c x \right ) c^{7} e^{3} h \,x^{6}}{6}+\frac {3 \arcsin \left (c x \right ) c^{7} d^{2} e i \,x^{5}}{5}+\frac {3 \arcsin \left (c x \right ) c^{7} d \,e^{2} h \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{7} e^{3} g \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{7} d^{3} i \,x^{4}}{4}+\frac {3 \arcsin \left (c x \right ) c^{7} d^{2} e h \,x^{4}}{4}+\frac {3 \arcsin \left (c x \right ) c^{7} d \,e^{2} g \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{7} e^{3} f \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{7} d^{3} h \,x^{3}}{3}+\arcsin \left (c x \right ) c^{7} d^{2} e g \,x^{3}+\arcsin \left (c x \right ) c^{7} d \,e^{2} f \,x^{3}+\frac {\arcsin \left (c x \right ) c^{7} d^{3} g \,x^{2}}{2}+\frac {3 \arcsin \left (c x \right ) c^{7} d^{2} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d^{3} c^{7} f x -\frac {c^{5} d^{2} \left (d g +3 e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}-\frac {c \,e^{2} \left (3 d i +e h \right ) \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}-\frac {c^{4} d \left (d^{2} h +3 d e g +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 )}{3}-\frac {c^{2} e \left (3 d^{2} i +3 d e h +e^{2} 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 )}{5}-\frac {c^{3} \left (d^{3} i +3 d^{2} e h +3 d \,e^{2} g +e^{3} 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 )}{4}-\frac {e^{3} i \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+d^{3} c^{6} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{6}}}{c}\) | \(893\) |
| default | \(\frac {\frac {a \left (\frac {e^{3} i \,c^{7} x^{7}}{7}+\frac {\left (3 c d \,e^{2} i +e^{3} c h \right ) c^{6} x^{6}}{6}+\frac {\left (3 c^{2} d^{2} e i +3 c^{2} d \,e^{2} h +e^{3} c^{2} g \right ) c^{5} x^{5}}{5}+\frac {\left (c^{3} d^{3} i +3 c^{3} d^{2} e h +3 c^{3} d \,e^{2} g +e^{3} f \,c^{3}\right ) c^{4} x^{4}}{4}+\frac {\left (c^{4} d^{3} h +3 c^{4} d^{2} e g +3 c^{4} d \,e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{5} d^{3} g +3 c^{5} d^{2} e f \right ) c^{2} x^{2}}{2}+d^{3} c^{7} f x \right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} i \,c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{7} d \,e^{2} i \,x^{6}}{2}+\frac {\arcsin \left (c x \right ) c^{7} e^{3} h \,x^{6}}{6}+\frac {3 \arcsin \left (c x \right ) c^{7} d^{2} e i \,x^{5}}{5}+\frac {3 \arcsin \left (c x \right ) c^{7} d \,e^{2} h \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{7} e^{3} g \,x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{7} d^{3} i \,x^{4}}{4}+\frac {3 \arcsin \left (c x \right ) c^{7} d^{2} e h \,x^{4}}{4}+\frac {3 \arcsin \left (c x \right ) c^{7} d \,e^{2} g \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{7} e^{3} f \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{7} d^{3} h \,x^{3}}{3}+\arcsin \left (c x \right ) c^{7} d^{2} e g \,x^{3}+\arcsin \left (c x \right ) c^{7} d \,e^{2} f \,x^{3}+\frac {\arcsin \left (c x \right ) c^{7} d^{3} g \,x^{2}}{2}+\frac {3 \arcsin \left (c x \right ) c^{7} d^{2} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d^{3} c^{7} f x -\frac {c^{5} d^{2} \left (d g +3 e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}-\frac {c \,e^{2} \left (3 d i +e h \right ) \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}-\frac {c^{4} d \left (d^{2} h +3 d e g +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 )}{3}-\frac {c^{2} e \left (3 d^{2} i +3 d e h +e^{2} 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 )}{5}-\frac {c^{3} \left (d^{3} i +3 d^{2} e h +3 d \,e^{2} g +e^{3} 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 )}{4}-\frac {e^{3} i \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+d^{3} c^{6} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{6}}}{c}\) | \(893\) |
| orering | \(\text {Expression too large to display}\) | \(2991\) |
Input:
int((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOS E)
Output:
a*(1/7*e^3*i*x^7+1/6*(3*d*e^2*i+e^3*h)*x^6+1/5*(3*d^2*e*i+3*d*e^2*h+e^3*g) *x^5+1/4*(d^3*i+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*(1/7*c*arcsin(c*x)*e^3*i*x ^7+1/2*c*arcsin(c*x)*x^6*d*e^2*i+1/6*c*arcsin(c*x)*e^3*h*x^6+3/5*c*arcsin( c*x)*x^5*d^2*e*i+3/5*c*arcsin(c*x)*x^5*d*e^2*h+1/5*c*arcsin(c*x)*x^5*e^3*g +1/4*c*arcsin(c*x)*x^4*d^3*i+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*a rcsin(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*arcsin(c*x)*x^2*d^2*e*f+arcsin(c*x)*d^3*f*c*x-1/420/c^6*(210*c^5* d^2*(d*g+3*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+70*c*e^2*(3* d*i+e*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))+140*c^4*d*(d^2*h+3*d*e*g+3*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))+84*c^2*e*(3 *d^2*i+3*d*e*h+e^2*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))+105*c^3*(d^3*i+3*d^2*e*h+3*d*e^2*g+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*arcs in(c*x))+60*e^3*i*(-1/7*c^6*x^6*(-c^2*x^2+1)^(1/2)-6/35*c^4*x^4*(-c^2*x^2+ 1)^(1/2)-8/35*c^2*x^2*(-c^2*x^2+1)^(1/2)-16/35*(-c^2*x^2+1)^(1/2))-420*d^3 *c^6*f*(-c^2*x^2+1)^(1/2)))
Time = 0.22 (sec) , antiderivative size = 936, normalized size of antiderivative = 1.36 \[ \int (d+e x)^3 \left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x)) \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="fr icas")
Output:
1/352800*(50400*a*c^7*e^3*i*x^7 + 352800*a*c^7*d^3*f*x + 58800*(a*c^7*e^3* h + 3*a*c^7*d*e^2*i)*x^6 + 70560*(a*c^7*e^3*g + 3*a*c^7*d*e^2*h + 3*a*c^7* d^2*e*i)*x^5 + 88200*(a*c^7*e^3*f + 3*a*c^7*d*e^2*g + 3*a*c^7*d^2*e*h + a* c^7*d^3*i)*x^4 + 117600*(3*a*c^7*d*e^2*f + 3*a*c^7*d^2*e*g + a*c^7*d^3*h)* x^3 + 176400*(3*a*c^7*d^2*e*f + a*c^7*d^3*g)*x^2 + 105*(480*b*c^7*e^3*i*x^ 7 + 3360*b*c^7*d^3*f*x + 560*(b*c^7*e^3*h + 3*b*c^7*d*e^2*i)*x^6 + 672*(b* c^7*e^3*g + 3*b*c^7*d*e^2*h + 3*b*c^7*d^2*e*i)*x^5 + 840*(b*c^7*e^3*f + 3* b*c^7*d*e^2*g + 3*b*c^7*d^2*e*h + b*c^7*d^3*i)*x^4 + 1120*(3*b*c^7*d*e^2*f + 3*b*c^7*d^2*e*g + b*c^7*d^3*h)*x^3 + 1680*(3*b*c^7*d^2*e*f + b*c^7*d^3* g)*x^2 - 315*(8*b*c^5*d^2*e + b*c^3*e^3)*f - 105*(8*b*c^5*d^3 + 9*b*c^3*d* e^2)*g - 35*(27*b*c^3*d^2*e + 5*b*c*e^3)*h - 105*(3*b*c^3*d^3 + 5*b*c*d*e^ 2)*i)*arcsin(c*x) + (7200*b*c^6*e^3*i*x^6 + 9800*(b*c^6*e^3*h + 3*b*c^6*d* e^2*i)*x^5 + 288*(49*b*c^6*e^3*g + 147*b*c^6*d*e^2*h + 3*(49*b*c^6*d^2*e + 10*b*c^4*e^3)*i)*x^4 + 2450*(9*b*c^6*e^3*f + 27*b*c^6*d*e^2*g + (27*b*c^6 *d^2*e + 5*b*c^4*e^3)*h + 3*(3*b*c^6*d^3 + 5*b*c^4*d*e^2)*i)*x^3 + 32*(367 5*b*c^6*d*e^2*f + 147*(25*b*c^6*d^2*e + 4*b*c^4*e^3)*g + 49*(25*b*c^6*d^3 + 36*b*c^4*d*e^2)*h + 36*(49*b*c^4*d^2*e + 10*b*c^2*e^3)*i)*x^2 + 117600*( 3*b*c^6*d^3 + 2*b*c^4*d*e^2)*f + 9408*(25*b*c^4*d^2*e + 4*b*c^2*e^3)*g + 3 136*(25*b*c^4*d^3 + 36*b*c^2*d*e^2)*h + 2304*(49*b*c^2*d^2*e + 10*b*e^3)*i + 3675*(9*(8*b*c^6*d^2*e + b*c^4*e^3)*f + 3*(8*b*c^6*d^3 + 9*b*c^4*d*e...
Leaf count of result is larger than twice the leaf count of optimal. 1809 vs. \(2 (694) = 1388\).
Time = 0.89 (sec) , antiderivative size = 1809, normalized size of antiderivative = 2.63 \[ \int (d+e x)^3 \left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x)) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)**3*(i*x**3+h*x**2+g*x+f)*(a+b*asin(c*x)),x)
Output:
Piecewise((a*d**3*f*x + a*d**3*g*x**2/2 + a*d**3*h*x**3/3 + a*d**3*i*x**4/ 4 + 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 + 3*a*d**2 *e*i*x**5/5 + 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*d*e**2*i*x**6/2 + a*e**3*f*x**4/4 + a*e**3*g*x**5/5 + a*e**3*h*x**6/6 + a*e**3*i*x**7/7 + 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 + b*d**3*i*x**4*asin(c*x)/4 + 3*b*d**2*e*f*x**2*asi n(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 + 3*b *d**2*e*i*x**5*asin(c*x)/5 + 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*d*e**2*i*x**6*asin(c*x)/2 + b*e**3*f*x**4*asin(c*x)/4 + b*e**3*g*x**5*asin(c*x)/5 + b*e**3*h*x**6*a sin(c*x)/6 + b*e**3*i*x**7*asin(c*x)/7 + 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) + b*d**3*i*x**3*sqrt(-c**2*x**2 + 1)/(16*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) + 3*b*d**2*e*i*x**4*sqrt(-c**2*x* *2 + 1)/(25*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*d*e**2*i*x**5*sqrt(-c**2*x**2 + 1)/(12*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*e**3*i*x**6*sqrt(-c**2*x**2 +...
Time = 0.15 (sec) , antiderivative size = 1231, normalized size of antiderivative = 1.79 \[ \int (d+e x)^3 \left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x)) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="ma xima")
Output:
1/7*a*e^3*i*x^7 + 1/6*a*e^3*h*x^6 + 1/2*a*d*e^2*i*x^6 + 1/5*a*e^3*g*x^5 + 3/5*a*d*e^2*h*x^5 + 3/5*a*d^2*e*i*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 + 1/4*a*d^3*i*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 *sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*e^3*f + 1/4*(2*x^2*arc sin(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*(s qrt(-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...
Leaf count of result is larger than twice the leaf count of optimal. 2010 vs. \(2 (647) = 1294\).
Time = 0.18 (sec) , antiderivative size = 2010, normalized size of antiderivative = 2.92 \[ \int (d+e x)^3 \left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x)) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="gi ac")
Output:
1/7*a*e^3*i*x^7 + 1/6*a*e^3*h*x^6 + 1/2*a*d*e^2*i*x^6 + 1/5*a*e^3*g*x^5 + 3/5*a*d*e^2*h*x^5 + 3/5*a*d^2*e*i*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 + 1/4*a*d^3*i*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*arcs in(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*arcs in(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* arcsin(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 + 3/5*(c^2*x^2 - 1)^2*b*d^2*e*i*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 - 1/16*(-c^2*x^2 + 1)^(3/2)*b*d^3*i*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 + 1/4*(c^2*x^2 - 1)^2*b*d^3*i*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 + 6/5*(c^2*x^2 - 1)*b*d^2*e*i*x*arcsin(c*x)/c^4 + ...
Timed out. \[ \int (d+e x)^3 \left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3\,\left (i\,x^3+h\,x^2+g\,x+f\right ) \,d x \] Input:
int((a + b*asin(c*x))*(d + e*x)^3*(f + g*x + h*x^2 + i*x^3),x)
Output:
int((a + b*asin(c*x))*(d + e*x)^3*(f + g*x + h*x^2 + i*x^3), x)
Time = 0.30 (sec) , antiderivative size = 1449, normalized size of antiderivative = 2.10 \[ \int (d+e x)^3 \left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x)) \, dx =\text {Too large to display} \] Input:
int((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*asin(c*x)),x)
Output:
(352800*asin(c*x)*b*c**7*d**3*f*x + 176400*asin(c*x)*b*c**7*d**3*g*x**2 + 117600*asin(c*x)*b*c**7*d**3*h*x**3 + 88200*asin(c*x)*b*c**7*d**3*i*x**4 + 529200*asin(c*x)*b*c**7*d**2*e*f*x**2 + 352800*asin(c*x)*b*c**7*d**2*e*g* x**3 + 264600*asin(c*x)*b*c**7*d**2*e*h*x**4 + 211680*asin(c*x)*b*c**7*d** 2*e*i*x**5 + 352800*asin(c*x)*b*c**7*d*e**2*f*x**3 + 264600*asin(c*x)*b*c* *7*d*e**2*g*x**4 + 211680*asin(c*x)*b*c**7*d*e**2*h*x**5 + 176400*asin(c*x )*b*c**7*d*e**2*i*x**6 + 88200*asin(c*x)*b*c**7*e**3*f*x**4 + 70560*asin(c *x)*b*c**7*e**3*g*x**5 + 58800*asin(c*x)*b*c**7*e**3*h*x**6 + 50400*asin(c *x)*b*c**7*e**3*i*x**7 - 88200*asin(c*x)*b*c**5*d**3*g - 264600*asin(c*x)* b*c**5*d**2*e*f - 33075*asin(c*x)*b*c**3*d**3*i - 99225*asin(c*x)*b*c**3*d **2*e*h - 99225*asin(c*x)*b*c**3*d*e**2*g - 33075*asin(c*x)*b*c**3*e**3*f - 55125*asin(c*x)*b*c*d*e**2*i - 18375*asin(c*x)*b*c*e**3*h + 352800*sqrt( - c**2*x**2 + 1)*b*c**6*d**3*f + 88200*sqrt( - c**2*x**2 + 1)*b*c**6*d**3 *g*x + 39200*sqrt( - c**2*x**2 + 1)*b*c**6*d**3*h*x**2 + 22050*sqrt( - c** 2*x**2 + 1)*b*c**6*d**3*i*x**3 + 264600*sqrt( - c**2*x**2 + 1)*b*c**6*d**2 *e*f*x + 117600*sqrt( - c**2*x**2 + 1)*b*c**6*d**2*e*g*x**2 + 66150*sqrt( - c**2*x**2 + 1)*b*c**6*d**2*e*h*x**3 + 42336*sqrt( - c**2*x**2 + 1)*b*c** 6*d**2*e*i*x**4 + 117600*sqrt( - c**2*x**2 + 1)*b*c**6*d*e**2*f*x**2 + 661 50*sqrt( - c**2*x**2 + 1)*b*c**6*d*e**2*g*x**3 + 42336*sqrt( - c**2*x**2 + 1)*b*c**6*d*e**2*h*x**4 + 29400*sqrt( - c**2*x**2 + 1)*b*c**6*d*e**2*i...