Integrand size = 21, antiderivative size = 358 \[ \int (d+e x)^3 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {b \left (5 c^4 d^3 f+e^3 g+5 c^2 d e (e f+d g)\right ) \sqrt {1-c^2 x^2}}{5 c^5}+\frac {b \left (8 c^2 d^2 (3 e f+d g)+3 e^2 (e f+3 d g)\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^2 (e f+3 d g) x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {b e \left (2 e^2 g+5 c^2 d (e f+d g)\right ) \left (1-c^2 x^2\right )^{3/2}}{15 c^5}+\frac {b e^3 g \left (1-c^2 x^2\right )^{5/2}}{25 c^5}-\frac {b \left (8 c^2 d^2 (3 e f+d g)+3 e^2 (e f+3 d g)\right ) \arcsin (c x)}{32 c^4}+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \arcsin (c x))+d e (e f+d g) x^3 (a+b \arcsin (c x))+\frac {1}{4} e^2 (e f+3 d g) x^4 (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x)) \] Output:
1/5*b*(5*c^4*d^3*f+e^3*g+5*c^2*d*e*(d*g+e*f))*(-c^2*x^2+1)^(1/2)/c^5+1/32* b*(8*c^2*d^2*(d*g+3*e*f)+3*e^2*(3*d*g+e*f))*x*(-c^2*x^2+1)^(1/2)/c^3+1/16* b*e^2*(3*d*g+e*f)*x^3*(-c^2*x^2+1)^(1/2)/c-1/15*b*e*(2*e^2*g+5*c^2*d*(d*g+ e*f))*(-c^2*x^2+1)^(3/2)/c^5+1/25*b*e^3*g*(-c^2*x^2+1)^(5/2)/c^5-1/32*b*(8 *c^2*d^2*(d*g+3*e*f)+3*e^2*(3*d*g+e*f))*arcsin(c*x)/c^4+d^3*f*x*(a+b*arcsi n(c*x))+1/2*d^2*(d*g+3*e*f)*x^2*(a+b*arcsin(c*x))+d*e*(d*g+e*f)*x^3*(a+b*a rcsin(c*x))+1/4*e^2*(3*d*g+e*f)*x^4*(a+b*arcsin(c*x))+1/5*e^3*g*x^5*(a+b*a rcsin(c*x))
Time = 0.24 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.85 \[ \int (d+e x)^3 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {120 a c^5 x \left (10 d^3 (2 f+g x)+10 d^2 e x (3 f+2 g x)+5 d e^2 x^2 (4 f+3 g x)+e^3 x^3 (5 f+4 g x)\right )+b \sqrt {1-c^2 x^2} \left (256 e^3 g+2 c^4 \left (300 d^3 (4 f+g x)+100 d^2 e x (9 f+4 g x)+25 d e^2 x^2 (16 f+9 g x)+3 e^3 x^3 (25 f+16 g x)\right )+c^2 e \left (1600 d^2 g+25 d e (64 f+27 g x)+e^2 x (225 f+128 g x)\right )\right )+15 b c \left (-40 c^2 d^2 (3 e f+d g)-15 e^2 (e f+3 d g)+8 c^4 x \left (10 d^3 (2 f+g x)+10 d^2 e x (3 f+2 g x)+5 d e^2 x^2 (4 f+3 g x)+e^3 x^3 (5 f+4 g x)\right )\right ) \arcsin (c x)}{2400 c^5} \] Input:
Integrate[(d + e*x)^3*(f + g*x)*(a + b*ArcSin[c*x]),x]
Output:
(120*a*c^5*x*(10*d^3*(2*f + g*x) + 10*d^2*e*x*(3*f + 2*g*x) + 5*d*e^2*x^2* (4*f + 3*g*x) + e^3*x^3*(5*f + 4*g*x)) + b*Sqrt[1 - c^2*x^2]*(256*e^3*g + 2*c^4*(300*d^3*(4*f + g*x) + 100*d^2*e*x*(9*f + 4*g*x) + 25*d*e^2*x^2*(16* f + 9*g*x) + 3*e^3*x^3*(25*f + 16*g*x)) + c^2*e*(1600*d^2*g + 25*d*e*(64*f + 27*g*x) + e^2*x*(225*f + 128*g*x))) + 15*b*c*(-40*c^2*d^2*(3*e*f + d*g) - 15*e^2*(e*f + 3*d*g) + 8*c^4*x*(10*d^3*(2*f + g*x) + 10*d^2*e*x*(3*f + 2*g*x) + 5*d*e^2*x^2*(4*f + 3*g*x) + e^3*x^3*(5*f + 4*g*x)))*ArcSin[c*x])/ (2400*c^5)
Time = 1.44 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {5248, 27, 2340, 25, 2340, 25, 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.
| \(\displaystyle \int (d+e x)^3 (f+g x) (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5248 |
| \(\displaystyle -b c \int \frac {x \left (4 e^3 g x^4+5 e^2 (e f+3 d g) x^3+20 d e (e f+d g) x^2+10 d^2 (3 e f+d g) x+20 d^3 f\right )}{20 \sqrt {1-c^2 x^2}}dx+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{20} b c \int \frac {x \left (4 e^3 g x^4+5 e^2 (e f+3 d g) x^3+20 d e (e f+d g) x^2+10 d^2 (3 e f+d g) x+20 d^3 f\right )}{\sqrt {1-c^2 x^2}}dx+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{20} b c \left (-\frac {\int -\frac {x \left (100 c^2 f d^3+50 c^2 (3 e f+d g) x d^2+25 c^2 e^2 (e f+3 d g) x^3+4 e \left (25 d (e f+d g) c^2+4 e^2 g\right ) x^2\right )}{\sqrt {1-c^2 x^2}}dx}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{20} b c \left (\frac {\int \frac {x \left (100 c^2 f d^3+50 c^2 (3 e f+d g) x d^2+25 c^2 e^2 (e f+3 d g) x^3+4 e \left (25 d (e f+d g) c^2+4 e^2 g\right ) x^2\right )}{\sqrt {1-c^2 x^2}}dx}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{20} b c \left (\frac {-\frac {\int -\frac {x \left (400 d^3 f c^4+16 e \left (25 d (e f+d g) c^2+4 e^2 g\right ) x^2 c^2+25 \left (8 c^2 (3 e f+d g) d^2+3 e^2 (e f+3 d g)\right ) x c^2\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {25}{4} e^2 x^3 \sqrt {1-c^2 x^2} (3 d g+e f)}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{20} b c \left (\frac {\frac {\int \frac {x \left (400 d^3 f c^4+16 e \left (25 d (e f+d g) c^2+4 e^2 g\right ) x^2 c^2+25 \left (8 c^2 (3 e f+d g) d^2+3 e^2 (e f+3 d g)\right ) x c^2\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {25}{4} e^2 x^3 \sqrt {1-c^2 x^2} (3 d g+e f)}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{20} b c \left (\frac {\frac {-\frac {\int -\frac {c^2 x \left (75 \left (8 c^2 (3 e f+d g) d^2+3 e^2 (e f+3 d g)\right ) x c^2+16 \left (75 d^3 f c^4+50 d e (e f+d g) c^2+8 e^3 g\right )\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d (d g+e f)+4 e^2 g\right )}{4 c^2}-\frac {25}{4} e^2 x^3 \sqrt {1-c^2 x^2} (3 d g+e f)}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{20} b c \left (\frac {\frac {\frac {\int \frac {c^2 x \left (75 \left (8 c^2 (3 e f+d g) d^2+3 e^2 (e f+3 d g)\right ) x c^2+16 \left (75 d^3 f c^4+50 d e (e f+d g) c^2+8 e^3 g\right )\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d (d g+e f)+4 e^2 g\right )}{4 c^2}-\frac {25}{4} e^2 x^3 \sqrt {1-c^2 x^2} (3 d g+e f)}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{20} b c \left (\frac {\frac {\frac {1}{3} \int \frac {x \left (75 \left (8 c^2 (3 e f+d g) d^2+3 e^2 (e f+3 d g)\right ) x c^2+16 \left (75 d^3 f c^4+50 d e (e f+d g) c^2+8 e^3 g\right )\right )}{\sqrt {1-c^2 x^2}}dx-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d (d g+e f)+4 e^2 g\right )}{4 c^2}-\frac {25}{4} e^2 x^3 \sqrt {1-c^2 x^2} (3 d g+e f)}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle -\frac {1}{20} b c \left (\frac {\frac {\frac {1}{3} \left (\frac {\int \frac {c^2 \left (75 \left (8 c^2 (3 e f+d g) d^2+3 e^2 (e f+3 d g)\right )+32 \left (75 d^3 f c^4+50 d e (e f+d g) c^2+8 e^3 g\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {75}{2} x \sqrt {1-c^2 x^2} \left (8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right )\right )-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d (d g+e f)+4 e^2 g\right )}{4 c^2}-\frac {25}{4} e^2 x^3 \sqrt {1-c^2 x^2} (3 d g+e f)}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{20} b c \left (\frac {\frac {\frac {1}{3} \left (\frac {1}{2} \int \frac {75 \left (8 c^2 (3 e f+d g) d^2+3 e^2 (e f+3 d g)\right )+32 \left (75 d^3 f c^4+50 d e (e f+d g) c^2+8 e^3 g\right ) x}{\sqrt {1-c^2 x^2}}dx-\frac {75}{2} x \sqrt {1-c^2 x^2} \left (8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right )\right )-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d (d g+e f)+4 e^2 g\right )}{4 c^2}-\frac {25}{4} e^2 x^3 \sqrt {1-c^2 x^2} (3 d g+e f)}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {1}{20} b c \left (\frac {\frac {\frac {1}{3} \left (\frac {1}{2} \left (75 \left (8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx-\frac {32 \sqrt {1-c^2 x^2} \left (75 c^4 d^3 f+50 c^2 d e (d g+e f)+8 e^3 g\right )}{c^2}\right )-\frac {75}{2} x \sqrt {1-c^2 x^2} \left (8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right )\right )-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d (d g+e f)+4 e^2 g\right )}{4 c^2}-\frac {25}{4} e^2 x^3 \sqrt {1-c^2 x^2} (3 d g+e f)}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )+d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle d^3 f x (a+b \arcsin (c x))+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 x^4 (3 d g+e f) (a+b \arcsin (c x))+d e x^3 (d g+e f) (a+b \arcsin (c x))+\frac {1}{5} e^3 g x^5 (a+b \arcsin (c x))-\frac {1}{20} b c \left (\frac {\frac {\frac {1}{3} \left (\frac {1}{2} \left (\frac {75 \arcsin (c x) \left (8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right )}{c}-\frac {32 \sqrt {1-c^2 x^2} \left (75 c^4 d^3 f+50 c^2 d e (d g+e f)+8 e^3 g\right )}{c^2}\right )-\frac {75}{2} x \sqrt {1-c^2 x^2} \left (8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right )\right )-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d (d g+e f)+4 e^2 g\right )}{4 c^2}-\frac {25}{4} e^2 x^3 \sqrt {1-c^2 x^2} (3 d g+e f)}{5 c^2}-\frac {4 e^3 g x^4 \sqrt {1-c^2 x^2}}{5 c^2}\right )\) |
Input:
Int[(d + e*x)^3*(f + g*x)*(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*e*(e*f + d*g)*x^3*(a + b*ArcSin[c*x]) + (e^2*(e*f + 3*d*g)*x^4*(a + b*ArcSin[c*x]))/4 + (e^3*g*x^5*(a + b*ArcSin[c*x]))/5 - (b*c*((-4*e^3*g*x^ 4*Sqrt[1 - c^2*x^2])/(5*c^2) + ((-25*e^2*(e*f + 3*d*g)*x^3*Sqrt[1 - c^2*x^ 2])/4 + ((-16*e*(4*e^2*g + 25*c^2*d*(e*f + d*g))*x^2*Sqrt[1 - c^2*x^2])/3 + ((-75*(8*c^2*d^2*(3*e*f + d*g) + 3*e^2*(e*f + 3*d*g))*x*Sqrt[1 - c^2*x^2 ])/2 + ((-32*(75*c^4*d^3*f + 8*e^3*g + 50*c^2*d*e*(e*f + d*g))*Sqrt[1 - c^ 2*x^2])/c^2 + (75*(8*c^2*d^2*(3*e*f + d*g) + 3*e^2*(e*f + 3*d*g))*ArcSin[c *x])/c)/2)/3)/(4*c^2))/(5*c^2)))/20
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 = 0.31 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.19
| method | result | size |
| parts | \(a \left (\frac {e^{3} g \,x^{5}}{5}+\frac {\left (3 d \,e^{2} g +e^{3} f \right ) x^{4}}{4}+\frac {\left (3 d^{2} e g +3 d \,e^{2} f \right ) x^{3}}{3}+\frac {\left (d^{3} g +3 d^{2} e f \right ) x^{2}}{2}+d^{3} f x \right )+\frac {b \left (\frac {c \arcsin \left (c x \right ) e^{3} g \,x^{5}}{5}+\frac {3 c \arcsin \left (c x \right ) x^{4} d \,e^{2} g}{4}+\frac {c \arcsin \left (c x \right ) x^{4} e^{3} f}{4}+c \arcsin \left (c x \right ) x^{3} d^{2} e g +c \arcsin \left (c x \right ) x^{3} d \,e^{2} f +\frac {c \arcsin \left (c x \right ) x^{2} d^{3} g}{2}+\frac {3 c \arcsin \left (c x \right ) x^{2} d^{2} e f}{2}+\arcsin \left (c x \right ) d^{3} f c x -\frac {15 c \,e^{2} \left (3 d g +e f \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+60 d \,c^{2} e \left (d g +e f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+30 c^{3} d^{2} \left (d g +3 e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+12 e^{3} g \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )-60 c^{4} d^{3} f \sqrt {-c^{2} x^{2}+1}}{60 c^{4}}\right )}{c}\) | \(426\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {e^{3} g \,c^{5} x^{5}}{5}+\frac {\left (3 c d \,e^{2} g +e^{3} f c \right ) c^{4} x^{4}}{4}+\frac {\left (3 e \,c^{2} d^{2} g +3 c^{2} d \,e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (d^{3} c^{3} g +3 e \,c^{3} d^{2} f \right ) c^{2} x^{2}}{2}+c^{5} d^{3} f x \right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} g \,c^{5} x^{5}}{5}+\frac {3 \arcsin \left (c x \right ) c^{5} d \,e^{2} g \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{5} e^{3} f \,x^{4}}{4}+\arcsin \left (c x \right ) c^{5} d^{2} e g \,x^{3}+\arcsin \left (c x \right ) c^{5} d \,e^{2} f \,x^{3}+\frac {\arcsin \left (c x \right ) c^{5} d^{3} g \,x^{2}}{2}+\frac {3 \arcsin \left (c x \right ) c^{5} d^{2} e f \,x^{2}}{2}+\arcsin \left (c x \right ) c^{5} d^{3} f x -\frac {c \,e^{2} \left (3 d g +e f \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-d \,c^{2} e \left (d g +e f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-\frac {c^{3} d^{2} \left (d g +3 e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}-\frac {e^{3} g \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+c^{4} d^{3} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{4}}}{c}\) | \(471\) |
| default | \(\frac {\frac {a \left (\frac {e^{3} g \,c^{5} x^{5}}{5}+\frac {\left (3 c d \,e^{2} g +e^{3} f c \right ) c^{4} x^{4}}{4}+\frac {\left (3 e \,c^{2} d^{2} g +3 c^{2} d \,e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (d^{3} c^{3} g +3 e \,c^{3} d^{2} f \right ) c^{2} x^{2}}{2}+c^{5} d^{3} f x \right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} g \,c^{5} x^{5}}{5}+\frac {3 \arcsin \left (c x \right ) c^{5} d \,e^{2} g \,x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{5} e^{3} f \,x^{4}}{4}+\arcsin \left (c x \right ) c^{5} d^{2} e g \,x^{3}+\arcsin \left (c x \right ) c^{5} d \,e^{2} f \,x^{3}+\frac {\arcsin \left (c x \right ) c^{5} d^{3} g \,x^{2}}{2}+\frac {3 \arcsin \left (c x \right ) c^{5} d^{2} e f \,x^{2}}{2}+\arcsin \left (c x \right ) c^{5} d^{3} f x -\frac {c \,e^{2} \left (3 d g +e f \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-d \,c^{2} e \left (d g +e f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-\frac {c^{3} d^{2} \left (d g +3 e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}-\frac {e^{3} g \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+c^{4} d^{3} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{4}}}{c}\) | \(471\) |
| orering | \(\frac {\left (864 c^{6} e^{4} g^{2} x^{7}+4176 c^{6} d \,e^{3} g^{2} x^{6}+1968 c^{6} e^{4} f g \,x^{6}+7850 c^{6} d^{2} e^{2} g^{2} x^{5}+9980 c^{6} d \,e^{3} f g \,x^{5}+1050 c^{6} e^{4} f^{2} x^{5}+6800 c^{6} d^{3} e \,g^{2} x^{4}+20600 c^{6} d^{2} e^{2} f g \,x^{4}+5400 c^{6} d \,e^{3} f^{2} x^{4}+1800 c^{6} d^{4} g^{2} x^{3}+22800 c^{6} d^{3} e f g \,x^{3}+11400 c^{6} d^{2} e^{2} f^{2} x^{3}+128 c^{4} e^{4} g^{2} x^{5}+6000 c^{6} d^{4} f g \,x^{2}+13200 c^{6} d^{3} e \,f^{2} x^{2}+932 c^{4} d \,e^{3} g^{2} x^{4}+396 c^{4} e^{4} f g \,x^{4}+2400 c^{6} d^{4} f^{2} x +3425 c^{4} d^{2} e^{2} g^{2} x^{3}+3950 c^{4} d \,e^{3} f g \,x^{3}+225 c^{4} e^{4} f^{2} x^{3}-2200 c^{4} d^{3} e \,g^{2} x^{2}-5800 c^{4} d^{2} e^{2} f g \,x^{2}+2400 c^{4} d \,e^{3} f^{2} x^{2}-1200 c^{4} d^{4} g^{2} x -15600 c^{4} d^{3} e f g x -7200 c^{4} d^{2} e^{2} f^{2} x +512 c^{2} e^{4} g^{2} x^{3}-3000 c^{4} d^{4} f g -9000 c^{4} d^{3} e \,f^{2}-3247 c^{2} d \,e^{3} g^{2} x^{2}-741 c^{2} e^{4} f g \,x^{2}-7750 c^{2} d^{2} e^{2} g^{2} x -9550 c^{2} d \,e^{3} f g x -900 c^{2} e^{4} f^{2} x -1600 c^{2} d^{3} e \,g^{2}-7075 c^{2} d^{2} e^{2} f g -5025 c^{2} d \,e^{3} f^{2}-1024 e^{4} g^{2} x -256 d \,e^{3} g^{2}-768 e^{4} f g \right ) \left (a +b \arcsin \left (c x \right )\right )}{2400 \left (e x +d \right ) \left (g x +f \right ) c^{6}}-\frac {\left (96 e^{3} g \,x^{4} c^{4}+450 c^{4} d \,e^{2} g \,x^{3}+150 c^{4} e^{3} f \,x^{3}+800 c^{4} d^{2} e g \,x^{2}+800 c^{4} d \,e^{2} f \,x^{2}+600 c^{4} d^{3} g x +1800 c^{4} d^{2} e f x +2400 c^{4} d^{3} f +128 c^{2} e^{3} g \,x^{2}+675 c^{2} d \,e^{2} g x +225 c^{2} e^{3} f x +1600 e \,c^{2} d^{2} g +1600 c^{2} d \,e^{2} f +256 e^{3} g \right ) \left (c x -1\right ) \left (c x +1\right ) \left (3 \left (e x +d \right )^{2} \left (g x +f \right ) \left (a +b \arcsin \left (c x \right )\right ) e +\left (e x +d \right )^{3} g \left (a +b \arcsin \left (c x \right )\right )+\frac {\left (e x +d \right )^{3} \left (g x +f \right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{2400 c^{6} \left (g x +f \right ) \left (e x +d \right )^{3}}\) | \(835\) |
Input:
int((e*x+d)^3*(g*x+f)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/5*e^3*g*x^5+1/4*(3*d*e^2*g+e^3*f)*x^4+1/3*(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*(1/5*c*arcsin(c*x)*e^3*g*x^5+3/4*c*a rcsin(c*x)*x^4*d*e^2*g+1/4*c*arcsin(c*x)*x^4*e^3*f+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*arcsin(c*x) *x^2*d^2*e*f+arcsin(c*x)*d^3*f*c*x-1/60/c^4*(15*c*e^2*(3*d*g+e*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))+60*d* c^2*e*(d*g+e*f)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+3 0*c^3*d^2*(d*g+3*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+12*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))-60*c^4*d^3*f*(-c^2*x^2+1)^(1/2)))
Time = 0.15 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.23 \[ \int (d+e x)^3 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {480 \, a c^{5} e^{3} g x^{5} + 2400 \, a c^{5} d^{3} f x + 600 \, {\left (a c^{5} e^{3} f + 3 \, a c^{5} d e^{2} g\right )} x^{4} + 2400 \, {\left (a c^{5} d e^{2} f + a c^{5} d^{2} e g\right )} x^{3} + 1200 \, {\left (3 \, a c^{5} d^{2} e f + a c^{5} d^{3} g\right )} x^{2} + 15 \, {\left (32 \, b c^{5} e^{3} g x^{5} + 160 \, b c^{5} d^{3} f x + 40 \, {\left (b c^{5} e^{3} f + 3 \, b c^{5} d e^{2} g\right )} x^{4} + 160 \, {\left (b c^{5} d e^{2} f + b c^{5} d^{2} e g\right )} x^{3} + 80 \, {\left (3 \, b c^{5} d^{2} e f + b c^{5} d^{3} g\right )} x^{2} - 15 \, {\left (8 \, b c^{3} d^{2} e + b c e^{3}\right )} f - 5 \, {\left (8 \, b c^{3} d^{3} + 9 \, b c d e^{2}\right )} g\right )} \arcsin \left (c x\right ) + {\left (96 \, b c^{4} e^{3} g x^{4} + 150 \, {\left (b c^{4} e^{3} f + 3 \, b c^{4} d e^{2} g\right )} x^{3} + 32 \, {\left (25 \, b c^{4} d e^{2} f + {\left (25 \, b c^{4} d^{2} e + 4 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 800 \, {\left (3 \, b c^{4} d^{3} + 2 \, b c^{2} d e^{2}\right )} f + 64 \, {\left (25 \, b c^{2} d^{2} e + 4 \, b e^{3}\right )} g + 75 \, {\left (3 \, {\left (8 \, b c^{4} d^{2} e + b c^{2} e^{3}\right )} f + {\left (8 \, b c^{4} d^{3} + 9 \, b c^{2} d e^{2}\right )} g\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{2400 \, c^{5}} \] Input:
integrate((e*x+d)^3*(g*x+f)*(a+b*arcsin(c*x)),x, algorithm="fricas")
Output:
1/2400*(480*a*c^5*e^3*g*x^5 + 2400*a*c^5*d^3*f*x + 600*(a*c^5*e^3*f + 3*a* c^5*d*e^2*g)*x^4 + 2400*(a*c^5*d*e^2*f + a*c^5*d^2*e*g)*x^3 + 1200*(3*a*c^ 5*d^2*e*f + a*c^5*d^3*g)*x^2 + 15*(32*b*c^5*e^3*g*x^5 + 160*b*c^5*d^3*f*x + 40*(b*c^5*e^3*f + 3*b*c^5*d*e^2*g)*x^4 + 160*(b*c^5*d*e^2*f + b*c^5*d^2* e*g)*x^3 + 80*(3*b*c^5*d^2*e*f + b*c^5*d^3*g)*x^2 - 15*(8*b*c^3*d^2*e + b* c*e^3)*f - 5*(8*b*c^3*d^3 + 9*b*c*d*e^2)*g)*arcsin(c*x) + (96*b*c^4*e^3*g* x^4 + 150*(b*c^4*e^3*f + 3*b*c^4*d*e^2*g)*x^3 + 32*(25*b*c^4*d*e^2*f + (25 *b*c^4*d^2*e + 4*b*c^2*e^3)*g)*x^2 + 800*(3*b*c^4*d^3 + 2*b*c^2*d*e^2)*f + 64*(25*b*c^2*d^2*e + 4*b*e^3)*g + 75*(3*(8*b*c^4*d^2*e + b*c^2*e^3)*f + ( 8*b*c^4*d^3 + 9*b*c^2*d*e^2)*g)*x)*sqrt(-c^2*x^2 + 1))/c^5
Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (345) = 690\).
Time = 0.46 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.15 \[ \int (d+e x)^3 (f+g x) (a+b \arcsin (c x)) \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)**3*(g*x+f)*(a+b*asin(c*x)),x)
Output:
Piecewise((a*d**3*f*x + a*d**3*g*x**2/2 + 3*a*d**2*e*f*x**2/2 + a*d**2*e*g *x**3 + a*d*e**2*f*x**3 + 3*a*d*e**2*g*x**4/4 + a*e**3*f*x**4/4 + a*e**3*g *x**5/5 + b*d**3*f*x*asin(c*x) + b*d**3*g*x**2*asin(c*x)/2 + 3*b*d**2*e*f* x**2*asin(c*x)/2 + b*d**2*e*g*x**3*asin(c*x) + b*d*e**2*f*x**3*asin(c*x) + 3*b*d*e**2*g*x**4*asin(c*x)/4 + b*e**3*f*x**4*asin(c*x)/4 + b*e**3*g*x**5 *asin(c*x)/5 + b*d**3*f*sqrt(-c**2*x**2 + 1)/c + b*d**3*g*x*sqrt(-c**2*x** 2 + 1)/(4*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) + 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) + 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*d**3*g*as in(c*x)/(4*c**2) - 3*b*d**2*e*f*asin(c*x)/(4*c**2) + 2*b*d**2*e*g*sqrt(-c* *2*x**2 + 1)/(3*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) + 3*b*e**3*f*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 4*b*e**3*g*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) - 9*b*d*e**2 *g*asin(c*x)/(32*c**4) - 3*b*e**3*f*asin(c*x)/(32*c**4) + 8*b*e**3*g*sqrt( -c**2*x**2 + 1)/(75*c**5), Ne(c, 0)), (a*(d**3*f*x + d**3*g*x**2/2 + 3*d** 2*e*f*x**2/2 + d**2*e*g*x**3 + d*e**2*f*x**3 + 3*d*e**2*g*x**4/4 + e**3*f* x**4/4 + e**3*g*x**5/5), True))
Time = 0.12 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.47 \[ \int (d+e x)^3 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {1}{5} \, a e^{3} g x^{5} + \frac {1}{4} \, a e^{3} f x^{4} + \frac {3}{4} \, a d e^{2} g x^{4} + a d e^{2} f x^{3} + a d^{2} e g x^{3} + \frac {3}{2} \, a d^{2} e f x^{2} + \frac {1}{2} \, a d^{3} g x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} e f + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d e^{2} f + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b e^{3} f + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} g + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} e g + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d e^{2} g + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b e^{3} g + a d^{3} f x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{3} f}{c} \] Input:
integrate((e*x+d)^3*(g*x+f)*(a+b*arcsin(c*x)),x, algorithm="maxima")
Output:
1/5*a*e^3*g*x^5 + 1/4*a*e^3*f*x^4 + 3/4*a*d*e^2*g*x^4 + a*d*e^2*f*x^3 + a* d^2*e*g*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*sqr t(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*e^3*f + 1/4*(2*x^2*arcsin( 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*s qrt(-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*a rcsin(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 + a*d^3*f*x + (c*x*arcsin(c*x) + sq rt(-c^2*x^2 + 1))*b*d^3*f/c
Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (330) = 660\).
Time = 0.15 (sec) , antiderivative size = 784, normalized size of antiderivative = 2.19 \[ \int (d+e x)^3 (f+g x) (a+b \arcsin (c x)) \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^3*(g*x+f)*(a+b*arcsin(c*x)),x, algorithm="giac")
Output:
1/5*a*e^3*g*x^5 + 1/4*a*e^3*f*x^4 + 3/4*a*d*e^2*g*x^4 + a*d*e^2*f*x^3 + a* d^2*e*g*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 + 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*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 + 3/2*(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 + 2/5*(c^2*x^2 - 1)*b*e^3*g*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 + 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 + 1/2*(c^2*x^2 - 1)*b*e^3*f*arcsin(c*x)/c^4 + 3/2*(c^2*x^2 - 1)*b*d*e^2*g*ar csin(c*x)/c^4 + 1/5*b*e^3*g*x*arcsin(c*x)/c^4 + sqrt(-c^2*x^2 + 1)*b*d*e^2 *f/c^3 + sqrt(-c^2*x^2 + 1)*b*d^2*e*g/c^3 + 1/25*(c^2*x^2 - 1)^2*sqrt(-c^2 *x^2 + 1)*b*e^3*g/c^5 + 5/32*b*e^3*f*arcsin(c*x)/c^4 + 15/32*b*d*e^2*g*arc sin(c*x)/c^4 - 2/15*(-c^2*x^2 + 1)^(3/2)*b*e^3*g/c^5 + 1/5*sqrt(-c^2*x^2 + 1)*b*e^3*g/c^5
Timed out. \[ \int (d+e x)^3 (f+g x) (a+b \arcsin (c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \] Input:
int((f + g*x)*(a + b*asin(c*x))*(d + e*x)^3,x)
Output:
int((f + g*x)*(a + b*asin(c*x))*(d + e*x)^3, x)
Time = 0.25 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.74 \[ \int (d+e x)^3 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {2400 \mathit {asin} \left (c x \right ) b \,c^{5} d^{3} f x +1200 \mathit {asin} \left (c x \right ) b \,c^{5} d^{3} g \,x^{2}+600 \mathit {asin} \left (c x \right ) b \,c^{5} e^{3} f \,x^{4}+480 \mathit {asin} \left (c x \right ) b \,c^{5} e^{3} g \,x^{5}-1800 \mathit {asin} \left (c x \right ) b \,c^{3} d^{2} e f -675 \mathit {asin} \left (c x \right ) b c d \,e^{2} g +600 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d^{3} g x +150 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} e^{3} f \,x^{3}+96 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} e^{3} g \,x^{4}+1600 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d^{2} e g +1600 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d \,e^{2} f +225 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} e^{3} f x +128 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} e^{3} g \,x^{2}+3600 a \,c^{5} d^{2} e f \,x^{2}+2400 a \,c^{5} d^{2} e g \,x^{3}+2400 a \,c^{5} d \,e^{2} f \,x^{3}+1800 a \,c^{5} d \,e^{2} g \,x^{4}+256 \sqrt {-c^{2} x^{2}+1}\, b \,e^{3} g +2400 \mathit {asin} \left (c x \right ) b \,c^{5} d^{2} e g \,x^{3}+2400 \mathit {asin} \left (c x \right ) b \,c^{5} d \,e^{2} f \,x^{3}+1800 \mathit {asin} \left (c x \right ) b \,c^{5} d \,e^{2} g \,x^{4}+1800 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d^{2} e f x +800 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d^{2} e g \,x^{2}+800 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d \,e^{2} f \,x^{2}+450 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d \,e^{2} g \,x^{3}+675 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d \,e^{2} g x +3600 \mathit {asin} \left (c x \right ) b \,c^{5} d^{2} e f \,x^{2}-600 \mathit {asin} \left (c x \right ) b \,c^{3} d^{3} g -225 \mathit {asin} \left (c x \right ) b c \,e^{3} f +2400 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d^{3} f +2400 a \,c^{5} d^{3} f x +1200 a \,c^{5} d^{3} g \,x^{2}+600 a \,c^{5} e^{3} f \,x^{4}+480 a \,c^{5} e^{3} g \,x^{5}}{2400 c^{5}} \] Input:
int((e*x+d)^3*(g*x+f)*(a+b*asin(c*x)),x)
Output:
(2400*asin(c*x)*b*c**5*d**3*f*x + 1200*asin(c*x)*b*c**5*d**3*g*x**2 + 3600 *asin(c*x)*b*c**5*d**2*e*f*x**2 + 2400*asin(c*x)*b*c**5*d**2*e*g*x**3 + 24 00*asin(c*x)*b*c**5*d*e**2*f*x**3 + 1800*asin(c*x)*b*c**5*d*e**2*g*x**4 + 600*asin(c*x)*b*c**5*e**3*f*x**4 + 480*asin(c*x)*b*c**5*e**3*g*x**5 - 600* asin(c*x)*b*c**3*d**3*g - 1800*asin(c*x)*b*c**3*d**2*e*f - 675*asin(c*x)*b *c*d*e**2*g - 225*asin(c*x)*b*c*e**3*f + 2400*sqrt( - c**2*x**2 + 1)*b*c** 4*d**3*f + 600*sqrt( - c**2*x**2 + 1)*b*c**4*d**3*g*x + 1800*sqrt( - c**2* x**2 + 1)*b*c**4*d**2*e*f*x + 800*sqrt( - c**2*x**2 + 1)*b*c**4*d**2*e*g*x **2 + 800*sqrt( - c**2*x**2 + 1)*b*c**4*d*e**2*f*x**2 + 450*sqrt( - c**2*x **2 + 1)*b*c**4*d*e**2*g*x**3 + 150*sqrt( - c**2*x**2 + 1)*b*c**4*e**3*f*x **3 + 96*sqrt( - c**2*x**2 + 1)*b*c**4*e**3*g*x**4 + 1600*sqrt( - c**2*x** 2 + 1)*b*c**2*d**2*e*g + 1600*sqrt( - c**2*x**2 + 1)*b*c**2*d*e**2*f + 675 *sqrt( - c**2*x**2 + 1)*b*c**2*d*e**2*g*x + 225*sqrt( - c**2*x**2 + 1)*b*c **2*e**3*f*x + 128*sqrt( - c**2*x**2 + 1)*b*c**2*e**3*g*x**2 + 256*sqrt( - c**2*x**2 + 1)*b*e**3*g + 2400*a*c**5*d**3*f*x + 1200*a*c**5*d**3*g*x**2 + 3600*a*c**5*d**2*e*f*x**2 + 2400*a*c**5*d**2*e*g*x**3 + 2400*a*c**5*d*e* *2*f*x**3 + 1800*a*c**5*d*e**2*g*x**4 + 600*a*c**5*e**3*f*x**4 + 480*a*c** 5*e**3*g*x**5)/(2400*c**5)