Integrand size = 21, antiderivative size = 457 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^6} \, dx=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt {1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c^3 \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) \sqrt {1-c^2 x^2}}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac {b c^3 \left (4 e^4 g-c^2 d e^2 (11 e f-18 d g)-c^4 d^3 (10 e f+d g)\right ) \sqrt {1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}+\frac {b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{40 e^2 \left (c^2 d^2-e^2\right )^{9/2}} \]
-1/5*(-d*g+e*f)*(a+b*arcsin(c*x))/e^2/(e*x+d)^5-1/4*g*(a+b*arcsin(c*x))/e^ 2/(e*x+d)^4+1/40*b*c^5*(c^2*d^2*e^2*(-19*d*g+24*e*f)+3*e^4*(-6*d*g+e*f)+2* c^4*d^4*(d*g+4*e*f))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^( 1/2))/e^2/(c^2*d^2-e^2)^(9/2)+1/20*b*c*(-d*g+e*f)*(-c^2*x^2+1)^(1/2)/e/(c^ 2*d^2-e^2)/(e*x+d)^4-1/60*b*c*(5*e^2*g-c^2*d*(-2*d*g+7*e*f))*(-c^2*x^2+1)^ (1/2)/e/(c^2*d^2-e^2)^2/(e*x+d)^3+1/120*b*c^3*(e^2*(-34*d*g+9*e*f)+c^2*d^2 *(-d*g+26*e*f))*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)^3/(e*x+d)^2-1/24*b*c^3* (4*e^4*g-c^2*d*e^2*(-18*d*g+11*e*f)-c^4*d^3*(d*g+10*e*f))*(-c^2*x^2+1)^(1/ 2)/e/(c^2*d^2-e^2)^4/(e*x+d)
Time = 1.31 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^6} \, dx=\frac {\frac {3 a (-8 e f+8 d g)}{(d+e x)^5}-\frac {30 a g}{(d+e x)^4}+\frac {b c e \sqrt {1-c^2 x^2} \left (-6 \left (-c^2 d^2+e^2\right )^3 (e f-d g)-2 \left (-c^2 d^2+e^2\right )^2 \left (5 e^2 g+c^2 d (-7 e f+2 d g)\right ) (d+e x)-c^2 \left (c^2 d^2-e^2\right ) \left (c^2 d^2 (-26 e f+d g)+e^2 (-9 e f+34 d g)\right ) (d+e x)^2+5 c^2 \left (-4 e^4 g+c^2 d e^2 (11 e f-18 d g)+c^4 d^3 (10 e f+d g)\right ) (d+e x)^3\right )}{\left (-c^2 d^2+e^2\right )^4 (d+e x)^4}-\frac {6 b (4 e f+d g+5 e g x) \arcsin (c x)}{(d+e x)^5}+\frac {3 b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right ) \log (d+e x)}{(-c d+e)^4 (c d+e)^4 \sqrt {-c^2 d^2+e^2}}-\frac {3 b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{(-c d+e)^4 (c d+e)^4 \sqrt {-c^2 d^2+e^2}}}{120 e^2} \]
((3*a*(-8*e*f + 8*d*g))/(d + e*x)^5 - (30*a*g)/(d + e*x)^4 + (b*c*e*Sqrt[1 - c^2*x^2]*(-6*(-(c^2*d^2) + e^2)^3*(e*f - d*g) - 2*(-(c^2*d^2) + e^2)^2* (5*e^2*g + c^2*d*(-7*e*f + 2*d*g))*(d + e*x) - c^2*(c^2*d^2 - e^2)*(c^2*d^ 2*(-26*e*f + d*g) + e^2*(-9*e*f + 34*d*g))*(d + e*x)^2 + 5*c^2*(-4*e^4*g + c^2*d*e^2*(11*e*f - 18*d*g) + c^4*d^3*(10*e*f + d*g))*(d + e*x)^3))/((-(c ^2*d^2) + e^2)^4*(d + e*x)^4) - (6*b*(4*e*f + d*g + 5*e*g*x)*ArcSin[c*x])/ (d + e*x)^5 + (3*b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g) + 3*e^4*(e*f - 6*d*g ) + 2*c^4*d^4*(4*e*f + d*g))*Log[d + e*x])/((-(c*d) + e)^4*(c*d + e)^4*Sqr t[-(c^2*d^2) + e^2]) - (3*b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g) + 3*e^4*(e* f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c*d) + e)^4*(c*d + e)^4*Sqrt[-(c^2*d^2) + e^2 ]))/(120*e^2)
Time = 0.91 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5252, 27, 688, 27, 688, 25, 27, 688, 25, 679, 488, 217}
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.
| \(\displaystyle \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 5252 |
| \(\displaystyle -b c \int -\frac {4 e f+d g+5 e g x}{20 e^2 (d+e x)^5 \sqrt {1-c^2 x^2}}dx-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {4 e f+d g+5 e g x}{(d+e x)^5 \sqrt {1-c^2 x^2}}dx}{20 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {b c \left (\frac {\int -\frac {4 \left (-d (4 e f+d g) c^2+3 e (e f-d g) x c^2+5 e^2 g\right )}{(d+e x)^4 \sqrt {1-c^2 x^2}}dx}{4 \left (c^2 d^2-e^2\right )}+\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^4}\right )}{20 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {\int \frac {-d (4 e f+d g) c^2+3 e (e f-d g) x c^2+5 e^2 g}{(d+e x)^4 \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}\right )}{20 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {\frac {\int -\frac {c^2 \left (3 \left (c^2 (4 e f+d g) d^2+e^2 (3 e f-8 d g)\right )+2 e \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) x\right )}{(d+e x)^3 \sqrt {1-c^2 x^2}}dx}{3 \left (c^2 d^2-e^2\right )}+\frac {e \sqrt {1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{3 \left (c^2 d^2-e^2\right ) (d+e x)^3}}{c^2 d^2-e^2}\right )}{20 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{3 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\int \frac {c^2 \left (3 \left (c^2 (4 e f+d g) d^2+e^2 (3 e f-8 d g)\right )+2 e \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) x\right )}{(d+e x)^3 \sqrt {1-c^2 x^2}}dx}{3 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{20 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{3 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {c^2 \int \frac {3 \left (c^2 (4 e f+d g) d^2+e^2 (3 e f-8 d g)\right )+2 e \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) x}{(d+e x)^3 \sqrt {1-c^2 x^2}}dx}{3 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{20 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{3 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {c^2 \left (\frac {\int -\frac {e \left (c^2 (26 e f-d g) d^2+e^2 (9 e f-34 d g)\right ) x c^2+2 \left (-3 d^3 (4 e f+d g) c^4-d e^2 (23 e f-28 d g) c^2+10 e^4 g\right )}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 \left (c^2 d^2-e^2\right )}+\frac {e \sqrt {1-c^2 x^2} \left (c^2 d^2 (26 e f-d g)+e^2 (9 e f-34 d g)\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}\right )}{3 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{20 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{3 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {c^2 \left (\frac {e \sqrt {1-c^2 x^2} \left (c^2 d^2 (26 e f-d g)+e^2 (9 e f-34 d g)\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\int \frac {e \left (c^2 (26 e f-d g) d^2+e^2 (9 e f-34 d g)\right ) x c^2+2 \left (-3 d^3 (4 e f+d g) c^4-d e^2 (23 e f-28 d g) c^2+10 e^4 g\right )}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 \left (c^2 d^2-e^2\right )}\right )}{3 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{20 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{3 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {c^2 \left (\frac {e \sqrt {1-c^2 x^2} \left (c^2 d^2 (26 e f-d g)+e^2 (9 e f-34 d g)\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\frac {5 e \sqrt {1-c^2 x^2} \left (c^4 \left (-d^3\right ) (d g+10 e f)-c^2 d e^2 (11 e f-18 d g)+4 e^4 g\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {3 c^2 \left (2 c^4 d^4 (d g+4 e f)+c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}}{2 \left (c^2 d^2-e^2\right )}\right )}{3 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{20 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{3 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {c^2 \left (\frac {e \sqrt {1-c^2 x^2} \left (c^2 d^2 (26 e f-d g)+e^2 (9 e f-34 d g)\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\frac {3 c^2 \left (2 c^4 d^4 (d g+4 e f)+c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)\right ) \int \frac {1}{-c^2 d^2+e^2-\frac {\left (d x c^2+e\right )^2}{1-c^2 x^2}}d\frac {d x c^2+e}{\sqrt {1-c^2 x^2}}}{c^2 d^2-e^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (c^4 \left (-d^3\right ) (d g+10 e f)-c^2 d e^2 (11 e f-18 d g)+4 e^4 g\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}}{2 \left (c^2 d^2-e^2\right )}\right )}{3 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{20 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {(e f-d g) (a+b \arcsin (c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}+\frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{3 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {c^2 \left (\frac {e \sqrt {1-c^2 x^2} \left (c^2 d^2 (26 e f-d g)+e^2 (9 e f-34 d g)\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\frac {5 e \sqrt {1-c^2 x^2} \left (c^4 \left (-d^3\right ) (d g+10 e f)-c^2 d e^2 (11 e f-18 d g)+4 e^4 g\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {3 c^2 \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (2 c^4 d^4 (d g+4 e f)+c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)\right )}{\left (c^2 d^2-e^2\right )^{3/2}}}{2 \left (c^2 d^2-e^2\right )}\right )}{3 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{20 e^2}\) |
-1/5*((e*f - d*g)*(a + b*ArcSin[c*x]))/(e^2*(d + e*x)^5) - (g*(a + b*ArcSi n[c*x]))/(4*e^2*(d + e*x)^4) + (b*c*((e*(e*f - d*g)*Sqrt[1 - c^2*x^2])/((c ^2*d^2 - e^2)*(d + e*x)^4) - ((e*(5*e^2*g - c^2*d*(7*e*f - 2*d*g))*Sqrt[1 - c^2*x^2])/(3*(c^2*d^2 - e^2)*(d + e*x)^3) - (c^2*((e*(e^2*(9*e*f - 34*d* g) + c^2*d^2*(26*e*f - d*g))*Sqrt[1 - c^2*x^2])/(2*(c^2*d^2 - e^2)*(d + e* x)^2) - ((5*e*(4*e^4*g - c^2*d*e^2*(11*e*f - 18*d*g) - c^4*d^3*(10*e*f + d *g))*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - (3*c^2*(c^2*d^2*e^2* (24*e*f - 19*d*g) + 3*e^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*ArcTan[ (e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3 /2))/(2*(c^2*d^2 - e^2))))/(3*(c^2*d^2 - e^2)))/(c^2*d^2 - e^2)))/(20*e^2)
3.1.96.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_ Symbol] :> With[{u = IntHide[Px*(d + e*x)^m, 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, d, e, m}, x] && PolynomialQ[Px, x]
Leaf count of result is larger than twice the leaf count of optimal. \(2405\) vs. \(2(429)=858\).
Time = 3.40 (sec) , antiderivative size = 2406, normalized size of antiderivative = 5.26
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2406\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2420\) |
| default | \(\text {Expression too large to display}\) | \(2420\) |
a*(-1/4*g/e^2/(e*x+d)^4-1/5*(-d*g+e*f)/e^2/(e*x+d)^5)-1/4*b*c^4*arcsin(c*x )*g/e^2/(c*e*x+c*d)^4+1/5*b*c^5*arcsin(c*x)/e^2/(c*e*x+c*d)^5*d*g-1/5*b*c^ 5*arcsin(c*x)/e/(c*e*x+c*d)^5*f+1/12*b*c^4/e^4*g/(c^2*d^2-e^2)/(c*x+d*c/e) ^3*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)+17/60*b*c^ 5/e^3*g*d/(c^2*d^2-e^2)^2/(c*x+d*c/e)^2*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e )-(c^2*d^2-e^2)/e^2)^(1/2)+13/12*b*c^6/e^2*g*d^2/(c^2*d^2-e^2)^3/(c*x+d*c/ e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)-11/8*b*c^7 /e^3*g*d^3/(c^2*d^2-e^2)^3/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2) /e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d* c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))+9/20*b*c^5/e^3*g*d/ (c^2*d^2-e^2)^2/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/ e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d* c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))-1/6*b*c^4/e^2*g/(c^2*d^2-e^2)^ 2/(c*x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2) -1/20*b*c^5/e^5/(c^2*d^2-e^2)/(c*x+d*c/e)^4*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d *c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*d*g+1/20*b*c^5/e^4/(c^2*d^2-e^2)/(c*x+d*c/e )^4*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-7/60*b* c^6/e^4*d^2/(c^2*d^2-e^2)^2/(c*x+d*c/e)^3*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c /e)-(c^2*d^2-e^2)/e^2)^(1/2)*g+7/60*b*c^6/e^3*d/(c^2*d^2-e^2)^2/(c*x+d*c/e )^3*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-7/24...
Leaf count of result is larger than twice the leaf count of optimal. 1939 vs. \(2 (426) = 852\).
Time = 175.63 (sec) , antiderivative size = 3904, normalized size of antiderivative = 8.54 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^6} \, dx=\text {Too large to display} \]
[-1/240*(60*(a*c^10*d^10*e - 5*a*c^8*d^8*e^3 + 10*a*c^6*d^6*e^5 - 10*a*c^4 *d^4*e^7 + 5*a*c^2*d^2*e^9 - a*e^11)*g*x - 3*(((8*b*c^9*d^4*e^6 + 24*b*c^7 *d^2*e^8 + 3*b*c^5*e^10)*f + (2*b*c^9*d^5*e^5 - 19*b*c^7*d^3*e^7 - 18*b*c^ 5*d*e^9)*g)*x^5 + 5*((8*b*c^9*d^5*e^5 + 24*b*c^7*d^3*e^7 + 3*b*c^5*d*e^9)* f + (2*b*c^9*d^6*e^4 - 19*b*c^7*d^4*e^6 - 18*b*c^5*d^2*e^8)*g)*x^4 + 10*(( 8*b*c^9*d^6*e^4 + 24*b*c^7*d^4*e^6 + 3*b*c^5*d^2*e^8)*f + (2*b*c^9*d^7*e^3 - 19*b*c^7*d^5*e^5 - 18*b*c^5*d^3*e^7)*g)*x^3 + 10*((8*b*c^9*d^7*e^3 + 24 *b*c^7*d^5*e^5 + 3*b*c^5*d^3*e^7)*f + (2*b*c^9*d^8*e^2 - 19*b*c^7*d^6*e^4 - 18*b*c^5*d^4*e^6)*g)*x^2 + (8*b*c^9*d^9*e + 24*b*c^7*d^7*e^3 + 3*b*c^5*d ^5*e^5)*f + (2*b*c^9*d^10 - 19*b*c^7*d^8*e^2 - 18*b*c^5*d^6*e^4)*g + 5*((8 *b*c^9*d^8*e^2 + 24*b*c^7*d^6*e^4 + 3*b*c^5*d^4*e^6)*f + (2*b*c^9*d^9*e - 19*b*c^7*d^7*e^3 - 18*b*c^5*d^5*e^5)*g)*x)*sqrt(-c^2*d^2 + e^2)*log((2*c^2 *d*e*x - c^2*d^2 + (2*c^4*d^2 - c^2*e^2)*x^2 + 2*sqrt(-c^2*d^2 + e^2)*(c^2 *d*x + e)*sqrt(-c^2*x^2 + 1) + 2*e^2)/(e^2*x^2 + 2*d*e*x + d^2)) + 48*(a*c ^10*d^10*e - 5*a*c^8*d^8*e^3 + 10*a*c^6*d^6*e^5 - 10*a*c^4*d^4*e^7 + 5*a*c ^2*d^2*e^9 - a*e^11)*f + 12*(a*c^10*d^11 - 5*a*c^8*d^9*e^2 + 10*a*c^6*d^7* e^4 - 10*a*c^4*d^5*e^6 + 5*a*c^2*d^3*e^8 - a*d*e^10)*g + 12*(5*(b*c^10*d^1 0*e - 5*b*c^8*d^8*e^3 + 10*b*c^6*d^6*e^5 - 10*b*c^4*d^4*e^7 + 5*b*c^2*d^2* e^9 - b*e^11)*g*x + 4*(b*c^10*d^10*e - 5*b*c^8*d^8*e^3 + 10*b*c^6*d^6*e^5 - 10*b*c^4*d^4*e^7 + 5*b*c^2*d^2*e^9 - b*e^11)*f + (b*c^10*d^11 - 5*b*c...
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^6} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (d + e x\right )^{6}}\, dx \]
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^6} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{6}} \,d x } \]
-1/20*(5*e*x + d)*a*g/(e^7*x^5 + 5*d*e^6*x^4 + 10*d^2*e^5*x^3 + 10*d^3*e^4 *x^2 + 5*d^4*e^3*x + d^5*e^2) - 1/5*a*f/(e^6*x^5 + 5*d*e^5*x^4 + 10*d^2*e^ 4*x^3 + 10*d^3*e^3*x^2 + 5*d^4*e^2*x + d^5*e) - 1/20*((5*b*e*g*x + 4*b*e*f + b*d*g)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 20*(e^7*x^5 + 5*d*e ^6*x^4 + 10*d^2*e^5*x^3 + 10*d^3*e^4*x^2 + 5*d^4*e^3*x + d^5*e^2)*integrat e(1/20*(5*b*c*e*g*x + 4*b*c*e*f + b*c*d*g)*e^(1/2*log(c*x + 1) + 1/2*log(- c*x + 1))/(c^4*e^7*x^9 + 5*c^4*d*e^6*x^8 - 5*c^2*d^4*e^3*x^3 - c^2*d^5*e^2 *x^2 + (10*c^4*d^2*e^5 - c^2*e^7)*x^7 + 5*(2*c^4*d^3*e^4 - c^2*d*e^6)*x^6 + 5*(c^4*d^4*e^3 - 2*c^2*d^2*e^5)*x^5 + (c^4*d^5*e^2 - 10*c^2*d^3*e^4)*x^4 + (c^2*e^7*x^7 + 5*c^2*d*e^6*x^6 - 5*d^4*e^3*x - d^5*e^2 + (10*c^2*d^2*e^ 5 - e^7)*x^5 + 5*(2*c^2*d^3*e^4 - d*e^6)*x^4 + 5*(c^2*d^4*e^3 - 2*d^2*e^5) *x^3 + (c^2*d^5*e^2 - 10*d^3*e^4)*x^2)*e^(log(c*x + 1) + log(-c*x + 1))), x))/(e^7*x^5 + 5*d*e^6*x^4 + 10*d^2*e^5*x^3 + 10*d^3*e^4*x^2 + 5*d^4*e^3*x + d^5*e^2)
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^6} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{6}} \,d x } \]
Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^6} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d+e\,x\right )}^6} \,d x \]