\(\int (f+g x)^2 (d-c^2 d x^2)^{5/2} (a+b \arcsin (c x))^2 \, dx\) [67]

Optimal result

Integrand size = 33, antiderivative size = 1533 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {64 b^2 d^2 f g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {245 b^2 d^2 f^2 x \sqrt {d-c^2 d x^2}}{1152}-\frac {359 b^2 d^2 g^2 x \sqrt {d-c^2 d x^2}}{36864 c^2}-\frac {1079 b^2 d^2 g^2 x^3 \sqrt {d-c^2 d x^2}}{55296}+\frac {209 b^2 c^2 d^2 g^2 x^5 \sqrt {d-c^2 d x^2}}{13824}-\frac {1}{256} b^2 c^4 d^2 g^2 x^7 \sqrt {d-c^2 d x^2}+\frac {32 b^2 d^2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2}-\frac {65 b^2 d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {24 b^2 d^2 f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}-\frac {1}{108} b^2 d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {4 b^2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {115 b^2 d^2 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {359 b^2 d^2 g^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{36864 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b d^2 f g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 g^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c \sqrt {1-c^2 x^2}}-\frac {4 b c d^2 f g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{384 \sqrt {1-c^2 x^2}}+\frac {12 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{144 \sqrt {1-c^2 x^2}}-\frac {4 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{32 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{384 b c^3 \sqrt {1-c^2 x^2}} \]

[Out]

-245/1152*b^2*d^2*f^2*x*(-c^2*d*x^2+d)^(1/2)-1079/55296*b^2*d^2*g^2*x^3*(-c^2*d*x^2+d)^(1/2)+5/16*d^2*f^2*x*(a
+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+5/64*d^2*g^2*x^3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+5/384*d^2*g^2
*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)+64/245*b^2*d^2*f*g*(-c^2*d*x^2+d)^(1/2)/c^2
-359/36864*b^2*d^2*g^2*x*(-c^2*d*x^2+d)^(1/2)/c^2+209/13824*b^2*c^2*d^2*g^2*x^5*(-c^2*d*x^2+d)^(1/2)-59/384*b*
c*d^2*g^2*x^4*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+17/144*b*c^3*d^2*g^2*x^6*(a+b*arcsin(c
*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/32*b*c^5*d^2*g^2*x^8*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c
^2*x^2+1)^(1/2)+1/18*b*d^2*f^2*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c-2/7*d^2*f*g*(-c^2*x
^2+1)^3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2+115/1152*b^2*d^2*f^2*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c/(
-c^2*x^2+1)^(1/2)+359/36864*b^2*d^2*g^2*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+5/48*d^2*f^2*(
a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)+24/1225*b^2*d^2*f*g*(-c^2*x^2+1)^2*(-c^2*d*x^2+
d)^(1/2)/c^2+4/343*b^2*d^2*f*g*(-c^2*x^2+1)^3*(-c^2*d*x^2+d)^(1/2)/c^2+5/48*b*d^2*f^2*(-c^2*x^2+1)^(3/2)*(a+b*
arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c+4/7*b*d^2*f*g*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2
)-4/7*b*c*d^2*f*g*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+12/35*b*c^3*d^2*f*g*x^5*(a+b*a
rcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-4/49*b*c^5*d^2*f*g*x^7*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1
/2)/(-c^2*x^2+1)^(1/2)-1/256*b^2*c^4*d^2*g^2*x^7*(-c^2*d*x^2+d)^(1/2)-65/1728*b^2*d^2*f^2*x*(-c^2*x^2+1)*(-c^2
*d*x^2+d)^(1/2)-1/108*b^2*d^2*f^2*x*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)-5/128*d^2*g^2*x*(a+b*arcsin(c*x))^2*(-
c^2*d*x^2+d)^(1/2)/c^2+5/24*d^2*f^2*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+5/48*d^2*g^2*x^3*(
-c^2*x^2+1)*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+1/6*d^2*f^2*x*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2*(-c^2*d*
x^2+d)^(1/2)+1/8*d^2*g^2*x^3*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)-5/16*b*c*d^2*f^2*x^2*(a+b
*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5/128*b*d^2*g^2*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/
2)/c/(-c^2*x^2+1)^(1/2)+32/735*b^2*d^2*f*g*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 1533, normalized size of antiderivative = 1.00, number of steps used = 50, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {4861, 4847, 4743, 4741, 4737, 4723, 327, 222, 4767, 201, 200, 4739, 12, 1813, 1864, 4787, 4783, 4795, 14, 4777, 470, 272, 45, 1281} \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {b c^5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^8}{32 \sqrt {1-c^2 x^2}}-\frac {4 b c^5 d^2 f g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^7}{49 \sqrt {1-c^2 x^2}}-\frac {1}{256} b^2 c^4 d^2 g^2 \sqrt {d-c^2 d x^2} x^7+\frac {17 b c^3 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^6}{144 \sqrt {1-c^2 x^2}}+\frac {12 b c^3 d^2 f g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^5}{35 \sqrt {1-c^2 x^2}}+\frac {209 b^2 c^2 d^2 g^2 \sqrt {d-c^2 d x^2} x^5}{13824}-\frac {59 b c d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^4}{384 \sqrt {1-c^2 x^2}}+\frac {5}{64} d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x^3+\frac {1}{8} d^2 g^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x^3+\frac {5}{48} d^2 g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x^3-\frac {4 b c d^2 f g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^3}{7 \sqrt {1-c^2 x^2}}-\frac {1079 b^2 d^2 g^2 \sqrt {d-c^2 d x^2} x^3}{55296}-\frac {5 b c d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^2}{16 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x^2}{128 c \sqrt {1-c^2 x^2}}+\frac {5}{16} d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x-\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x}{128 c^2}+\frac {1}{6} d^2 f^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x+\frac {5}{24} d^2 f^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x+\frac {4 b d^2 f g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) x}{7 c \sqrt {1-c^2 x^2}}-\frac {245 b^2 d^2 f^2 \sqrt {d-c^2 d x^2} x}{1152}-\frac {359 b^2 d^2 g^2 \sqrt {d-c^2 d x^2} x}{36864 c^2}-\frac {1}{108} b^2 d^2 f^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} x-\frac {65 b^2 d^2 f^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x}{1728}+\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{384 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {115 b^2 d^2 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {359 b^2 d^2 g^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{36864 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5 b d^2 f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {4 b^2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {24 b^2 d^2 f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {64 b^2 d^2 f g \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {32 b^2 d^2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2} \]

[In]

Int[(f + g*x)^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(64*b^2*d^2*f*g*Sqrt[d - c^2*d*x^2])/(245*c^2) - (245*b^2*d^2*f^2*x*Sqrt[d - c^2*d*x^2])/1152 - (359*b^2*d^2*g
^2*x*Sqrt[d - c^2*d*x^2])/(36864*c^2) - (1079*b^2*d^2*g^2*x^3*Sqrt[d - c^2*d*x^2])/55296 + (209*b^2*c^2*d^2*g^
2*x^5*Sqrt[d - c^2*d*x^2])/13824 - (b^2*c^4*d^2*g^2*x^7*Sqrt[d - c^2*d*x^2])/256 + (32*b^2*d^2*f*g*(1 - c^2*x^
2)*Sqrt[d - c^2*d*x^2])/(735*c^2) - (65*b^2*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/1728 + (24*b^2*d^2*f*
g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1225*c^2) - (b^2*d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/108 +
(4*b^2*d^2*f*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^2) + (115*b^2*d^2*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*
x])/(1152*c*Sqrt[1 - c^2*x^2]) + (359*b^2*d^2*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(36864*c^3*Sqrt[1 - c^2*x^2
]) + (4*b*d^2*f*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c*Sqrt[1 - c^2*x^2]) - (5*b*c*d^2*f^2*x^2*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) + (5*b*d^2*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcS
in[c*x]))/(128*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d^2*f*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 -
c^2*x^2]) - (59*b*c*d^2*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(384*Sqrt[1 - c^2*x^2]) + (12*b*c^3*d
^2*f*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(35*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*g^2*x^6*Sqrt[d - c^
2*d*x^2]*(a + b*ArcSin[c*x]))/(144*Sqrt[1 - c^2*x^2]) - (4*b*c^5*d^2*f*g*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin
[c*x]))/(49*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*g^2*x^8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(32*Sqrt[1 - c^2*
x^2]) + (5*b*d^2*f^2*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (b*d^2*f^2*(1 - c^2
*x^2)^(5/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*c) + (5*d^2*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c
*x])^2)/16 - (5*d^2*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(128*c^2) + (5*d^2*g^2*x^3*Sqrt[d - c^2*d
*x^2]*(a + b*ArcSin[c*x])^2)/64 + (5*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/24 + (
5*d^2*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/48 + (d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/6 + (d^2*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2
)/8 - (2*d^2*f*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(7*c^2) + (5*d^2*f^2*Sqrt[d - c^2*
d*x^2]*(a + b*ArcSin[c*x])^3)/(48*b*c*Sqrt[1 - c^2*x^2]) + (5*d^2*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^
3)/(384*b*c^3*Sqrt[1 - c^2*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 222

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]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 1281

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*((
a + b*ArcSin[c*x])^n/2), x] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[x*(a + b*ArcSin[c*x])^(
n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4743

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((
a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (Dist[2*d*(p/(2*p + 1)), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n,
x], x] - Dist[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcS
in[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]

Rule 4767

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*
d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4777

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4783

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f
*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/(f*(m + 2))), x] + (Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/S
qrt[1 - c^2*x^2]], Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x] - Dist[b*c*(n/(f*(m + 2)))*Si
mp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b,
c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 4787

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(f*(m + 2*p + 1))), x] + (Dist[2*d*(p/(m + 2*p + 1)), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(
1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b
, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 4847

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g},
x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 ||
(n == 1 && p > -1) || (m == 2 && p < -2))

Rule 4861

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; F
reeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] &&  !GtQ[d, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (f+g x)^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (f^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2+2 f g x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2+g^2 x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {\left (d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (2 d^2 f g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {\left (5 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx}{6 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x)) \, dx}{3 \sqrt {1-c^2 x^2}}+\frac {\left (4 b d^2 f g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x)) \, dx}{7 c \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x)) \, dx}{4 \sqrt {1-c^2 x^2}} \\ & = \frac {4 b d^2 f g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {4 b c d^2 f g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}-\frac {b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}+\frac {12 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{12 \sqrt {1-c^2 x^2}}-\frac {4 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{32 \sqrt {1-c^2 x^2}}+\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {\left (5 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{18 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x)) \, dx}{12 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 d^2 f g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt {1-c^2 x^2}} \, dx}{7 \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x)) \, dx}{24 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}} \\ & = -\frac {1}{108} b^2 d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {4 b d^2 f g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {4 b c d^2 f g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}-\frac {11 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{96 \sqrt {1-c^2 x^2}}+\frac {12 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{144 \sqrt {1-c^2 x^2}}-\frac {4 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{32 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {\left (5 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{108 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{48 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int x (a+b \arcsin (c x)) \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 d^2 f g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx}{245 \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 (a+b \arcsin (c x)) \, dx}{32 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{96 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}} \, dx}{24 \sqrt {1-c^2 x^2}} \\ & = -\frac {1}{256} b^2 c^4 d^2 g^2 x^7 \sqrt {d-c^2 d x^2}-\frac {65 b^2 d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {4 b d^2 f g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}-\frac {4 b c d^2 f g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{384 \sqrt {1-c^2 x^2}}+\frac {12 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{144 \sqrt {1-c^2 x^2}}-\frac {4 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{32 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{144 \sqrt {1-c^2 x^2}}-\frac {\left (5 b^2 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 f g \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (-48 c^2+43 c^4 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{768 \sqrt {1-c^2 x^2}}+\frac {\left (5 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (5 b d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int x (a+b \arcsin (c x)) \, dx}{64 c \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{288 \sqrt {1-c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{128 \sqrt {1-c^2 x^2}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 742, normalized size of antiderivative = 0.48 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (12348000 a^3 \left (8 c^2 f^2+g^2\right )-3360 a b^2 c^2 x \left (1960 c^2 f^2 x \left (99-39 c^2 x^2+8 c^4 x^4\right )+4608 f g \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+245 g^2 x \left (-45+177 c^2 x^2-136 c^4 x^4+36 c^6 x^6\right )\right )+352800 a^2 b c \sqrt {1-c^2 x^2} \left (768 f g \left (-1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33-26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right )-b^3 c \sqrt {1-c^2 x^2} \left (274400 c^2 f^2 x \left (897-194 c^2 x^2+32 c^4 x^4\right )+147456 f g \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )+8575 g^2 x \left (1077+2158 c^2 x^2-1672 c^4 x^4+432 c^6 x^6\right )\right )+105 b \left (352800 a^2 \left (8 c^2 f^2+g^2\right )+b^2 \left (87955 g^2+1120 c^2 \left (2093 f^2+4608 f g x+315 g^2 x^2\right )-3360 c^4 x^2 \left (1848 f^2+1536 f g x+413 g^2 x^2\right )-640 c^8 x^6 \left (784 f^2+1152 f g x+441 g^2 x^2\right )+1792 c^6 x^4 \left (1365 f^2+1728 f g x+595 g^2 x^2\right )\right )+6720 a b c \sqrt {1-c^2 x^2} \left (768 f g \left (-1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33-26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right )\right ) \arcsin (c x)+352800 b^2 \left (105 a \left (8 c^2 f^2+g^2\right )+b c \sqrt {1-c^2 x^2} \left (768 f g \left (-1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33-26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right )\right ) \arcsin (c x)^2+12348000 b^3 \left (8 c^2 f^2+g^2\right ) \arcsin (c x)^3\right )}{948326400 b c^3 \sqrt {1-c^2 x^2}} \]

[In]

Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(12348000*a^3*(8*c^2*f^2 + g^2) - 3360*a*b^2*c^2*x*(1960*c^2*f^2*x*(99 - 39*c^2*x^2 +
8*c^4*x^4) + 4608*f*g*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 245*g^2*x*(-45 + 177*c^2*x^2 - 136*c^4*x^
4 + 36*c^6*x^6)) + 352800*a^2*b*c*Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2
+ 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)) - b^3*c*Sqrt[1 - c^2*x^2]*(274400*c^2*f
^2*x*(897 - 194*c^2*x^2 + 32*c^4*x^4) + 147456*f*g*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6) + 8575*g^2
*x*(1077 + 2158*c^2*x^2 - 1672*c^4*x^4 + 432*c^6*x^6)) + 105*b*(352800*a^2*(8*c^2*f^2 + g^2) + b^2*(87955*g^2
+ 1120*c^2*(2093*f^2 + 4608*f*g*x + 315*g^2*x^2) - 3360*c^4*x^2*(1848*f^2 + 1536*f*g*x + 413*g^2*x^2) - 640*c^
8*x^6*(784*f^2 + 1152*f*g*x + 441*g^2*x^2) + 1792*c^6*x^4*(1365*f^2 + 1728*f*g*x + 595*g^2*x^2)) + 6720*a*b*c*
Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*
c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)))*ArcSin[c*x] + 352800*b^2*(105*a*(8*c^2*f^2 + g^2) + b*c*Sqrt[1 - c^2*x^2
]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^
4*x^4 + 48*c^6*x^6)))*ArcSin[c*x]^2 + 12348000*b^3*(8*c^2*f^2 + g^2)*ArcSin[c*x]^3))/(948326400*b*c^3*Sqrt[1 -
c^2*x^2])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.06 (sec) , antiderivative size = 4170, normalized size of antiderivative = 2.72

method	result	size
default	\(\text {Expression too large to display}\)	\(4170\)
parts	\(\text {Expression too large to display}\)	\(4170\)

[In]

int((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

a^2*(f^2*(1/6*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d
/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))))+g^2*(-1/8*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/8/c^2*
(1/6*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^
(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))))))-2/7*f*g*(-c^2*d*x^2+d)^(7/2)/c^2/d)+b^2*(-5/384*(-d*(c^
2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^3*(8*c^2*f^2+g^2)*d^2+1/21952*(-d*(c^2*x^2-1))^
(1/2)*(64*c^8*x^8-144*c^6*x^6-64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*
c^2*x^2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*f*g*(14*I*arcsin(c*x)+49*arcsin(c*x)^2-2
)*d^2/c^2/(c^2*x^2-1)-3/274400*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*f*g*(630*I*arcsin(c
*x)+1225*arcsin(c*x)^2-106)*sin(6*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+1/6912*(-d*(c^2*x^2-1))^(1/2)*(-32*I*(-c^2*
x^2+1)^(1/2)*c^6*x^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*
c^3*x^3+I*(-c^2*x^2+1)^(1/2)-6*c*x)*(6*I*arcsin(c*x)*c^2*f^2+18*arcsin(c*x)^2*c^2*f^2-6*I*arcsin(c*x)*g^2-18*a
rcsin(c*x)^2*g^2-c^2*f^2+g^2)*d^2/c^3/(c^2*x^2-1)+1/65536*(-d*(c^2*x^2-1))^(1/2)*(-128*I*(-c^2*x^2+1)^(1/2)*x^
8*c^8+128*c^9*x^9+256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7-160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+272*c^5*x^5+32
*I*(-c^2*x^2+1)^(1/2)*c^2*x^2-88*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+8*c*x)*g^2*(8*I*arcsin(c*x)+32*arcsin(c*x)^2-1)*
d^2/c^3/(c^2*x^2-1)-5/64*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*f*g*(arcsin(c*x)^2-2+2*I*
arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-5/64*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*g*(arcsin(
c*x)^2-2-2*I*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+1/192*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c
^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*f*g*(-6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)*d^2/c^2/(c^2*x^2-1)-1/
55296*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(696*I*arcsin(c*x)*c^2*f^2+1152*arcsin(c*x)^
2*c^2*f^2-156*I*arcsin(c*x)*g^2-72*arcsin(c*x)^2*g^2-154*c^2*f^2+19*g^2)*cos(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1
)-1/68600*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*g*(385*I*arcsin(c*x)+1225*arcsin(c*x)^
2-92)*cos(6*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^
3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*(30*arcsin(c*x)^2*c^2*f^2+2*arcsin(c*x)^2*g^2-15*c^2*f^2-30*I*arcsin(c*x)*c^
2*f^2-g^2-2*I*arcsin(c*x)*g^2)*d^2/c^3/(c^2*x^2-1)-3/2048*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(
1/2)-I)*(88*I*arcsin(c*x)*c^2*f^2+64*arcsin(c*x)^2*c^2*f^2+4*I*arcsin(c*x)*g^2+8*arcsin(c*x)^2*g^2-38*c^2*f^2-
3*g^2)*cos(3*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)+5/55296*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2
-1)*(120*I*arcsin(c*x)*c^2*f^2+288*arcsin(c*x)^2*c^2*f^2-12*I*arcsin(c*x)*g^2-72*arcsin(c*x)^2*g^2-34*c^2*f^2+
7*g^2)*sin(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-1/1200*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-
1)*f*g*(30*I*arcsin(c*x)+75*arcsin(c*x)^2-14)*cos(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/2400*(-d*(c^2*x^2-1))^(
1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*f*g*(90*I*arcsin(c*x)+75*arcsin(c*x)^2-22)*sin(4*arcsin(c*x))*d^2/c^
2/(c^2*x^2-1)+1/65536*(-d*(c^2*x^2-1))^(1/2)*(128*I*(-c^2*x^2+1)^(1/2)*x^8*c^8+128*c^9*x^9-256*I*(-c^2*x^2+1)^
(1/2)*x^6*c^6-320*c^7*x^7+160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+272*c^5*x^5-32*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-88*c^3*
x^3+I*(-c^2*x^2+1)^(1/2)+8*c*x)*g^2*(-8*I*arcsin(c*x)+32*arcsin(c*x)^2-1)*d^2/c^3/(c^2*x^2-1)+1/2048*(-d*(c^2*
x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(216*I*arcsin(c*x)*c^2*f^2+288*arcsin(c*x)^2*c^2*f^2+20*I*a
rcsin(c*x)*g^2+8*arcsin(c*x)^2*g^2-126*c^2*f^2-7*g^2)*sin(3*arcsin(c*x))*d^2/c^3/(c^2*x^2-1))+2*a*b*(-5/256*(-
d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^2*(8*c^2*f^2+g^2)*d^2+1/3136*(-d*(c^2*x^2-
1))^(1/2)*(64*c^8*x^8-144*c^6*x^6-64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5
-25*c^2*x^2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*f*g*(I+7*arcsin(c*x))*d^2/c^2/(c^2*x
^2-1)+1/2304*(-d*(c^2*x^2-1))^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2)*c^6*x^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c
^4-64*c^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-6*c*x)*(6*arcsin(c*x)*c^2*f^2+I*
c^2*f^2-6*arcsin(c*x)*g^2-I*g^2)*d^2/c^3/(c^2*x^2-1)-3/1024*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)
^(1/2)-I)*(22*I*c^2*f^2+32*arcsin(c*x)*c^2*f^2+I*g^2+4*arcsin(c*x)*g^2)*cos(3*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)
-5/64*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*f*g*(arcsin(c*x)+I)*d^2/c^2/(c^2*x^2-1)-5/64
*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*g*(arcsin(c*x)-I)*d^2/c^2/(c^2*x^2-1)+1/64*(-d*
(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*f*g*(-I+3
*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-1/160*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*f*g*(3*I+5
*arcsin(c*x))*sin(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+1/16384*(-d*(c^2*x^2-1))^(1/2)*(128*I*(-c^2*x^2+1)^(1/2)*
x^8*c^8+128*c^9*x^9-256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7+160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+272*c^5*x^5-
32*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-88*c^3*x^3+I*(-c^2*x^2+1)^(1/2)+8*c*x)*g^2*(-I+8*arcsin(c*x))*d^2/c^3/(c^2*x^2
-1)-1/3920*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*g*(11*I+70*arcsin(c*x))*cos(6*arcsin(
c*x))*d^2/c^2/(c^2*x^2-1)-1/9216*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(58*I*c^2*f^2+192
*arcsin(c*x)*c^2*f^2-13*I*g^2-12*arcsin(c*x)*g^2)*cos(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-3/7840*(-d*(c^2*x^2-1
))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*f*g*(9*I+35*arcsin(c*x))*sin(6*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+
5/9216*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(10*I*c^2*f^2+48*arcsin(c*x)*c^2*f^2-I*g^2-
12*arcsin(c*x)*g^2)*sin(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-1/80*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x
*c+c^2*x^2-1)*f*g*(I+5*arcsin(c*x))*cos(4*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(2*I*(
-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*(30*arcsin(c*x)*c^2*f^2+2*arcsin(c*x)*g^2-15*I
*c^2*f^2-I*g^2)*d^2/c^3/(c^2*x^2-1)+1/16384*(-d*(c^2*x^2-1))^(1/2)*(-128*I*(-c^2*x^2+1)^(1/2)*x^8*c^8+128*c^9*
x^9+256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7-160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+272*c^5*x^5+32*I*(-c^2*x^2+1
)^(1/2)*c^2*x^2-88*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+8*c*x)*g^2*(8*arcsin(c*x)+I)*d^2/c^3/(c^2*x^2-1)+1/1024*(-d*(c
^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(54*I*c^2*f^2+144*arcsin(c*x)*c^2*f^2+5*I*g^2+4*arcsin(c
*x)*g^2)*sin(3*arcsin(c*x))*d^2/c^3/(c^2*x^2-1))

Fricas [F]

\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral((a^2*c^4*d^2*g^2*x^6 + 2*a^2*c^4*d^2*f*g*x^5 - 4*a^2*c^2*d^2*f*g*x^3 + 2*a^2*d^2*f*g*x + a^2*d^2*f^2
+ (a^2*c^4*d^2*f^2 - 2*a^2*c^2*d^2*g^2)*x^4 - (2*a^2*c^2*d^2*f^2 - a^2*d^2*g^2)*x^2 + (b^2*c^4*d^2*g^2*x^6 + 2
*b^2*c^4*d^2*f*g*x^5 - 4*b^2*c^2*d^2*f*g*x^3 + 2*b^2*d^2*f*g*x + b^2*d^2*f^2 + (b^2*c^4*d^2*f^2 - 2*b^2*c^2*d^
2*g^2)*x^4 - (2*b^2*c^2*d^2*f^2 - b^2*d^2*g^2)*x^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*g^2*x^6 + 2*a*b*c^4*d^2*f*g
*x^5 - 4*a*b*c^2*d^2*f*g*x^3 + 2*a*b*d^2*f*g*x + a*b*d^2*f^2 + (a*b*c^4*d^2*f^2 - 2*a*b*c^2*d^2*g^2)*x^4 - (2*
a*b*c^2*d^2*f^2 - a*b*d^2*g^2)*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)

Sympy [F(-1)]

Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)**2*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2,x)

[Out]

Timed out

Maxima [F]

\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*
arcsin(c*x)/c)*a^2*f^2 + 1/384*(8*(-c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d) + 10*(-c^
2*d*x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/2)*arcsin(c*x)/c^3)*a^2*g^2 - 2/7*(-c
^2*d*x^2 + d)^(7/2)*a^2*f*g/(c^2*d) + sqrt(d)*integrate(((b^2*c^4*d^2*g^2*x^6 + 2*b^2*c^4*d^2*f*g*x^5 - 4*b^2*
c^2*d^2*f*g*x^3 + 2*b^2*d^2*f*g*x + b^2*d^2*f^2 + (b^2*c^4*d^2*f^2 - 2*b^2*c^2*d^2*g^2)*x^4 - (2*b^2*c^2*d^2*f
^2 - b^2*d^2*g^2)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*g^2*x^6 + 2*a*b*c^4*d^2*f
*g*x^5 - 4*a*b*c^2*d^2*f*g*x^3 + 2*a*b*d^2*f*g*x + a*b*d^2*f^2 + (a*b*c^4*d^2*f^2 - 2*a*b*c^2*d^2*g^2)*x^4 - (
2*a*b*c^2*d^2*f^2 - a*b*d^2*g^2)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)
, x)

Giac [F(-2)]

Exception generated. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

[In]

int((f + g*x)^2*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2),x)

[Out]

int((f + g*x)^2*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2), x)