Integrand size = 30, antiderivative size = 352 \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=-\frac {5 b c d^2 f^2 x^2 \sqrt {d+c d x} \sqrt {f-c f x}}{32 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^2 \sqrt {d+c d x} \sqrt {f-c f x} \left (1-c^2 x^2\right )^{3/2}}{96 c}+\frac {b d^2 f^2 \sqrt {d+c d x} \sqrt {f-c f x} \left (1-c^2 x^2\right )^{5/2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c d x} \sqrt {f-c f x} (a+b \arcsin (c x))+\frac {5}{24} d^2 f^2 x \sqrt {d+c d x} \sqrt {f-c f x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))+\frac {1}{6} d^2 f^2 x \sqrt {d+c d x} \sqrt {f-c f x} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))+\frac {5 d^2 f^2 \sqrt {d+c d x} \sqrt {f-c f x} (a+b \arcsin (c x))^2}{32 b c \sqrt {1-c^2 x^2}} \] Output:
-5/32*b*c*d^2*f^2*x^2*(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)/(-c^2*x^2+1)^(1/2)+ 5/96*b*d^2*f^2*(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)*(-c^2*x^2+1)^(3/2)/c+1/36* b*d^2*f^2*(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)*(-c^2*x^2+1)^(5/2)/c+5/16*d^2*f ^2*x*(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)*(a+b*arcsin(c*x))+5/24*d^2*f^2*x*(c* d*x+d)^(1/2)*(-c*f*x+f)^(1/2)*(-c^2*x^2+1)*(a+b*arcsin(c*x))+1/6*d^2*f^2*x *(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))+5/32*d^ 2*f^2*(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)*(a+b*arcsin(c*x))^2/b/c/(-c^2*x^2+1 )^(1/2)
Time = 2.56 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.86 \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\frac {d^2 f^2 \left (360 b \sqrt {d+c d x} \sqrt {f-c f x} \arcsin (c x)^2-720 a \sqrt {d} \sqrt {f} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )+\sqrt {d+c d x} \sqrt {f-c f x} \left (1584 a c x \sqrt {1-c^2 x^2}-1248 a c^3 x^3 \sqrt {1-c^2 x^2}+384 a c^5 x^5 \sqrt {1-c^2 x^2}+270 b \cos (2 \arcsin (c x))+27 b \cos (4 \arcsin (c x))+2 b \cos (6 \arcsin (c x))\right )+12 b \sqrt {d+c d x} \sqrt {f-c f x} \arcsin (c x) (45 \sin (2 \arcsin (c x))+9 \sin (4 \arcsin (c x))+\sin (6 \arcsin (c x)))\right )}{2304 c \sqrt {1-c^2 x^2}} \] Input:
Integrate[(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]),x]
Output:
(d^2*f^2*(360*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 720*a*Sqrt [d]*Sqrt[f]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) /(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(1584 *a*c*x*Sqrt[1 - c^2*x^2] - 1248*a*c^3*x^3*Sqrt[1 - c^2*x^2] + 384*a*c^5*x^ 5*Sqrt[1 - c^2*x^2] + 270*b*Cos[2*ArcSin[c*x]] + 27*b*Cos[4*ArcSin[c*x]] + 2*b*Cos[6*ArcSin[c*x]]) + 12*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x ]*(45*Sin[2*ArcSin[c*x]] + 9*Sin[4*ArcSin[c*x]] + Sin[6*ArcSin[c*x]])))/(2 304*c*Sqrt[1 - c^2*x^2])
Time = 0.84 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.57, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5178, 5158, 241, 5158, 244, 2009, 5156, 15, 5152}
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.
| \(\displaystyle \int (c d x+d)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5178 |
| \(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \int \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx}{\left (1-c^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {1}{6} b c \int x \left (1-c^2 x^2\right )^2dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))\right )}{\left (1-c^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {(c d x+d)^{5/2} (f-c f x)^{5/2} \left (\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5}{6} \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{\left (1-c^2 x^2\right )^{5/2}}\) |
Input:
Int[(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]),x]
Output:
((d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*((b*(1 - c^2*x^2)^3)/(36*c) + (x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/6 + (5*(-1/4*(b*c*(x^2/2 - (c^2*x^4)/ 4)) + (x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/4 + (3*(-1/4*(b*c*x^2) + (x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/2 + (a + b*ArcSin[c*x])^2/(4*b* c)))/4))/6))/(1 - c^2*x^2)^(5/2)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcSin[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] - Simp[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]
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] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[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*ArcSin[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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 2)^q) Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Result contains complex when optimal does not.
Time = 7.45 (sec) , antiderivative size = 897, normalized size of antiderivative = 2.55
| method | result | size |
| default | \(-\frac {a \left (c d x +d \right )^{\frac {5}{2}} \left (-c f x +f \right )^{\frac {7}{2}}}{6 f c}-\frac {a d \left (c d x +d \right )^{\frac {3}{2}} \left (-c f x +f \right )^{\frac {7}{2}}}{6 f c}-\frac {a \,d^{2} \sqrt {c d x +d}\, \left (-c f x +f \right )^{\frac {7}{2}}}{8 f c}+\frac {a \,d^{2} \left (-c f x +f \right )^{\frac {5}{2}} \sqrt {c d x +d}}{24 c}+\frac {5 a \,d^{2} f \left (-c f x +f \right )^{\frac {3}{2}} \sqrt {c d x +d}}{48 c}+\frac {5 a \,d^{2} f^{2} \sqrt {-c f x +f}\, \sqrt {c d x +d}}{16 c}+\frac {5 a \,d^{3} f^{3} \sqrt {\left (-c f x +f \right ) \left (c d x +d \right )}\, \arctan \left (\frac {\sqrt {c^{2} d f}\, x}{\sqrt {-c^{2} d f \,x^{2}+d f}}\right )}{16 \sqrt {-c f x +f}\, \sqrt {c d x +d}\, \sqrt {c^{2} d f}}+b \left (-\frac {5 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d^{2} f^{2}}{32 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-32 i \sqrt {-c^{2} x^{2}+1}\, x^{6} c^{6}+32 c^{7} x^{7}+48 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}-64 c^{5} x^{5}-18 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+38 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-6 c x \right ) \left (i+6 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{2304 c \left (c^{2} x^{2}-1\right )}+\frac {15 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{256 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (29 i+96 \arcsin \left (c x \right )\right ) \cos \left (5 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{4608 c \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (5 i+24 \arcsin \left (c x \right )\right ) \sin \left (5 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{4608 c \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (11 i+16 \arcsin \left (c x \right )\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{512 c \left (c^{2} x^{2}-1\right )}+\frac {9 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (3 i+8 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{512 c \left (c^{2} x^{2}-1\right )}\right )\) | \(897\) |
| parts | \(-\frac {a \left (c d x +d \right )^{\frac {5}{2}} \left (-c f x +f \right )^{\frac {7}{2}}}{6 f c}-\frac {a d \left (c d x +d \right )^{\frac {3}{2}} \left (-c f x +f \right )^{\frac {7}{2}}}{6 f c}-\frac {a \,d^{2} \sqrt {c d x +d}\, \left (-c f x +f \right )^{\frac {7}{2}}}{8 f c}+\frac {a \,d^{2} \left (-c f x +f \right )^{\frac {5}{2}} \sqrt {c d x +d}}{24 c}+\frac {5 a \,d^{2} f \left (-c f x +f \right )^{\frac {3}{2}} \sqrt {c d x +d}}{48 c}+\frac {5 a \,d^{2} f^{2} \sqrt {-c f x +f}\, \sqrt {c d x +d}}{16 c}+\frac {5 a \,d^{3} f^{3} \sqrt {\left (-c f x +f \right ) \left (c d x +d \right )}\, \arctan \left (\frac {\sqrt {c^{2} d f}\, x}{\sqrt {-c^{2} d f \,x^{2}+d f}}\right )}{16 \sqrt {-c f x +f}\, \sqrt {c d x +d}\, \sqrt {c^{2} d f}}+b \left (-\frac {5 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d^{2} f^{2}}{32 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-32 i \sqrt {-c^{2} x^{2}+1}\, x^{6} c^{6}+32 c^{7} x^{7}+48 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}-64 c^{5} x^{5}-18 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+38 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-6 c x \right ) \left (i+6 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{2304 c \left (c^{2} x^{2}-1\right )}+\frac {15 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{256 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (29 i+96 \arcsin \left (c x \right )\right ) \cos \left (5 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{4608 c \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (5 i+24 \arcsin \left (c x \right )\right ) \sin \left (5 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{4608 c \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (11 i+16 \arcsin \left (c x \right )\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{512 c \left (c^{2} x^{2}-1\right )}+\frac {9 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (3 i+8 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d^{2} f^{2}}{512 c \left (c^{2} x^{2}-1\right )}\right )\) | \(897\) |
Input:
int((c*d*x+d)^(5/2)*(-c*f*x+f)^(5/2)*(a+b*arcsin(c*x)),x,method=_RETURNVER BOSE)
Output:
-1/6*a/f/c*(c*d*x+d)^(5/2)*(-c*f*x+f)^(7/2)-1/6*a*d/f/c*(c*d*x+d)^(3/2)*(- c*f*x+f)^(7/2)-1/8*a*d^2/f/c*(c*d*x+d)^(1/2)*(-c*f*x+f)^(7/2)+1/24*a*d^2/c *(-c*f*x+f)^(5/2)*(c*d*x+d)^(1/2)+5/48*a*d^2*f/c*(-c*f*x+f)^(3/2)*(c*d*x+d )^(1/2)+5/16*a*d^2*f^2/c*(-c*f*x+f)^(1/2)*(c*d*x+d)^(1/2)+5/16*a*d^3*f^3*( (-c*f*x+f)*(c*d*x+d))^(1/2)/(-c*f*x+f)^(1/2)/(c*d*x+d)^(1/2)/(c^2*d*f)^(1/ 2)*arctan((c^2*d*f)^(1/2)*x/(-c^2*d*f*x^2+d*f)^(1/2))+b*(-5/32*(-f*(c*x-1) )^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^2*d ^2*f^2+1/2304*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-32*I*(-c^2*x^2+1)^(1/ 2)*x^6*c^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)*(I+6*arcsin( c*x))*d^2*f^2/c/(c^2*x^2-1)+15/256*(d*(c*x+1))^(1/2)*(-f*(c*x-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)*(-I+2* arcsin(c*x))*d^2*f^2/c/(c^2*x^2-1)-1/4608*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^( 1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(29*I+96*arcsin(c*x))*cos(5*arcs in(c*x))*d^2*f^2/c/(c^2*x^2-1)+5/4608*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2) *(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*(5*I+24*arcsin(c*x))*sin(5*arcsin(c* x))*d^2*f^2/c/(c^2*x^2-1)-3/512*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(I*c^ 2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(11*I+16*arcsin(c*x))*cos(3*arcsin(c*x))*d ^2*f^2/c/(c^2*x^2-1)+9/512*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(I*(-c^2*x ^2+1)^(1/2)*c*x+c^2*x^2-1)*(3*I+8*arcsin(c*x))*sin(3*arcsin(c*x))*d^2*f...
\[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \] Input:
integrate((c*d*x+d)^(5/2)*(-c*f*x+f)^(5/2)*(a+b*arcsin(c*x)),x, algorithm= "fricas")
Output:
integral((a*c^4*d^2*f^2*x^4 - 2*a*c^2*d^2*f^2*x^2 + a*d^2*f^2 + (b*c^4*d^2 *f^2*x^4 - 2*b*c^2*d^2*f^2*x^2 + b*d^2*f^2)*arcsin(c*x))*sqrt(c*d*x + d)*s qrt(-c*f*x + f), x)
Timed out. \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \] Input:
integrate((c*d*x+d)**(5/2)*(-c*f*x+f)**(5/2)*(a+b*asin(c*x)),x)
Output:
Timed out
\[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \] Input:
integrate((c*d*x+d)^(5/2)*(-c*f*x+f)^(5/2)*(a+b*arcsin(c*x)),x, algorithm= "maxima")
Output:
b*sqrt(d)*sqrt(f)*integrate((c^4*d^2*f^2*x^4 - 2*c^2*d^2*f^2*x^2 + d^2*f^2 )*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + 1/48*(15*sqrt(-c^2*d*f*x^2 + d*f)*d^2*f^2*x + 15*d^3*f^3*arcsin(c*x) /(sqrt(d*f)*c) + 10*(-c^2*d*f*x^2 + d*f)^(3/2)*d*f*x + 8*(-c^2*d*f*x^2 + d *f)^(5/2)*x)*a
\[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \] Input:
integrate((c*d*x+d)^(5/2)*(-c*f*x+f)^(5/2)*(a+b*arcsin(c*x)),x, algorithm= "giac")
Output:
integrate((c*d*x + d)^(5/2)*(-c*f*x + f)^(5/2)*(b*arcsin(c*x) + a), x)
Timed out. \[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^{5/2}\,{\left (f-c\,f\,x\right )}^{5/2} \,d x \] Input:
int((a + b*asin(c*x))*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2),x)
Output:
int((a + b*asin(c*x))*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2), x)
\[ \int (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {f}\, \sqrt {d}\, d^{2} f^{2} \left (-30 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a +8 \sqrt {c x +1}\, \sqrt {-c x +1}\, a \,c^{5} x^{5}-26 \sqrt {c x +1}\, \sqrt {-c x +1}\, a \,c^{3} x^{3}+33 \sqrt {c x +1}\, \sqrt {-c x +1}\, a c x +48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x^{4}d x \right ) b \,c^{5}-96 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) b \,c^{3}+48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )d x \right ) b c \right )}{48 c} \] Input:
int((c*d*x+d)^(5/2)*(-c*f*x+f)^(5/2)*(a+b*asin(c*x)),x)
Output:
(sqrt(f)*sqrt(d)*d**2*f**2*( - 30*asin(sqrt( - c*x + 1)/sqrt(2))*a + 8*sqr t(c*x + 1)*sqrt( - c*x + 1)*a*c**5*x**5 - 26*sqrt(c*x + 1)*sqrt( - c*x + 1 )*a*c**3*x**3 + 33*sqrt(c*x + 1)*sqrt( - c*x + 1)*a*c*x + 48*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x**4,x)*b*c**5 - 96*int(sqrt(c*x + 1)*sqrt ( - c*x + 1)*asin(c*x)*x**2,x)*b*c**3 + 48*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x),x)*b*c))/(48*c)