Integrand size = 30, antiderivative size = 376 \[ \int \sqrt {d+c d x} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=-\frac {2 b f^2 x \sqrt {d+c d x} \sqrt {f-c f x}}{3 \sqrt {1-c^2 x^2}}-\frac {3 b c f^2 x^2 \sqrt {d+c d x} \sqrt {f-c f x}}{16 \sqrt {1-c^2 x^2}}+\frac {2 b c^2 f^2 x^3 \sqrt {d+c d x} \sqrt {f-c f x}}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 f^2 x^4 \sqrt {d+c d x} \sqrt {f-c f x}}{16 \sqrt {1-c^2 x^2}}+\frac {3}{8} f^2 x \sqrt {d+c d x} \sqrt {f-c f x} (a+b \arccos (c x))+\frac {1}{4} c^2 f^2 x^3 \sqrt {d+c d x} \sqrt {f-c f x} (a+b \arccos (c x))+\frac {2 f^2 \sqrt {d+c d x} \sqrt {f-c f x} \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{3 c}+\frac {5 f^2 \sqrt {d+c d x} \sqrt {f-c f x} (a+b \arccos (c x))^2}{16 b c \sqrt {1-c^2 x^2}} \] Output:
-2/3*b*f^2*x*(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)/(-c^2*x^2+1)^(1/2)-3/16*b*c* f^2*x^2*(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)/(-c^2*x^2+1)^(1/2)+2/9*b*c^2*f^2* x^3*(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)/(-c^2*x^2+1)^(1/2)-1/16*b*c^3*f^2*x^4 *(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)/(-c^2*x^2+1)^(1/2)+3/8*f^2*x*(c*d*x+d)^( 1/2)*(-c*f*x+f)^(1/2)*(a+b*arccos(c*x))+1/4*c^2*f^2*x^3*(c*d*x+d)^(1/2)*(- c*f*x+f)^(1/2)*(a+b*arccos(c*x))+2/3*f^2*(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2)* (-c^2*x^2+1)*(a+b*arccos(c*x))/c+5/16*f^2*(c*d*x+d)^(1/2)*(-c*f*x+f)^(1/2) *(a+b*arccos(c*x))^2/b/c/(-c^2*x^2+1)^(1/2)
Time = 2.24 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.78 \[ \int \sqrt {d+c d x} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=\frac {-360 b f^2 \sqrt {d+c d x} \sqrt {f-c f x} \arccos (c x)^2-720 a \sqrt {d} f^{5/2} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )+f^2 \sqrt {d+c d x} \sqrt {f-c f x} \left (-256 b c x \left (-3+c^2 x^2\right )+48 a \sqrt {1-c^2 x^2} \left (16+9 c x-16 c^2 x^2+6 c^3 x^3\right )+144 b \cos (2 \arccos (c x))+9 b \cos (4 \arccos (c x))\right )+12 b f^2 \sqrt {d+c d x} \sqrt {f-c f x} \arccos (c x) \left (64 \left (1-c^2 x^2\right )^{3/2}+24 \sin (2 \arccos (c x))+3 \sin (4 \arccos (c x))\right )}{1152 c \sqrt {1-c^2 x^2}} \] Input:
Integrate[Sqrt[d + c*d*x]*(f - c*f*x)^(5/2)*(a + b*ArcCos[c*x]),x]
Output:
(-360*b*f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcCos[c*x]^2 - 720*a*Sqrt[d]* f^(5/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sq rt[d]*Sqrt[f]*(-1 + c^2*x^2))] + f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-256 *b*c*x*(-3 + c^2*x^2) + 48*a*Sqrt[1 - c^2*x^2]*(16 + 9*c*x - 16*c^2*x^2 + 6*c^3*x^3) + 144*b*Cos[2*ArcCos[c*x]] + 9*b*Cos[4*ArcCos[c*x]]) + 12*b*f^2 *Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcCos[c*x]*(64*(1 - c^2*x^2)^(3/2) + 24* Sin[2*ArcCos[c*x]] + 3*Sin[4*ArcCos[c*x]]))/(1152*c*Sqrt[1 - c^2*x^2])
Time = 0.80 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.49, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5179, 27, 5263, 2009}
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.
| \(\displaystyle \int \sqrt {c d x+d} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5179 |
| \(\displaystyle \frac {\sqrt {c d x+d} \sqrt {f-c f x} \int f^2 (1-c x)^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f^2 \sqrt {c d x+d} \sqrt {f-c f x} \int (1-c x)^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5263 |
| \(\displaystyle \frac {f^2 \sqrt {c d x+d} \sqrt {f-c f x} \int \left (c^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) x^2-2 c \sqrt {1-c^2 x^2} (a+b \arccos (c x)) x+\sqrt {1-c^2 x^2} (a+b \arccos (c x))\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f^2 \sqrt {c d x+d} \sqrt {f-c f x} \left (\frac {3}{8} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c}+\frac {1}{4} c^2 x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {5 (a+b \arccos (c x))^2}{16 b c}+\frac {1}{16} b c^3 x^4-\frac {2}{9} b c^2 x^3+\frac {3}{16} b c x^2+\frac {2 b x}{3}\right )}{\sqrt {1-c^2 x^2}}\) |
Input:
Int[Sqrt[d + c*d*x]*(f - c*f*x)^(5/2)*(a + b*ArcCos[c*x]),x]
Output:
(f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*((2*b*x)/3 + (3*b*c*x^2)/16 - (2*b*c^ 2*x^3)/9 + (b*c^3*x^4)/16 + (3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/8 + (c^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/4 + (2*(1 - c^2*x^2)^(3/ 2)*(a + b*ArcCos[c*x]))/(3*c) - (5*(a + b*ArcCos[c*x])^2)/(16*b*c)))/Sqrt[ 1 - c^2*x^2]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcCos[(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*ArcCos[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]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcCos[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))
Result contains complex when optimal does not.
Time = 1.71 (sec) , antiderivative size = 952, normalized size of antiderivative = 2.53
| method | result | size |
| default | \(-\frac {a \sqrt {c d x +d}\, \left (-c f x +f \right )^{\frac {7}{2}}}{4 c f}+\frac {a \left (-c f x +f \right )^{\frac {5}{2}} \sqrt {c d x +d}}{12 c}+\frac {5 a f \left (-c f x +f \right )^{\frac {3}{2}} \sqrt {c d x +d}}{24 c}+\frac {5 a \,f^{2} \sqrt {-c f x +f}\, \sqrt {c d x +d}}{8 c}+\frac {5 a d \,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 )}{8 \sqrt {-c f x +f}\, \sqrt {c d x +d}\, \sqrt {c^{2} d f}}+b \left (\frac {5 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f^{2}}{16 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+4 \arccos \left (c x \right )\right ) f^{2}}{256 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-3 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) \left (i+3 \arccos \left (c x \right )\right ) f^{2}}{36 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right ) f^{2}}{4 \left (c^{2} x^{2}-1\right ) c}+\frac {\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 \arccos \left (c x \right )\right ) f^{2}}{16 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (17 i+28 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) f^{2}}{256 \left (c^{2} x^{2}-1\right ) c}-\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 (5 i+12 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) f^{2}}{256 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (2 i+3 \arccos \left (c x \right )\right ) \cos \left (2 \arccos \left (c x \right )\right ) f^{2}}{9 \left (c^{2} x^{2}-1\right ) c}-\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 (5 i+3 \arccos \left (c x \right )\right ) \sin \left (2 \arccos \left (c x \right )\right ) f^{2}}{18 \left (c^{2} x^{2}-1\right ) c}\right )\) | \(952\) |
| parts | \(-\frac {a \sqrt {c d x +d}\, \left (-c f x +f \right )^{\frac {7}{2}}}{4 c f}+\frac {a \left (-c f x +f \right )^{\frac {5}{2}} \sqrt {c d x +d}}{12 c}+\frac {5 a f \left (-c f x +f \right )^{\frac {3}{2}} \sqrt {c d x +d}}{24 c}+\frac {5 a \,f^{2} \sqrt {-c f x +f}\, \sqrt {c d x +d}}{8 c}+\frac {5 a d \,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 )}{8 \sqrt {-c f x +f}\, \sqrt {c d x +d}\, \sqrt {c^{2} d f}}+b \left (\frac {5 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f^{2}}{16 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+4 \arccos \left (c x \right )\right ) f^{2}}{256 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-3 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) \left (i+3 \arccos \left (c x \right )\right ) f^{2}}{36 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right ) f^{2}}{4 \left (c^{2} x^{2}-1\right ) c}+\frac {\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 \arccos \left (c x \right )\right ) f^{2}}{16 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (17 i+28 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) f^{2}}{256 \left (c^{2} x^{2}-1\right ) c}-\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 (5 i+12 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) f^{2}}{256 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (2 i+3 \arccos \left (c x \right )\right ) \cos \left (2 \arccos \left (c x \right )\right ) f^{2}}{9 \left (c^{2} x^{2}-1\right ) c}-\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 (5 i+3 \arccos \left (c x \right )\right ) \sin \left (2 \arccos \left (c x \right )\right ) f^{2}}{18 \left (c^{2} x^{2}-1\right ) c}\right )\) | \(952\) |
Input:
int((c*d*x+d)^(1/2)*(-c*f*x+f)^(5/2)*(a+b*arccos(c*x)),x,method=_RETURNVER BOSE)
Output:
-1/4*a/c/f*(c*d*x+d)^(1/2)*(-c*f*x+f)^(7/2)+1/12*a/c*(-c*f*x+f)^(5/2)*(c*d *x+d)^(1/2)+5/24*a*f/c*(-c*f*x+f)^(3/2)*(c*d*x+d)^(1/2)+5/8*a*f^2/c*(-c*f* x+f)^(1/2)*(c*d*x+d)^(1/2)+5/8*a*d*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*(5/16*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-c^2*x^2 +1)^(1/2)/(c^2*x^2-1)/c*arccos(c*x)^2*f^2+1/256*(d*(c*x+1))^(1/2)*(-f*(c*x -1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+4*c*x-8*I* (-c^2*x^2+1)^(1/2)*x^2*c^2+I*(-c^2*x^2+1)^(1/2))*(I+4*arccos(c*x))*f^2/(c^ 2*x^2-1)/c-1/36*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+ 4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-3*I*(-c^2*x^2+1)^(1/2)*c*x+1)*(I+3*arccos(c *x))*f^2/(c^2*x^2-1)/c+1/4*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-I*(-c^2* x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arccos(c*x)-I)*f^2/(c^2*x^2-1)/c+1/16*(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*arccos(c*x))*f^2/(c^2*x^2-1)/c-1/256*(d*( c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(17 *I+28*arccos(c*x))*cos(3*arccos(c*x))*f^2/(c^2*x^2-1)/c-3/256*(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)*(5*I+12*arcc os(c*x))*sin(3*arccos(c*x))*f^2/(c^2*x^2-1)/c-1/9*(d*(c*x+1))^(1/2)*(-f*(c *x-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(2*I+3*arccos(c*x))*cos (2*arccos(c*x))*f^2/(c^2*x^2-1)/c-1/18*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(...
\[ \int \sqrt {d+c d x} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=\int { \sqrt {c d x + d} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \] Input:
integrate((c*d*x+d)^(1/2)*(-c*f*x+f)^(5/2)*(a+b*arccos(c*x)),x, algorithm= "fricas")
Output:
integral((a*c^2*f^2*x^2 - 2*a*c*f^2*x + a*f^2 + (b*c^2*f^2*x^2 - 2*b*c*f^2 *x + b*f^2)*arccos(c*x))*sqrt(c*d*x + d)*sqrt(-c*f*x + f), x)
Timed out. \[ \int \sqrt {d+c d x} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=\text {Timed out} \] Input:
integrate((c*d*x+d)**(1/2)*(-c*f*x+f)**(5/2)*(a+b*acos(c*x)),x)
Output:
Timed out
\[ \int \sqrt {d+c d x} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=\int { \sqrt {c d x + d} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \] Input:
integrate((c*d*x+d)^(1/2)*(-c*f*x+f)^(5/2)*(a+b*arccos(c*x)),x, algorithm= "maxima")
Output:
b*sqrt(d)*sqrt(f)*integrate((c^2*f^2*x^2 - 2*c*f^2*x + f^2)*sqrt(c*x + 1)* sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x), x) + 1/24*(15*s qrt(-c^2*d*f*x^2 + d*f)*f^2*x + 15*d*f^3*arcsin(c*x)/(sqrt(d*f)*c) - 6*(-c ^2*d*f*x^2 + d*f)^(3/2)*f*x/d + 16*(-c^2*d*f*x^2 + d*f)^(3/2)*f/(c*d))*a
\[ \int \sqrt {d+c d x} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=\int { \sqrt {c d x + d} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \] Input:
integrate((c*d*x+d)^(1/2)*(-c*f*x+f)^(5/2)*(a+b*arccos(c*x)),x, algorithm= "giac")
Output:
integrate(sqrt(c*d*x + d)*(-c*f*x + f)^(5/2)*(b*arccos(c*x) + a), x)
Timed out. \[ \int \sqrt {d+c d x} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=\int \left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d+c\,d\,x}\,{\left (f-c\,f\,x\right )}^{5/2} \,d x \] Input:
int((a + b*acos(c*x))*(d + c*d*x)^(1/2)*(f - c*f*x)^(5/2),x)
Output:
int((a + b*acos(c*x))*(d + c*d*x)^(1/2)*(f - c*f*x)^(5/2), x)
\[ \int \sqrt {d+c d x} (f-c f x)^{5/2} (a+b \arccos (c x)) \, dx=\frac {\sqrt {f}\, \sqrt {d}\, f^{2} \left (-30 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a +6 \sqrt {c x +1}\, \sqrt {-c x +1}\, a \,c^{3} x^{3}-16 \sqrt {c x +1}\, \sqrt {-c x +1}\, a \,c^{2} x^{2}+9 \sqrt {c x +1}\, \sqrt {-c x +1}\, a c x +16 \sqrt {c x +1}\, \sqrt {-c x +1}\, a +24 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{3}-48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x d x \right ) b \,c^{2}+24 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right )d x \right ) b c \right )}{24 c} \] Input:
int((c*d*x+d)^(1/2)*(-c*f*x+f)^(5/2)*(a+b*acos(c*x)),x)
Output:
(sqrt(f)*sqrt(d)*f**2*( - 30*asin(sqrt( - c*x + 1)/sqrt(2))*a + 6*sqrt(c*x + 1)*sqrt( - c*x + 1)*a*c**3*x**3 - 16*sqrt(c*x + 1)*sqrt( - c*x + 1)*a*c **2*x**2 + 9*sqrt(c*x + 1)*sqrt( - c*x + 1)*a*c*x + 16*sqrt(c*x + 1)*sqrt( - c*x + 1)*a + 24*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*acos(c*x)*x**2,x)*b* c**3 - 48*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*acos(c*x)*x,x)*b*c**2 + 24*in t(sqrt(c*x + 1)*sqrt( - c*x + 1)*acos(c*x),x)*b*c))/(24*c)