Integrand size = 30, antiderivative size = 345 \[ \int \frac {(f-c f x)^{5/2} (a+b \arccos (c x))}{\sqrt {d+c d x}} \, dx=-\frac {11 b f^3 x \sqrt {1-c^2 x^2}}{3 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {3 b c f^3 x^2 \sqrt {1-c^2 x^2}}{4 \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {b c^2 f^3 x^3 \sqrt {1-c^2 x^2}}{9 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {11 f^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{3 c \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {3 f^3 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{2 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {c f^3 x^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{3 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {5 f^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 b c \sqrt {d+c d x} \sqrt {f-c f x}} \] Output:
-11/3*b*f^3*x*(-c^2*x^2+1)^(1/2)/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)+3/4*b*c* f^3*x^2*(-c^2*x^2+1)^(1/2)/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)-1/9*b*c^2*f^3* x^3*(-c^2*x^2+1)^(1/2)/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)+11/3*f^3*(-c^2*x^2 +1)*(a+b*arccos(c*x))/c/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)-3/2*f^3*x*(-c^2*x ^2+1)*(a+b*arccos(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)+1/3*c*f^3*x^2*(-c ^2*x^2+1)*(a+b*arccos(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)+5/4*f^3*(-c^2 *x^2+1)^(1/2)*(a+b*arccos(c*x))^2/b/c/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)
Time = 5.94 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.83 \[ \int \frac {(f-c f x)^{5/2} (a+b \arccos (c x))}{\sqrt {d+c d x}} \, dx=\frac {f^2 (-1+c x) \csc ^2\left (\frac {1}{2} \arccos (c x)\right ) \left (90 b \sqrt {d+c d x} \sqrt {f-c f x} \arccos (c x)^2+180 a \sqrt {d} \sqrt {f} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )+\sqrt {d+c d x} \sqrt {f-c f x} \left (-8 b c x \left (33+c^2 x^2\right )-12 a \sqrt {1-c^2 x^2} \left (22-9 c x+2 c^2 x^2\right )+27 b \cos (2 \arccos (c x))\right )-6 b \sqrt {d+c d x} \sqrt {f-c f x} \arccos (c x) \left (45 \sqrt {1-c^2 x^2}-9 \sin (2 \arccos (c x))+\sin (3 \arccos (c x))\right )\right )}{144 c d \sqrt {1-c^2 x^2}} \] Input:
Integrate[((f - c*f*x)^(5/2)*(a + b*ArcCos[c*x]))/Sqrt[d + c*d*x],x]
Output:
(f^2*(-1 + c*x)*Csc[ArcCos[c*x]/2]^2*(90*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x] *ArcCos[c*x]^2 + 180*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]*(-8*b*c*x*(33 + c^2*x^2) - 12*a*Sqrt[1 - c^2*x^2]*(2 2 - 9*c*x + 2*c^2*x^2) + 27*b*Cos[2*ArcCos[c*x]]) - 6*b*Sqrt[d + c*d*x]*Sq rt[f - c*f*x]*ArcCos[c*x]*(45*Sqrt[1 - c^2*x^2] - 9*Sin[2*ArcCos[c*x]] + S in[3*ArcCos[c*x]])))/(144*c*d*Sqrt[1 - c^2*x^2])
Time = 0.88 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.50, 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.
| \(\displaystyle \int \frac {(f-c f x)^{5/2} (a+b \arccos (c x))}{\sqrt {c d x+d}} \, dx\) |
| \(\Big \downarrow \) 5179 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {f^3 (1-c x)^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{\sqrt {c d x+d} \sqrt {f-c f x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f^3 \sqrt {1-c^2 x^2} \int \frac {(1-c x)^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{\sqrt {c d x+d} \sqrt {f-c f x}}\) |
| \(\Big \downarrow \) 5263 |
| \(\displaystyle \frac {f^3 \sqrt {1-c^2 x^2} \int \left (-\frac {c^3 (a+b \arccos (c x)) x^3}{\sqrt {1-c^2 x^2}}+\frac {3 c^2 (a+b \arccos (c x)) x^2}{\sqrt {1-c^2 x^2}}-\frac {3 c (a+b \arccos (c x)) x}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}\right )dx}{\sqrt {c d x+d} \sqrt {f-c f x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f^3 \sqrt {1-c^2 x^2} \left (\frac {1}{3} c x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {3}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {11 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c}-\frac {5 (a+b \arccos (c x))^2}{4 b c}+\frac {1}{9} b c^2 x^3-\frac {3}{4} b c x^2+\frac {11 b x}{3}\right )}{\sqrt {c d x+d} \sqrt {f-c f x}}\) |
Input:
Int[((f - c*f*x)^(5/2)*(a + b*ArcCos[c*x]))/Sqrt[d + c*d*x],x]
Output:
(f^3*Sqrt[1 - c^2*x^2]*((11*b*x)/3 - (3*b*c*x^2)/4 + (b*c^2*x^3)/9 + (11*S qrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(3*c) - (3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/2 + (c*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/3 - (5*( a + b*ArcCos[c*x])^2)/(4*b*c)))/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x])
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 = 9.13 (sec) , antiderivative size = 947, normalized size of antiderivative = 2.74
| method | result | size |
| default | \(\frac {a \left (-c f x +f \right )^{\frac {5}{2}} \sqrt {c d x +d}}{3 d c}+\frac {5 a f \left (-c f x +f \right )^{\frac {3}{2}} \sqrt {c d x +d}}{6 d c}+\frac {5 a \,f^{2} \sqrt {-c f x +f}\, \sqrt {c d x +d}}{2 d c}+\frac {5 a \,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 )}{2 \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}}{4 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-4 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+8 c^{4} x^{4}+3 c x -4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x c -8 c^{2} x^{2}+i \sqrt {-c^{2} x^{2}+1}+1\right ) \left (i+3 \arccos \left (c x \right )\right ) f^{2}}{144 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {15 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (\arccos \left (c x \right )+i\right ) f^{2}}{16 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {15 \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}}{8 \left (c x +1\right ) d c \left (c x -1\right )}-\frac {3 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}+i \sqrt {-c^{2} x^{2}+1}-c x -1\right ) \left (-i+2 \arccos \left (c x \right )\right ) f^{2}}{32 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (23 i+54 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) f^{2}}{288 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i c x -i+\sqrt {-c^{2} x^{2}+1}\right ) \left (9 i+14 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) f^{2}}{96 \left (c x +1\right ) d c \left (c x -1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (109 i+132 \arccos \left (c x \right )\right ) \cos \left (2 \arccos \left (c x \right )\right ) f^{2}}{144 \left (c x +1\right ) d c \left (c x -1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i c x -i+\sqrt {-c^{2} x^{2}+1}\right ) \left (67 i+42 \arccos \left (c x \right )\right ) \sin \left (2 \arccos \left (c x \right )\right ) f^{2}}{72 \left (c x +1\right ) d c \left (c x -1\right )}\right )\) | \(947\) |
| parts | \(\frac {a \left (-c f x +f \right )^{\frac {5}{2}} \sqrt {c d x +d}}{3 d c}+\frac {5 a f \left (-c f x +f \right )^{\frac {3}{2}} \sqrt {c d x +d}}{6 d c}+\frac {5 a \,f^{2} \sqrt {-c f x +f}\, \sqrt {c d x +d}}{2 d c}+\frac {5 a \,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 )}{2 \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}}{4 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-4 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+8 c^{4} x^{4}+3 c x -4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x c -8 c^{2} x^{2}+i \sqrt {-c^{2} x^{2}+1}+1\right ) \left (i+3 \arccos \left (c x \right )\right ) f^{2}}{144 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {15 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (\arccos \left (c x \right )+i\right ) f^{2}}{16 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {15 \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}}{8 \left (c x +1\right ) d c \left (c x -1\right )}-\frac {3 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}+i \sqrt {-c^{2} x^{2}+1}-c x -1\right ) \left (-i+2 \arccos \left (c x \right )\right ) f^{2}}{32 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (23 i+54 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) f^{2}}{288 \left (c x +1\right ) d c \left (c x -1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i c x -i+\sqrt {-c^{2} x^{2}+1}\right ) \left (9 i+14 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) f^{2}}{96 \left (c x +1\right ) d c \left (c x -1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (109 i+132 \arccos \left (c x \right )\right ) \cos \left (2 \arccos \left (c x \right )\right ) f^{2}}{144 \left (c x +1\right ) d c \left (c x -1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i c x -i+\sqrt {-c^{2} x^{2}+1}\right ) \left (67 i+42 \arccos \left (c x \right )\right ) \sin \left (2 \arccos \left (c x \right )\right ) f^{2}}{72 \left (c x +1\right ) d c \left (c x -1\right )}\right )\) | \(947\) |
Input:
int((-c*f*x+f)^(5/2)*(a+b*arccos(c*x))/(c*d*x+d)^(1/2),x,method=_RETURNVER BOSE)
Output:
1/3*a/d/c*(-c*f*x+f)^(5/2)*(c*d*x+d)^(1/2)+5/6*a*f/d/c*(-c*f*x+f)^(3/2)*(c *d*x+d)^(1/2)+5/2*a*f^2/d/c*(-c*f*x+f)^(1/2)*(c*d*x+d)^(1/2)+5/2*a*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*(5/4*(d*(c*x+1))^(1/ 2)*(-f*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x+1)/d/c/(c*x-1)*arccos(c*x)^2 *f^2+1/144*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-4*c^3*x^3+8*I*(-c^2*x^2+ 1)^(1/2)*x^3*c^3+8*c^4*x^4+3*c*x-4*I*(-c^2*x^2+1)^(1/2)*c^2*x^2-4*I*(-c^2* x^2+1)^(1/2)*x*c-8*c^2*x^2+I*(-c^2*x^2+1)^(1/2)+1)*(I+3*arccos(c*x))*f^2/( c*x+1)/d/c/(c*x-1)+15/16*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(I*(-c^2*x^2 +1)^(1/2)+c*x-1)*(arccos(c*x)+I)*f^2/(c*x+1)/d/c/(c*x-1)+15/8*(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*x+1)/d/c/(c*x-1)-3/32*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-2 *I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2+I*(-c^2*x^2+1)^(1/2)-c*x-1)*(-I+2*arcc os(c*x))*f^2/(c*x+1)/d/c/(c*x-1)+1/288*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2 )*(-I*(-c^2*x^2+1)^(1/2)+c*x-1)*(23*I+54*arccos(c*x))*cos(3*arccos(c*x))*f ^2/(c*x+1)/d/c/(c*x-1)+1/96*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(I*c*x-I+ (-c^2*x^2+1)^(1/2))*(9*I+14*arccos(c*x))*sin(3*arccos(c*x))*f^2/(c*x+1)/d/ c/(c*x-1)-1/144*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2 )+c*x-1)*(109*I+132*arccos(c*x))*cos(2*arccos(c*x))*f^2/(c*x+1)/d/c/(c*x-1 )-1/72*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(I*c*x-I+(-c^2*x^2+1)^(1/2)...
\[ \int \frac {(f-c f x)^{5/2} (a+b \arccos (c x))}{\sqrt {d+c d x}} \, dx=\int { \frac {{\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {c d x + d}} \,d x } \] Input:
integrate((-c*f*x+f)^(5/2)*(a+b*arccos(c*x))/(c*d*x+d)^(1/2),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*f*x + f)/sqrt(c*d*x + d), x)
Timed out. \[ \int \frac {(f-c f x)^{5/2} (a+b \arccos (c x))}{\sqrt {d+c d x}} \, dx=\text {Timed out} \] Input:
integrate((-c*f*x+f)**(5/2)*(a+b*acos(c*x))/(c*d*x+d)**(1/2),x)
Output:
Timed out
\[ \int \frac {(f-c f x)^{5/2} (a+b \arccos (c x))}{\sqrt {d+c d x}} \, dx=\int { \frac {{\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {c d x + d}} \,d x } \] Input:
integrate((-c*f*x+f)^(5/2)*(a+b*arccos(c*x))/(c*d*x+d)^(1/2),x, algorithm= "maxima")
Output:
1/6*(2*sqrt(-c^2*d*f*x^2 + d*f)*c*f^2*x^2/d - 9*sqrt(-c^2*d*f*x^2 + d*f)*f ^2*x/d + 15*f^3*arcsin(c*x)/(sqrt(d*f)*c) + 22*sqrt(-c^2*d*f*x^2 + d*f)*f^ 2/(c*d))*a + b*sqrt(f)*integrate((c^2*f^2*x^2 - 2*c*f^2*x + f^2)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/sqrt(c*x + 1), x)/sqrt(d)
\[ \int \frac {(f-c f x)^{5/2} (a+b \arccos (c x))}{\sqrt {d+c d x}} \, dx=\int { \frac {{\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {c d x + d}} \,d x } \] Input:
integrate((-c*f*x+f)^(5/2)*(a+b*arccos(c*x))/(c*d*x+d)^(1/2),x, algorithm= "giac")
Output:
integrate((-c*f*x + f)^(5/2)*(b*arccos(c*x) + a)/sqrt(c*d*x + d), x)
Timed out. \[ \int \frac {(f-c f x)^{5/2} (a+b \arccos (c x))}{\sqrt {d+c d x}} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (f-c\,f\,x\right )}^{5/2}}{\sqrt {d+c\,d\,x}} \,d x \] Input:
int(((a + b*acos(c*x))*(f - c*f*x)^(5/2))/(d + c*d*x)^(1/2),x)
Output:
int(((a + b*acos(c*x))*(f - c*f*x)^(5/2))/(d + c*d*x)^(1/2), x)
\[ \int \frac {(f-c f x)^{5/2} (a+b \arccos (c x))}{\sqrt {d+c d x}} \, dx=\frac {\sqrt {f}\, f^{2} \left (-30 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a +2 \sqrt {c x +1}\, \sqrt {-c x +1}\, a \,c^{2} x^{2}-9 \sqrt {c x +1}\, \sqrt {-c x +1}\, a c x +22 \sqrt {c x +1}\, \sqrt {-c x +1}\, a +6 \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x^{2}}{\sqrt {c x +1}}d x \right ) b \,c^{3}-12 \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x}{\sqrt {c x +1}}d x \right ) b \,c^{2}+6 \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )}{\sqrt {c x +1}}d x \right ) b c \right )}{6 \sqrt {d}\, c} \] Input:
int((-c*f*x+f)^(5/2)*(a+b*acos(c*x))/(c*d*x+d)^(1/2),x)
Output:
(sqrt(f)*f**2*( - 30*asin(sqrt( - c*x + 1)/sqrt(2))*a + 2*sqrt(c*x + 1)*sq rt( - c*x + 1)*a*c**2*x**2 - 9*sqrt(c*x + 1)*sqrt( - c*x + 1)*a*c*x + 22*s qrt(c*x + 1)*sqrt( - c*x + 1)*a + 6*int((sqrt( - c*x + 1)*acos(c*x)*x**2)/ sqrt(c*x + 1),x)*b*c**3 - 12*int((sqrt( - c*x + 1)*acos(c*x)*x)/sqrt(c*x + 1),x)*b*c**2 + 6*int((sqrt( - c*x + 1)*acos(c*x))/sqrt(c*x + 1),x)*b*c))/ (6*sqrt(d)*c)