\(\int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{\sqrt {f-c f x}} \, dx\) [525]

Optimal result

Integrand size = 30, antiderivative size = 242 \[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{\sqrt {f-c f x}} \, dx=\frac {2 b d^2 x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {f-c f x}}+\frac {b c d^2 x^2 \sqrt {1-c^2 x^2}}{4 \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{c \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{2 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {3 d^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 b c \sqrt {d+c d x} \sqrt {f-c f x}} \] Output:

2*b*d^2*x*(-c^2*x^2+1)^(1/2)/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)+1/4*b*c*d^2* 
x^2*(-c^2*x^2+1)^(1/2)/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)-2*d^2*(-c^2*x^2+1) 
*(a+b*arccos(c*x))/c/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)-1/2*d^2*x*(-c^2*x^2+ 
1)*(a+b*arccos(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2)+3/4*d^2*(-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)
 

Mathematica [A] (verified)

Time = 3.16 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.06 \[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{\sqrt {f-c f x}} \, dx=-\frac {d (1+c x) \sec ^2\left (\frac {1}{2} \arccos (c x)\right ) \left (6 b \sqrt {d+c d x} \sqrt {f-c f x} \arccos (c x)^2+12 a \sqrt {d} \sqrt {f} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )+\sqrt {d+c d x} \sqrt {f-c f x} \left (16 b c x+4 a (4+c x) \sqrt {1-c^2 x^2}+b \cos (2 \arccos (c x))\right )+2 b \sqrt {d+c d x} \sqrt {f-c f x} \arccos (c x) \left (8 \sqrt {1-c^2 x^2}+\sin (2 \arccos (c x))\right )\right )}{16 c f \sqrt {1-c^2 x^2}} \] Input:

Integrate[((d + c*d*x)^(3/2)*(a + b*ArcCos[c*x]))/Sqrt[f - c*f*x],x]
 

Output:

-1/16*(d*(1 + c*x)*Sec[ArcCos[c*x]/2]^2*(6*b*Sqrt[d + c*d*x]*Sqrt[f - c*f* 
x]*ArcCos[c*x]^2 + 12*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]*(16*b*c*x + 4*a*(4 + c*x)*Sqrt[1 - c^2*x^2] + b*Cos 
[2*ArcCos[c*x]]) + 2*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcCos[c*x]*(8*Sqrt 
[1 - c^2*x^2] + Sin[2*ArcCos[c*x]])))/(c*f*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.52, 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.

\(\displaystyle \int \frac {(c d x+d)^{3/2} (a+b \arccos (c x))}{\sqrt {f-c f x}} \, dx\)

\(\Big \downarrow \) 5179

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {d^2 (c x+1)^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{\sqrt {c d x+d} \sqrt {f-c f x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \sqrt {1-c^2 x^2} \int \frac {(c x+1)^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{\sqrt {c d x+d} \sqrt {f-c f x}}\)

\(\Big \downarrow \) 5263

\(\displaystyle \frac {d^2 \sqrt {1-c^2 x^2} \int \left (\frac {c^2 (a+b \arccos (c x)) x^2}{\sqrt {1-c^2 x^2}}+\frac {2 c (a+b \arccos (c x)) x}{\sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}\right )dx}{\sqrt {c d x+d} \sqrt {f-c f x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {1-c^2 x^2} \left (-\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c}-\frac {3 (a+b \arccos (c x))^2}{4 b c}-\frac {1}{4} b c x^2-2 b x\right )}{\sqrt {c d x+d} \sqrt {f-c f x}}\)

Input:

Int[((d + c*d*x)^(3/2)*(a + b*ArcCos[c*x]))/Sqrt[f - c*f*x],x]
 

Output:

(d^2*Sqrt[1 - c^2*x^2]*(-2*b*x - (b*c*x^2)/4 - (2*Sqrt[1 - c^2*x^2]*(a + b 
*ArcCos[c*x]))/c - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/2 - (3*(a + b 
*ArcCos[c*x])^2)/(4*b*c)))/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5179

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]
 

rule 5263

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))
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.01 (sec) , antiderivative size = 714, normalized size of antiderivative = 2.95

method	result	size
default	\(-\frac {a \left (c d x +d \right )^{\frac {3}{2}} \sqrt {-c f x +f}}{2 c f}-\frac {3 a d \sqrt {c d x +d}\, \sqrt {-c f x +f}}{2 c f}+\frac {3 a \,d^{2} \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 d x +d}\, \sqrt {-c f x +f}\, \sqrt {c^{2} d f}}+b \left (\frac {3 \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} d}{4 \left (c x -1\right ) f c \left (c x +1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (2 c^{2} x^{2}+4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+4 c^{3} x^{3}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c -i \sqrt {-c^{2} x^{2}+1}-3 c x \right ) \left (i+2 \arccos \left (c x \right )\right ) d}{32 \left (c x -1\right ) f c \left (c x +1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) \left (\arccos \left (c x \right )+i\right ) d}{2 \left (c x -1\right ) f 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}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right ) d}{\left (c x -1\right ) f c \left (c x +1\right )}+\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 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 ) d}{32 \left (c x -1\right ) f 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 (7 i+8 \arccos \left (c x \right )\right ) \cos \left (2 \arccos \left (c x \right )\right ) d}{16 \left (c x -1\right ) f 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 +\sqrt {-c^{2} x^{2}+1}+i\right ) \left (4 i+3 \arccos \left (c x \right )\right ) \sin \left (2 \arccos \left (c x \right )\right ) d}{8 \left (c x -1\right ) f c \left (c x +1\right )}\right )\)	\(714\)
parts	\(-\frac {a \left (c d x +d \right )^{\frac {3}{2}} \sqrt {-c f x +f}}{2 c f}-\frac {3 a d \sqrt {c d x +d}\, \sqrt {-c f x +f}}{2 c f}+\frac {3 a \,d^{2} \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 d x +d}\, \sqrt {-c f x +f}\, \sqrt {c^{2} d f}}+b \left (\frac {3 \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} d}{4 \left (c x -1\right ) f c \left (c x +1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (2 c^{2} x^{2}+4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+4 c^{3} x^{3}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c -i \sqrt {-c^{2} x^{2}+1}-3 c x \right ) \left (i+2 \arccos \left (c x \right )\right ) d}{32 \left (c x -1\right ) f c \left (c x +1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) \left (\arccos \left (c x \right )+i\right ) d}{2 \left (c x -1\right ) f 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}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right ) d}{\left (c x -1\right ) f c \left (c x +1\right )}+\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 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 ) d}{32 \left (c x -1\right ) f 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 (7 i+8 \arccos \left (c x \right )\right ) \cos \left (2 \arccos \left (c x \right )\right ) d}{16 \left (c x -1\right ) f 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 +\sqrt {-c^{2} x^{2}+1}+i\right ) \left (4 i+3 \arccos \left (c x \right )\right ) \sin \left (2 \arccos \left (c x \right )\right ) d}{8 \left (c x -1\right ) f c \left (c x +1\right )}\right )\)	\(714\)

Input:

int((c*d*x+d)^(3/2)*(a+b*arccos(c*x))/(-c*f*x+f)^(1/2),x,method=_RETURNVER 
BOSE)
 

Output:

-1/2*a/c/f*(c*d*x+d)^(3/2)*(-c*f*x+f)^(1/2)-3/2*a*d/c/f*(c*d*x+d)^(1/2)*(- 
c*f*x+f)^(1/2)+3/2*a*d^2*((-c*f*x+f)*(c*d*x+d))^(1/2)/(c*d*x+d)^(1/2)/(-c* 
f*x+f)^(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*(3/4*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x- 
1)/f/c/(c*x+1)*arccos(c*x)^2*d-1/32*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*( 
2*c^2*x^2+4*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+4*c^3*x^3-1+2*I*(-c^2*x^2+1)^(1/2 
)*c*x-I*(-c^2*x^2+1)^(1/2)-3*c*x)*(I+2*arccos(c*x))*d/(c*x-1)/f/c/(c*x+1)+ 
1/2*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(1+c*x+I*(-c^2*x^2+1)^(1/2))*(arc 
cos(c*x)+I)*d/(c*x-1)/f/c/(c*x+1)-(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)*d/(c*x-1)/f/c/(c*x+1)+1 
/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*arccos(c*x))*d/(c*x-1)/f/c/(c*x+1) 
-1/16*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)+c*x+1)*( 
7*I+8*arccos(c*x))*cos(2*arccos(c*x))*d/(c*x-1)/f/c/(c*x+1)-1/8*(d*(c*x+1) 
)^(1/2)*(-f*(c*x-1))^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2)+I)*(4*I+3*arccos(c*x) 
)*sin(2*arccos(c*x))*d/(c*x-1)/f/c/(c*x+1))
 

Fricas [F]

\[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{\sqrt {f-c f x}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {-c f x + f}} \,d x } \] Input:

integrate((c*d*x+d)^(3/2)*(a+b*arccos(c*x))/(-c*f*x+f)^(1/2),x, algorithm= 
"fricas")
 

Output:

integral(-(a*c*d*x + a*d + (b*c*d*x + b*d)*arccos(c*x))*sqrt(c*d*x + d)*sq 
rt(-c*f*x + f)/(c*f*x - f), x)
 

Sympy [F]

\[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{\sqrt {f-c f x}} \, dx=\int \frac {\left (d \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{\sqrt {- f \left (c x - 1\right )}}\, dx \] Input:

integrate((c*d*x+d)**(3/2)*(a+b*acos(c*x))/(-c*f*x+f)**(1/2),x)
 

Output:

Integral((d*(c*x + 1))**(3/2)*(a + b*acos(c*x))/sqrt(-f*(c*x - 1)), x)
 

Maxima [F]

\[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{\sqrt {f-c f x}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {-c f x + f}} \,d x } \] Input:

integrate((c*d*x+d)^(3/2)*(a+b*arccos(c*x))/(-c*f*x+f)^(1/2),x, algorithm= 
"maxima")
 

Output:

-1/2*(sqrt(-c^2*d*f*x^2 + d*f)*d*x/f - 3*d^2*arcsin(c*x)/(sqrt(d*f)*c) + 4 
*sqrt(-c^2*d*f*x^2 + d*f)*d/(c*f))*a - b*sqrt(d)*integrate((c*d*x + d)*sqr 
t(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/(c*x 
- 1), x)/sqrt(f)
 

Giac [F]

\[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{\sqrt {f-c f x}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {-c f x + f}} \,d x } \] Input:

integrate((c*d*x+d)^(3/2)*(a+b*arccos(c*x))/(-c*f*x+f)^(1/2),x, algorithm= 
"giac")
 

Output:

integrate((c*d*x + d)^(3/2)*(b*arccos(c*x) + a)/sqrt(-c*f*x + f), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{\sqrt {f-c f x}} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^{3/2}}{\sqrt {f-c\,f\,x}} \,d x \] Input:

int(((a + b*acos(c*x))*(d + c*d*x)^(3/2))/(f - c*f*x)^(1/2),x)
 

Output:

int(((a + b*acos(c*x))*(d + c*d*x)^(3/2))/(f - c*f*x)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{\sqrt {f-c f x}} \, dx=\frac {\sqrt {d}\, d \left (-6 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a -\sqrt {c x +1}\, \sqrt {-c x +1}\, a c x -4 \sqrt {c x +1}\, \sqrt {-c x +1}\, a +2 \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right ) x}{\sqrt {-c x +1}}d x \right ) b \,c^{2}+2 \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right )}{\sqrt {-c x +1}}d x \right ) b c \right )}{2 \sqrt {f}\, c} \] Input:

int((c*d*x+d)^(3/2)*(a+b*acos(c*x))/(-c*f*x+f)^(1/2),x)
 

Output:

(sqrt(d)*d*( - 6*asin(sqrt( - c*x + 1)/sqrt(2))*a - sqrt(c*x + 1)*sqrt( - 
c*x + 1)*a*c*x - 4*sqrt(c*x + 1)*sqrt( - c*x + 1)*a + 2*int((sqrt(c*x + 1) 
*acos(c*x)*x)/sqrt( - c*x + 1),x)*b*c**2 + 2*int((sqrt(c*x + 1)*acos(c*x)) 
/sqrt( - c*x + 1),x)*b*c))/(2*sqrt(f)*c)