\(\int \frac {a+b \arccos (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx\) [533]

Optimal result

Integrand size = 30, antiderivative size = 98 \[ \int \frac {a+b \arccos (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=\frac {d (1+c x) \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {b d \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (d+c d x)^{3/2} (f-c f x)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \arccos (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=-\frac {\sqrt {d+c d x} \sqrt {f-c f x} \left (a \sqrt {1-c^2 x^2}+b \sqrt {1-c^2 x^2} \arccos (c x)+b (-1+c x) \log (f-c f x)\right )}{c d f^2 (-1+c x) \sqrt {1-c^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5179, 27, 5261, 27, 451, 16}

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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5261

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 451

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

\(\Big \downarrow \) 16

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 451

Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c^2/a   In 
t[1/(c - d*x), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
 

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 5261

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, 
 x]}, Simp[(a + b*ArcCos[c*x])   u, x] + Simp[b*c   Int[1/Sqrt[1 - c^2*x^2] 
   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IG 
tQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3] 
)
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.35

method	result	size
default	\(\frac {a \sqrt {c d x +d}}{f d c \sqrt {-c f x +f}}+b \left (-\frac {2 i \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )}{d \,f^{2} c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \arccos \left (c x \right )}{d \,f^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{d \,f^{2} c \left (c^{2} x^{2}-1\right )}\right )\)	\(230\)
parts	\(\frac {a \sqrt {c d x +d}}{f d c \sqrt {-c f x +f}}+b \left (-\frac {2 i \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )}{d \,f^{2} c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \arccos \left (c x \right )}{d \,f^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{d \,f^{2} c \left (c^{2} x^{2}-1\right )}\right )\)	\(230\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 353, normalized size of antiderivative = 3.60 \[ \int \frac {a+b \arccos (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=\left [\frac {{\left (b c x - b\right )} \sqrt {d f} \log \left (\frac {c^{6} d f x^{6} - 4 \, c^{5} d f x^{5} + 5 \, c^{4} d f x^{4} - 4 \, c^{2} d f x^{2} + 4 \, c d f x + {\left (c^{4} x^{4} - 4 \, c^{3} x^{3} + 6 \, c^{2} x^{2} - 4 \, c x\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {c d x + d} \sqrt {-c f x + f} \sqrt {d f} - 2 \, d f}{c^{4} x^{4} - 2 \, c^{3} x^{3} + 2 \, c x - 1}\right ) - 2 \, \sqrt {c d x + d} \sqrt {-c f x + f} {\left (b \arccos \left (c x\right ) + a\right )}}{2 \, {\left (c^{2} d f^{2} x - c d f^{2}\right )}}, -\frac {{\left (b c x - b\right )} \sqrt {-d f} \arctan \left (\frac {{\left (c^{2} x^{2} - 2 \, c x + 2\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {c d x + d} \sqrt {-c f x + f} \sqrt {-d f}}{c^{4} d f x^{4} - 2 \, c^{3} d f x^{3} - c^{2} d f x^{2} + 2 \, c d f x}\right ) + \sqrt {c d x + d} \sqrt {-c f x + f} {\left (b \arccos \left (c x\right ) + a\right )}}{c^{2} d f^{2} x - c d f^{2}}\right ] \] Input:

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

Output:

[1/2*((b*c*x - b)*sqrt(d*f)*log((c^6*d*f*x^6 - 4*c^5*d*f*x^5 + 5*c^4*d*f*x 
^4 - 4*c^2*d*f*x^2 + 4*c*d*f*x + (c^4*x^4 - 4*c^3*x^3 + 6*c^2*x^2 - 4*c*x) 
*sqrt(-c^2*x^2 + 1)*sqrt(c*d*x + d)*sqrt(-c*f*x + f)*sqrt(d*f) - 2*d*f)/(c 
^4*x^4 - 2*c^3*x^3 + 2*c*x - 1)) - 2*sqrt(c*d*x + d)*sqrt(-c*f*x + f)*(b*a 
rccos(c*x) + a))/(c^2*d*f^2*x - c*d*f^2), -((b*c*x - b)*sqrt(-d*f)*arctan( 
(c^2*x^2 - 2*c*x + 2)*sqrt(-c^2*x^2 + 1)*sqrt(c*d*x + d)*sqrt(-c*f*x + f)* 
sqrt(-d*f)/(c^4*d*f*x^4 - 2*c^3*d*f*x^3 - c^2*d*f*x^2 + 2*c*d*f*x)) + sqrt 
(c*d*x + d)*sqrt(-c*f*x + f)*(b*arccos(c*x) + a))/(c^2*d*f^2*x - c*d*f^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \arccos (c x)}{\sqrt {d+c d x} (f-c f x)^{3/2}} \, dx=-\frac {\sqrt {-c^{2} d f x^{2} + d f} b \arccos \left (c x\right )}{c^{2} d f^{2} x - c d f^{2}} - \frac {\sqrt {-c^{2} d f x^{2} + d f} a}{c^{2} d f^{2} x - c d f^{2}} - \frac {b \log \left (c x - 1\right )}{c \sqrt {d} f^{\frac {3}{2}}} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt( - c*x + 1)*int(acos(c*x)/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c*x - s 
qrt(c*x + 1)*sqrt( - c*x + 1)),x)*b*c + sqrt(c*x + 1)*a)/(sqrt(f)*sqrt(d)* 
sqrt( - c*x + 1)*c*f)