\(\int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx\) [531]

Optimal result

Integrand size = 30, antiderivative size = 252 \[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=-\frac {b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {4 d^3 (1+c x) \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {d^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {3 d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{2 b c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {4 b d^3 \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (d+c d x)^{3/2} (f-c f x)^{3/2}} \] Output:

-b*d^3*x*(-c^2*x^2+1)^(3/2)/(c*d*x+d)^(3/2)/(-c*f*x+f)^(3/2)+4*d^3*(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)+d^3*(-c 
^2*x^2+1)^2*(a+b*arccos(c*x))/c/(c*d*x+d)^(3/2)/(-c*f*x+f)^(3/2)-3/2*d^3*( 
-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))^2/b/c/(c*d*x+d)^(3/2)/(-c*f*x+f)^(3/2) 
+4*b*d^3*(-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 = 3.63 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.10 \[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=\frac {d (1+c x) \left (2 b (-5+c x) \sqrt {d+c d x} \sqrt {f-c f x} \sqrt {1-c^2 x^2} \arccos (c x)+3 b (-1+c x) \sqrt {d+c d x} \sqrt {f-c f x} \arccos (c x)^2+6 a \sqrt {d} \sqrt {f} (-1+c x) \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )+2 \sqrt {d+c d x} \sqrt {f-c f x} \left (b c x (-1+c x)+a (-5+c x) \sqrt {1-c^2 x^2}-8 b (-1+c x) \log \left (\sin \left (\frac {1}{2} \arccos (c x)\right )\right )\right )\right ) \sin ^2\left (\frac {1}{2} \arccos (c x)\right )}{c f^2 (-1+c x) \left (1-c^2 x^2\right )^{3/2}} \] Input:

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

Output:

(d*(1 + c*x)*(2*b*(-5 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*Sqrt[1 - c^2* 
x^2]*ArcCos[c*x] + 3*b*(-1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcCos[c 
*x]^2 + 6*a*Sqrt[d]*Sqrt[f]*(-1 + c*x)*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))] + 2*Sqrt[d + 
 c*d*x]*Sqrt[f - c*f*x]*(b*c*x*(-1 + c*x) + a*(-5 + c*x)*Sqrt[1 - c^2*x^2] 
 - 8*b*(-1 + c*x)*Log[Sin[ArcCos[c*x]/2]]))*Sin[ArcCos[c*x]/2]^2)/(c*f^2*( 
-1 + c*x)*(1 - c^2*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.53, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5179, 27, 5275, 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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5275

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 5275

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[c*x] 
)^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, 
 b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 
 0] && GtQ[d, 0] && IGtQ[n, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 33.14 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.94

method	result	size
default	\(-\frac {3 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, b \arccos \left (c x \right )^{2} d}{2 \left (c x -1\right ) f^{2} c \left (c x +1\right )}-\frac {3 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, a \arccos \left (c x \right ) d}{\left (c x -1\right ) f^{2} c \left (c x +1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (b \arccos \left (c x \right )+a +i b \right ) d}{2 \left (c x -1\right ) f^{2} c \left (c x +1\right )}+\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (b \arccos \left (c x \right )+a -i b \right ) d}{2 \left (c x -1\right ) f^{2} c \left (c x +1\right )}-\frac {8 i \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, b \arccos \left (c x \right ) d}{\left (c x -1\right ) f^{2} c \left (c x +1\right )}-\frac {4 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \left (a +b \arccos \left (c x \right )\right ) d}{\left (c x -1\right ) f^{2} c \left (c x +1\right )}+\frac {8 \sqrt {d \left (c x +1\right )}\, \sqrt {-f \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, b \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) d}{\left (c x -1\right ) f^{2} c \left (c x +1\right )}\)	\(488\)

Input:

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

Output:

-3/2*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x-1)/f^2/c 
/(c*x+1)*b*arccos(c*x)^2*d-3*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-c^2*x^ 
2+1)^(1/2)/(c*x-1)/f^2/c/(c*x+1)*a*arccos(c*x)*d+1/2*(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)*(b*arccos(c*x)+a+I*b) 
*d/(c*x-1)/f^2/c/(c*x+1)+1/2*(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)*(b*arccos(c*x)+a-I*b)*d/(c*x-1)/f^2/c/(c*x+1 
)-8*I*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x-1)/f^2/ 
c/(c*x+1)*b*arccos(c*x)*d-4*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*(-I*(-c^2 
*x^2+1)^(1/2)+c*x+1)*(a+b*arccos(c*x))*d/(c*x-1)/f^2/c/(c*x+1)+8*(d*(c*x+1 
))^(1/2)*(-f*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x-1)/f^2/c/(c*x+1)*b*ln( 
I*(-c^2*x^2+1)^(1/2)+c*x-1)*d
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

b*sqrt(d)*sqrt(f)*integrate((c*d*x + d)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arcta 
n2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/(c^2*f^2*x^2 - 2*c*f^2*x + f^2), x) 
- a*((-c^2*d*f*x^2 + d*f)^(3/2)/(c^3*f^3*x^2 - 2*c^2*f^3*x + c*f^3) + 6*sq 
rt(-c^2*d*f*x^2 + d*f)*d/(c^2*f^2*x - c*f^2) + 3*d^2*arcsin(c*x)/(c*f^2*sq 
rt(d/f)))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d+c d x)^{3/2} (a+b \arccos (c x))}{(f-c f x)^{3/2}} \, dx=\frac {\sqrt {d}\, d \left (6 \sqrt {-c x +1}\, \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a -\sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right ) x}{\sqrt {-c x +1}\, c x -\sqrt {-c x +1}}d x \right ) b \,c^{2}-\sqrt {-c x +1}\, \left (\int \frac {\sqrt {c x +1}\, \mathit {acos} \left (c x \right )}{\sqrt {-c x +1}\, c x -\sqrt {-c x +1}}d x \right ) b c -\sqrt {c x +1}\, a c x +5 \sqrt {c x +1}\, a \right )}{\sqrt {f}\, \sqrt {-c x +1}\, c f} \] Input:

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

Output:

(sqrt(d)*d*(6*sqrt( - c*x + 1)*asin(sqrt( - c*x + 1)/sqrt(2))*a - sqrt( - 
c*x + 1)*int((sqrt(c*x + 1)*acos(c*x)*x)/(sqrt( - c*x + 1)*c*x - sqrt( - c 
*x + 1)),x)*b*c**2 - sqrt( - c*x + 1)*int((sqrt(c*x + 1)*acos(c*x))/(sqrt( 
 - c*x + 1)*c*x - sqrt( - c*x + 1)),x)*b*c - sqrt(c*x + 1)*a*c*x + 5*sqrt( 
c*x + 1)*a))/(sqrt(f)*sqrt( - c*x + 1)*c*f)