\(\int \frac {(f-c f x)^{3/2} (a+b \arcsin (c x))}{\sqrt {d+c d x}} \, dx\) [51]

Optimal result

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

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

Mathematica [A] (verified)

Time = 4.42 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.98 \[ \int \frac {(f-c f x)^{3/2} (a+b \arcsin (c x))}{\sqrt {d+c d x}} \, dx=\frac {-4 b f (-4+c x) \sqrt {d+c d x} \sqrt {f-c f x} \sqrt {1-c^2 x^2} \arcsin (c x)+6 b f \sqrt {d+c d x} \sqrt {f-c f x} \arcsin (c x)^2-12 a \sqrt {d} f^{3/2} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )-f \sqrt {d+c d x} \sqrt {f-c f x} \left (16 b c x+4 a (-4+c x) \sqrt {1-c^2 x^2}+b \cos (2 \arcsin (c x))\right )}{8 c d \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-4*b*f*(-4 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*Sqrt[1 - c^2*x^2]*ArcSi 
n[c*x] + 6*b*f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 12*a*Sqrt[d 
]*f^(3/2)*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))] - f*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*ArcSin[c*x]]))/(8*c*d*Sq 
rt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.66 (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 = {5178, 27, 5262, 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.

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

\(\Big \downarrow \) 5178

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5262

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(f^2*Sqrt[1 - c^2*x^2]*(-2*b*x + (b*c*x^2)/4 + (2*Sqrt[1 - c^2*x^2]*(a + b 
*ArcSin[c*x]))/c - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/2 + (3*(a + b 
*ArcSin[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 5178

Int[((a_.) + ArcSin[(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*ArcSin[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 5262

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + 
 b*ArcSin[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 = 2.24 (sec) , antiderivative size = 716, normalized size of antiderivative = 2.96

method	result	size
default	\(\frac {a \left (-c f x +f \right )^{\frac {3}{2}} \sqrt {c d x +d}}{2 c d}+\frac {3 a f \sqrt {-c f x +f}\, \sqrt {c d x +d}}{2 c d}+\frac {3 a \,f^{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 f x +f}\, \sqrt {c d x +d}\, \sqrt {c^{2} d f}}+b \left (-\frac {3 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{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}-2 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-3 c x +2 i \sqrt {-c^{2} x^{2}+1}\, c x +1+i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+2 \arcsin \left (c x \right )\right ) f}{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 (\arcsin \left (c x \right )+i\right ) f}{2 \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 +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) f}{\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 (2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}-c x -1\right ) \left (-i+2 \arcsin \left (c x \right )\right ) f}{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 (7 i+8 \arcsin \left (c x \right )\right ) \cos \left (2 \arcsin \left (c x \right )\right ) f}{16 \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 (4 i+3 \arcsin \left (c x \right )\right ) \sin \left (2 \arcsin \left (c x \right )\right ) f}{8 \left (c x +1\right ) d c \left (c x -1\right )}\right )\)	\(716\)
parts	\(\frac {a \left (-c f x +f \right )^{\frac {3}{2}} \sqrt {c d x +d}}{2 c d}+\frac {3 a f \sqrt {-c f x +f}\, \sqrt {c d x +d}}{2 c d}+\frac {3 a \,f^{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 f x +f}\, \sqrt {c d x +d}\, \sqrt {c^{2} d f}}+b \left (-\frac {3 \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{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}-2 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-3 c x +2 i \sqrt {-c^{2} x^{2}+1}\, c x +1+i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+2 \arcsin \left (c x \right )\right ) f}{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 (\arcsin \left (c x \right )+i\right ) f}{2 \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 +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) f}{\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 (2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}-c x -1\right ) \left (-i+2 \arcsin \left (c x \right )\right ) f}{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 (7 i+8 \arcsin \left (c x \right )\right ) \cos \left (2 \arcsin \left (c x \right )\right ) f}{16 \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 (4 i+3 \arcsin \left (c x \right )\right ) \sin \left (2 \arcsin \left (c x \right )\right ) f}{8 \left (c x +1\right ) d c \left (c x -1\right )}\right )\)	\(716\)

Input:

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

Output:

1/2*a/c/d*(-c*f*x+f)^(3/2)*(c*d*x+d)^(1/2)+3/2*a*f/c/d*(-c*f*x+f)^(1/2)*(c 
*d*x+d)^(1/2)+3/2*a*f^2*((-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*(-3/4*(-f*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x+ 
1)/d/c/(c*x-1)*arcsin(c*x)^2*f-1/32*(d*(c*x+1))^(1/2)*(-f*(c*x-1))^(1/2)*( 
4*c^3*x^3-2*c^2*x^2-4*I*x^2*c^2*(-c^2*x^2+1)^(1/2)-3*c*x+2*I*(-c^2*x^2+1)^ 
(1/2)*c*x+1+I*(-c^2*x^2+1)^(1/2))*(I+2*arcsin(c*x))*f/(c*x+1)/d/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)+c*x-1)*(ar 
csin(c*x)+I)*f/(c*x+1)/d/c/(c*x-1)+(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)*(arcsin(c*x)-I)*f/(c*x+1)/d/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*arcsin(c*x))*f/(c*x+1)/d/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*arcsin(c*x))*cos(2*arcsin(c*x))*f/(c*x+1)/d/c/(c*x-1)+1/8*(d*(c*x+1))^ 
(1/2)*(-f*(c*x-1))^(1/2)*(I*c*x-I-(-c^2*x^2+1)^(1/2))*(4*I+3*arcsin(c*x))* 
sin(2*arcsin(c*x))*f/(c*x+1)/d/c/(c*x-1))
 

Fricas [F]

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

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

Output:

integral(-(a*c*f*x - a*f + (b*c*f*x - b*f)*arcsin(c*x))*sqrt(-c*f*x + f)/s 
qrt(c*d*x + d), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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