\(\int \frac {(f-c f x)^{3/2} (a+b \arcsin (c x))}{(d+c d x)^{3/2}} \, dx\) [52]

Optimal result

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

b*f^3*x*(-c^2*x^2+1)^(3/2)/(c*d*x+d)^(3/2)/(-c*f*x+f)^(3/2)-4*f^3*(-c*x+1) 
*(-c^2*x^2+1)*(a+b*arcsin(c*x))/c/(c*d*x+d)^(3/2)/(-c*f*x+f)^(3/2)-f^3*(-c 
^2*x^2+1)^2*(a+b*arcsin(c*x))/c/(c*d*x+d)^(3/2)/(-c*f*x+f)^(3/2)-3/2*f^3*( 
-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))^2/b/c/(c*d*x+d)^(3/2)/(-c*f*x+f)^(3/2) 
+4*b*f^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 = 6.17 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.15 \[ \int \frac {(f-c f x)^{3/2} (a+b \arcsin (c x))}{(d+c d x)^{3/2}} \, dx=\frac {f \left (6 a \sqrt {d} \sqrt {f} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )-\frac {\sqrt {d+c d x} \sqrt {f-c f x} \csc ^2\left (\frac {1}{2} \arcsin (c x)\right ) \left (2 b (5+c x) \left (-1+c x+\sqrt {1-c^2 x^2}\right ) \arcsin (c x)-3 b \left (-1-c x+\sqrt {1-c^2 x^2}\right ) \arcsin (c x)^2+2 \left (b c x \left (-1-c x+\sqrt {1-c^2 x^2}\right )+a (5+c x) \left (-1+c x+\sqrt {1-c^2 x^2}\right )+8 b \left (-1-c x+\sqrt {1-c^2 x^2}\right ) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )\right )}{2 \sqrt {1-c^2 x^2} \left (1+\cot \left (\frac {1}{2} \arcsin (c x)\right )\right )}\right )}{2 c d^2} \] Input:

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

Output:

(f*(6*a*Sqrt[d]*Sqrt[f]*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]*Csc[ArcSin 
[c*x]/2]^2*(2*b*(5 + c*x)*(-1 + c*x + Sqrt[1 - c^2*x^2])*ArcSin[c*x] - 3*b 
*(-1 - c*x + Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 2*(b*c*x*(-1 - c*x + Sqrt[ 
1 - c^2*x^2]) + a*(5 + c*x)*(-1 + c*x + Sqrt[1 - c^2*x^2]) + 8*b*(-1 - c*x 
 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])))/(2*S 
qrt[1 - c^2*x^2]*(1 + Cot[ArcSin[c*x]/2]))))/(2*c*d^2)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.54, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5178, 27, 5274, 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.

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

\(\Big \downarrow \) 5178

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5274

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(f^3*(1 - c^2*x^2)^(3/2)*(b*x - (4*(1 - c*x)*(a + b*ArcSin[c*x]))/(c*Sqrt[ 
1 - c^2*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c - (3*(a + b*ArcS 
in[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 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 5274

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[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 = 4.16 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.05

method	result	size
default	\(\frac {\sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (8 i \arcsin \left (c x \right ) b c x +3 b c x \arcsin \left (c x \right )^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, b c x -2 x^{2} c^{2} b -8 i a c x +6 \arcsin \left (c x \right ) a c x +2 \sqrt {-c^{2} x^{2}+1}\, a c x -16 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right ) b c x -8 i a +3 \arcsin \left (c x \right )^{2} b +10 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, b -2 x b c +8 i b \arcsin \left (c x \right )+6 \arcsin \left (c x \right ) a +10 \sqrt {-c^{2} x^{2}+1}\, a -16 b \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )\right ) f}{2 d^{2} c \left (c^{3} x^{3}+c^{2} x^{2}-c x -1\right )}\)	\(264\)

Input:

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

Output:

1/2*(-f*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c/(c^3*x^3 
+c^2*x^2-c*x-1)*(8*I*arcsin(c*x)*b*c*x+3*b*c*x*arcsin(c*x)^2+2*arcsin(c*x) 
*(-c^2*x^2+1)^(1/2)*b*c*x-2*x^2*c^2*b-8*I*a*c*x+6*arcsin(c*x)*a*c*x+2*(-c^ 
2*x^2+1)^(1/2)*a*c*x-16*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)*b*c*x-8*I*a+3*arcsi 
n(c*x)^2*b+10*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*b-2*x*b*c+8*I*b*arcsin(c*x)+6 
*arcsin(c*x)*a+10*(-c^2*x^2+1)^(1/2)*a-16*b*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I) 
)*f
 

Fricas [F]

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

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

Output:

integral(-(a*c*f*x - a*f + (b*c*f*x - b*f)*arcsin(c*x))*sqrt(c*d*x + d)*sq 
rt(-c*f*x + f)/(c^2*d^2*x^2 + 2*c*d^2*x + d^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-b*sqrt(d)*sqrt(f)*integrate((c*f*x - f)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arct 
an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*d^2*x^2 + 2*c*d^2*x + d^2), x) 
 + a*((-c^2*d*f*x^2 + d*f)^(3/2)/(c^3*d^3*x^2 + 2*c^2*d^3*x + c*d^3) - 6*s 
qrt(-c^2*d*f*x^2 + d*f)*f/(c^2*d^2*x + c*d^2) - 3*f^2*arcsin(c*x)/(c*d^2*s 
qrt(f/d)))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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