\(\int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx\) [64]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx=\frac {\sqrt {d+c d x} \left (2 a c x+2 b c x \arcsin (c x)+b \sqrt {1-c^2 x^2} \log (-f (1+c x))+b \sqrt {1-c^2 x^2} \log (f-c f x)\right )}{2 c d^2 f (1+c x) \sqrt {f-c f x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5178, 5160, 240}

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.

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

\(\Big \downarrow \) 5178

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

\(\Big \downarrow \) 5160

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

\(\Big \downarrow \) 240

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 5160

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[x*((a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp[b 
*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcSin[c*x 
])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d 
 + e, 0] && GtQ[n, 0]
 

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.83 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.19

method	result	size
default	\(a \left (-\frac {1}{f d c \sqrt {c d x +d}\, \sqrt {-c f x +f}}+\frac {\sqrt {c d x +d}}{f c \,d^{2} \sqrt {-c f x +f}}\right )+\frac {b \left (i \arcsin \left (c x \right ) x^{2} c^{2}-\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -i \arcsin \left (c x \right )+\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right ) \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}}{\left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c \,d^{2} f^{2}}\)	\(215\)
parts	\(a \left (-\frac {1}{f d c \sqrt {c d x +d}\, \sqrt {-c f x +f}}+\frac {\sqrt {c d x +d}}{f c \,d^{2} \sqrt {-c f x +f}}\right )+\frac {b \left (i \arcsin \left (c x \right ) x^{2} c^{2}-\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -i \arcsin \left (c x \right )+\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right ) \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}}{\left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c \,d^{2} f^{2}}\)	\(215\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(c*d*x + d)*sqrt(-c*f*x + f)*(b*arcsin(c*x) + a)/(c^4*d^2*f^2 
*x^4 - 2*c^2*d^2*f^2*x^2 + d^2*f^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \arcsin (c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx=\frac {b x \arcsin \left (c x\right )}{\sqrt {-c^{2} d f x^{2} + d f} d f} + \frac {a x}{\sqrt {-c^{2} d f x^{2} + d f} d f} - \frac {b \sqrt {\frac {1}{d f}} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{2 \, c d f} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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