\(\int \frac {(f+g x) (a+b \arcsin (c x))}{(d-c^2 d x^2)^{5/2}} \, dx\) [129]

Optimal result

Integrand size = 29, antiderivative size = 216 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b (f+g x)}{6 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {\left (g+c^2 f x\right ) (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 f x (a+b \arcsin (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b g \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{6 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b f \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \] Output:

-1/6*b*(g*x+f)/c/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/3*(c^2*f*x+ 
g)*(a+b*arcsin(c*x))/c^2/d/(-c^2*d*x^2+d)^(3/2)+2/3*f*x*(a+b*arcsin(c*x))/ 
d^2/(-c^2*d*x^2+d)^(1/2)-1/6*b*g*(-c^2*x^2+1)^(1/2)*arctanh(c*x)/c^2/d^2/( 
-c^2*d*x^2+d)^(1/2)+1/3*b*f*(-c^2*x^2+1)^(1/2)*ln(-c^2*x^2+1)/c/d^2/(-c^2* 
d*x^2+d)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.47 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.96 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} \left (i b c g \left (1-c^2 x^2\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),1\right )+\sqrt {-c^2} \left (2 a g+6 a c^2 f x-4 a c^4 f x^3-b c f \sqrt {1-c^2 x^2}-b c g x \sqrt {1-c^2 x^2}+2 b \left (g+c^2 f x \left (3-2 c^2 x^2\right )\right ) \arcsin (c x)+2 b c f \left (1-c^2 x^2\right )^{3/2} \log \left (-1+c^2 x^2\right )\right )\right )}{6 \left (-c^2\right )^{3/2} d^3 \left (-1+c^2 x^2\right )^2} \] Input:

Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]
 

Output:

-1/6*(Sqrt[d - c^2*d*x^2]*(I*b*c*g*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh 
[Sqrt[-c^2]*x], 1] + Sqrt[-c^2]*(2*a*g + 6*a*c^2*f*x - 4*a*c^4*f*x^3 - b*c 
*f*Sqrt[1 - c^2*x^2] - b*c*g*x*Sqrt[1 - c^2*x^2] + 2*b*(g + c^2*f*x*(3 - 2 
*c^2*x^2))*ArcSin[c*x] + 2*b*c*f*(1 - c^2*x^2)^(3/2)*Log[-1 + c^2*x^2])))/ 
((-c^2)^(3/2)*d^3*(-1 + c^2*x^2)^2)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.74, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {5276, 5260, 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.

\(\displaystyle \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5276

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5260

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (\frac {\left (c^2 f x+g\right ) (a+b \arcsin (c x))}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 f x (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}-b c \left (\frac {g \text {arctanh}(c x)}{6 c^3}+\frac {f+g x}{6 c^2 \left (1-c^2 x^2\right )}-\frac {f \log \left (1-c^2 x^2\right )}{3 c^2}\right )\right )}{d^2 \sqrt {d-c^2 d x^2}}\)

Input:

Int[((f + g*x)*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]
 

Output:

(Sqrt[1 - c^2*x^2]*(((g + c^2*f*x)*(a + b*ArcSin[c*x]))/(3*c^2*(1 - c^2*x^ 
2)^(3/2)) + (2*f*x*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) - b*c*((f + 
g*x)/(6*c^2*(1 - c^2*x^2)) + (g*ArcTanh[c*x])/(6*c^3) - (f*Log[1 - c^2*x^2 
])/(3*c^2))))/(d^2*Sqrt[d - c^2*d*x^2])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5260

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

rule 5276

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.78 (sec) , antiderivative size = 2237, normalized size of antiderivative = 10.36

method	result	size
default	\(\text {Expression too large to display}\)	\(2237\)
parts	\(\text {Expression too large to display}\)	\(2237\)

Input:

int((g*x+f)*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE 
)
 

Output:

a*(f*(1/3*x/d/(-c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(-c^2*d*x^2+d)^(1/2))+1/3*g/c 
^2/d/(-c^2*d*x^2+d)^(3/2))+4/3*I*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/ 
2)/c/d^3/(c^2*x^2-1)*f*arcsin(c*x)-4/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6 
*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*arcsin(c*x)*(-c^2*x^2+1)*x^4*g+2/3*I*b*( 
-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4*(-c^2*x^ 
2+1)*x^6*g+2/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2 
*x^2-4)*c^4*(-c^2*x^2+1)*x^5*f-5/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x 
^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*(-c^2*x^2+1)*x^4*g-5/3*I*b*(-d*(c^2*x^2-1) 
)^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*(-c^2*x^2+1)*x^3*f-8/3 
*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c*arcs 
in(c*x)*(-c^2*x^2+1)^(1/2)*f-2*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-1 
0*c^4*x^4+11*c^2*x^2-4)*c^3*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^4*f+14/3*I*b* 
(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c*arcsin(c* 
x)*(-c^2*x^2+1)^(1/2)*x^2*f-4/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10 
*c^4*x^4+11*c^2*x^2-4)/c^2*arcsin(c*x)*(-c^2*x^2+1)*g+2/3*b*(-d*(c^2*x^2-1 
))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c*(-c^2*x^2+1)^(1/2)*x*g- 
2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4*arc 
sin(c*x)*x^5*f+4*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6-10*c^4*x^4+11*c^2 
*x^2-4)*c^2*arcsin(c*x)*x^4*g+17/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^6*x^6 
-10*c^4*x^4+11*c^2*x^2-4)*c^2*arcsin(c*x)*x^3*f-2/3*b*(-d*(c^2*x^2-1))^...
 

Fricas [F]

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

integrate((g*x+f)*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fri 
cas")
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(a*g*x + a*f + (b*g*x + b*f)*arcsin(c*x))/( 
c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
 

Sympy [F]

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

integrate((g*x+f)*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral((a + b*asin(c*x))*(f + g*x)/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
 

Maxima [F]

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

integrate((g*x+f)*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="max 
ima")
 

Output:

1/6*b*c*f*(1/(c^4*d^(5/2)*x^2 - c^2*d^(5/2)) + 2*log(c*x + 1)/(c^2*d^(5/2) 
) + 2*log(c*x - 1)/(c^2*d^(5/2))) + 1/3*b*f*(2*x/(sqrt(-c^2*d*x^2 + d)*d^2 
) + x/((-c^2*d*x^2 + d)^(3/2)*d))*arcsin(c*x) + 1/3*a*f*(2*x/(sqrt(-c^2*d* 
x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d)) + b*g*integrate(x*arctan2(c* 
x, sqrt(c*x + 1)*sqrt(-c*x + 1))/((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt 
(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d) + 1/3*a*g/((-c^2*d*x^2 + d)^(3/2)*c^ 
2*d)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((g*x+f)*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="gia 
c")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:

int(((f + g*x)*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2),x)
 

Output:

int(((f + g*x)*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} f \,x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{2} f +3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right ) x}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} g \,x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right ) x}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{2} g +2 a \,c^{4} f \,x^{3}-3 a \,c^{2} f x -a g}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{2} d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int((g*x+f)*(a+b*asin(c*x))/(-c^2*d*x^2+d)^(5/2),x)
 

Output:

(3*sqrt( - c**2*x**2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 
- 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b*c**4*f 
*x**2 - 3*sqrt( - c**2*x**2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c** 
4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b 
*c**2*f + 3*sqrt( - c**2*x**2 + 1)*int((asin(c*x)*x)/(sqrt( - c**2*x**2 + 
1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1) 
),x)*b*c**4*g*x**2 - 3*sqrt( - c**2*x**2 + 1)*int((asin(c*x)*x)/(sqrt( - c 
**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c** 
2*x**2 + 1)),x)*b*c**2*g + 2*a*c**4*f*x**3 - 3*a*c**2*f*x - a*g)/(3*sqrt(d 
)*sqrt( - c**2*x**2 + 1)*c**2*d**2*(c**2*x**2 - 1))