\(\int (f+g x) (d-c^2 d x^2)^{5/2} (a+b \arcsin (c x)) \, dx\) [115]

Optimal result

Integrand size = 29, antiderivative size = 475 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}}{96 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {5}{24} d f x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{6} f x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {g \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^2 d}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{32 b c \sqrt {1-c^2 x^2}} \] Output:

1/7*b*d^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-5/32*b*c*d^2*f*x^2 
*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/7*b*c*d^2*g*x^3*(-c^2*d*x^2+d)^ 
(1/2)/(-c^2*x^2+1)^(1/2)+3/35*b*c^3*d^2*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x 
^2+1)^(1/2)-1/49*b*c^5*d^2*g*x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5 
/96*b*d^2*f*(-c^2*x^2+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)/c+1/36*b*d^2*f*(-c^2*x 
^2+1)^(5/2)*(-c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f*x*(-c^2*d*x^2+d)^(1/2)*(a+b* 
arcsin(c*x))+5/24*d*f*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))+1/6*f*x*(-c 
^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))-1/7*g*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin 
(c*x))/c^2/d+5/32*d^2*f*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/b/c/(-c^2 
*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.53 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (11025 a^2 c f+210 a b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )+b^2 c x \left (-245 c^2 f x \left (99-39 c^2 x^2+8 c^4 x^4\right )-288 g \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )\right )+210 b \left (105 a c f+b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )\right ) \arcsin (c x)+11025 b^2 c f \arcsin (c x)^2\right )}{70560 b c^2 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(d^2*Sqrt[d - c^2*d*x^2]*(11025*a^2*c*f + 210*a*b*Sqrt[1 - c^2*x^2]*(48*g* 
(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)) + b^2*c*x*(-24 
5*c^2*f*x*(99 - 39*c^2*x^2 + 8*c^4*x^4) - 288*g*(-35 + 35*c^2*x^2 - 21*c^4 
*x^4 + 5*c^6*x^6)) + 210*b*(105*a*c*f + b*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^ 
2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x] + 11025*b 
^2*c*f*ArcSin[c*x]^2))/(70560*b*c^2*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.54, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {5276, 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.

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

\(\Big \downarrow \) 5276

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

\(\Big \downarrow \) 5262

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{6} f x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5}{24} f x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {5}{16} f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {g \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^2}+\frac {5 f (a+b \arcsin (c x))^2}{32 b c}-\frac {1}{49} b c^5 g x^7+\frac {5}{96} b c^3 f x^4+\frac {3}{35} b c^3 g x^5+\frac {b f \left (1-c^2 x^2\right )^3}{36 c}-\frac {25}{96} b c f x^2-\frac {1}{7} b c g x^3+\frac {b g x}{7 c}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(d^2*Sqrt[d - c^2*d*x^2]*((b*g*x)/(7*c) - (25*b*c*f*x^2)/96 - (b*c*g*x^3)/ 
7 + (5*b*c^3*f*x^4)/96 + (3*b*c^3*g*x^5)/35 - (b*c^5*g*x^7)/49 + (b*f*(1 - 
 c^2*x^2)^3)/(36*c) + (5*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/16 + ( 
5*f*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/24 + (f*x*(1 - c^2*x^2)^(5/ 
2)*(a + b*ArcSin[c*x]))/6 - (g*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7 
*c^2) + (5*f*(a + b*ArcSin[c*x])^2)/(32*b*c)))/Sqrt[1 - c^2*x^2]
 

Defintions of rubi rules used

rule 2009

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

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))
 

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 = 1.30 (sec) , antiderivative size = 1423, normalized size of antiderivative = 3.00

method	result	size
default	\(\text {Expression too large to display}\)	\(1423\)
parts	\(\text {Expression too large to display}\)	\(1423\)

Input:

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

Output:

1/6*a*f*x*(-c^2*d*x^2+d)^(5/2)+5/24*a*f*(-c^2*d*x^2+d)^(3/2)*x*d+5/16*a*f* 
(-c^2*d*x^2+d)^(1/2)*x*d^2+5/16*a*f*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2) 
*x/(-c^2*d*x^2+d)^(1/2))-1/7*a*g*(-c^2*d*x^2+d)^(7/2)/c^2/d+b*(-5/32*(-d*( 
c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^2*d^2*f+1/6 
272*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6-64*I*c^7*x^7*(-c^2*x^2+ 
1)^(1/2)+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x^2-56*I*(-c^ 
2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*g*(I+7*arcsin(c*x))*d 
^2/c^2/(c^2*x^2-1)+1/2304*(-d*(c^2*x^2-1))^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2) 
*x^6*c^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c^5*x^5-18*I*(-c^2* 
x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-6*c*x)*f*(I+6*arcsin( 
c*x))*d^2/c/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*c*x*(-c^2* 
x^2+1)^(1/2)-1)*g*(arcsin(c*x)+I)*d^2/c^2/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1 
))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*g*(arcsin(c*x)-I)*d^2/c^2/(c 
^2*x^2-1)+15/256*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2* 
c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*(-I+2*arcsin(c*x))*d^2*f/c/(c^2*x^2-1) 
+1/128*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3* 
I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*g*(-I+3*arcsin(c*x))*d^2/c^2/(c^2*x^ 
2-1)-1/7840*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*g* 
(11*I+70*arcsin(c*x))*cos(6*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-3/15680*(-d*( 
c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*g*(9*I+35*arcsin...
 

Fricas [F]

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

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

Output:

integral((a*c^4*d^2*g*x^5 + a*c^4*d^2*f*x^4 - 2*a*c^2*d^2*g*x^3 - 2*a*c^2* 
d^2*f*x^2 + a*d^2*g*x + a*d^2*f + (b*c^4*d^2*g*x^5 + b*c^4*d^2*f*x^4 - 2*b 
*c^2*d^2*g*x^3 - 2*b*c^2*d^2*f*x^2 + b*d^2*g*x + b*d^2*f)*arcsin(c*x))*sqr 
t(-c^2*d*x^2 + d), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt 
(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*arcsin(c*x)/c)*a*f - 1/7*(-c^2*d*x^2 + 
 d)^(7/2)*a*g/(c^2*d) + sqrt(d)*integrate((b*c^4*d^2*g*x^5 + b*c^4*d^2*f*x 
^4 - 2*b*c^2*d^2*g*x^3 - 2*b*c^2*d^2*f*x^2 + b*d^2*g*x + b*d^2*f)*sqrt(c*x 
 + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x)
 

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (105 \mathit {asin} \left (c x \right ) a c f +56 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} f \,x^{5}+48 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} g \,x^{6}-182 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f \,x^{3}-144 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} g \,x^{4}+231 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f x +144 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g \,x^{2}-48 \sqrt {-c^{2} x^{2}+1}\, a g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{5}d x \right ) b \,c^{6} g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{4}d x \right ) b \,c^{6} f -672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{3}d x \right ) b \,c^{4} g -672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) b \,c^{4} f +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x d x \right ) b \,c^{2} g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )d x \right ) b \,c^{2} f +48 a g \right )}{336 c^{2}} \] Input:

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

Output:

(sqrt(d)*d**2*(105*asin(c*x)*a*c*f + 56*sqrt( - c**2*x**2 + 1)*a*c**6*f*x* 
*5 + 48*sqrt( - c**2*x**2 + 1)*a*c**6*g*x**6 - 182*sqrt( - c**2*x**2 + 1)* 
a*c**4*f*x**3 - 144*sqrt( - c**2*x**2 + 1)*a*c**4*g*x**4 + 231*sqrt( - c** 
2*x**2 + 1)*a*c**2*f*x + 144*sqrt( - c**2*x**2 + 1)*a*c**2*g*x**2 - 48*sqr 
t( - c**2*x**2 + 1)*a*g + 336*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**5,x) 
*b*c**6*g + 336*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**4,x)*b*c**6*f - 67 
2*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**3,x)*b*c**4*g - 672*int(sqrt( - 
c**2*x**2 + 1)*asin(c*x)*x**2,x)*b*c**4*f + 336*int(sqrt( - c**2*x**2 + 1) 
*asin(c*x)*x,x)*b*c**2*g + 336*int(sqrt( - c**2*x**2 + 1)*asin(c*x),x)*b*c 
**2*f + 48*a*g))/(336*c**2)