\(\int x^7 (d-c^2 d x^2)^{3/2} (a+b \arcsin (c x)) \, dx\) [77]

Optimal result

Integrand size = 27, antiderivative size = 375 \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {16 b d x \sqrt {d-c^2 d x^2}}{1155 c^7 \sqrt {1-c^2 x^2}}+\frac {8 b d x^3 \sqrt {d-c^2 d x^2}}{3465 c^5 \sqrt {1-c^2 x^2}}+\frac {2 b d x^5 \sqrt {d-c^2 d x^2}}{1925 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^7 \sqrt {d-c^2 d x^2}}{1617 c \sqrt {1-c^2 x^2}}-\frac {4 b c d x^9 \sqrt {d-c^2 d x^2}}{297 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^{11} \sqrt {d-c^2 d x^2}}{121 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^8 d}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{3 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^8 d^4} \] Output:

16/1155*b*d*x*(-c^2*d*x^2+d)^(1/2)/c^7/(-c^2*x^2+1)^(1/2)+8/3465*b*d*x^3*( 
-c^2*d*x^2+d)^(1/2)/c^5/(-c^2*x^2+1)^(1/2)+2/1925*b*d*x^5*(-c^2*d*x^2+d)^( 
1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/1617*b*d*x^7*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^ 
2+1)^(1/2)-4/297*b*c*d*x^9*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/121*b 
*c^3*d*x^11*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/5*(-c^2*d*x^2+d)^(5/ 
2)*(a+b*arcsin(c*x))/c^8/d+3/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin(c*x))/c^8/ 
d^2-1/3*(-c^2*d*x^2+d)^(9/2)*(a+b*arcsin(c*x))/c^8/d^3+1/11*(-c^2*d*x^2+d) 
^(11/2)*(a+b*arcsin(c*x))/c^8/d^4
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.46 \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (-3465 a \left (1-c^2 x^2\right )^{5/2} \left (16+40 c^2 x^2+70 c^4 x^4+105 c^6 x^6\right )+b c x \left (55440+9240 c^2 x^2+4158 c^4 x^4+2475 c^6 x^6-53900 c^8 x^8+33075 c^{10} x^{10}\right )-3465 b \left (1-c^2 x^2\right )^{5/2} \left (16+40 c^2 x^2+70 c^4 x^4+105 c^6 x^6\right ) \arcsin (c x)\right )}{4002075 c^8 \sqrt {1-c^2 x^2}} \] Input:

Integrate[x^7*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]
 

Output:

(d*Sqrt[d - c^2*d*x^2]*(-3465*a*(1 - c^2*x^2)^(5/2)*(16 + 40*c^2*x^2 + 70* 
c^4*x^4 + 105*c^6*x^6) + b*c*x*(55440 + 9240*c^2*x^2 + 4158*c^4*x^4 + 2475 
*c^6*x^6 - 53900*c^8*x^8 + 33075*c^10*x^10) - 3465*b*(1 - c^2*x^2)^(5/2)*( 
16 + 40*c^2*x^2 + 70*c^4*x^4 + 105*c^6*x^6)*ArcSin[c*x]))/(4002075*c^8*Sqr 
t[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.60, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5194, 27, 2341, 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.

\(\displaystyle \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx\)

\(\Big \downarrow \) 5194

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d \left (1-c^2 x^2\right )^2 \left (105 c^6 x^6+70 c^4 x^4+40 c^2 x^2+16\right )}{1155 c^8}dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^8 d^4}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{3 c^8 d^3}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^8 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^2 \left (105 c^6 x^6+70 c^4 x^4+40 c^2 x^2+16\right )dx}{1155 c^7 \sqrt {1-c^2 x^2}}+\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^8 d^4}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{3 c^8 d^3}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^8 d}\)

\(\Big \downarrow \) 2341

\(\displaystyle \frac {b d \sqrt {d-c^2 d x^2} \int \left (105 c^{10} x^{10}-140 c^8 x^8+5 c^6 x^6+6 c^4 x^4+8 c^2 x^2+16\right )dx}{1155 c^7 \sqrt {1-c^2 x^2}}+\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^8 d^4}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{3 c^8 d^3}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^8 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^8 d^4}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{3 c^8 d^3}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^8 d}+\frac {b d \left (\frac {105 c^{10} x^{11}}{11}-\frac {140 c^8 x^9}{9}+\frac {5 c^6 x^7}{7}+\frac {6 c^4 x^5}{5}+\frac {8 c^2 x^3}{3}+16 x\right ) \sqrt {d-c^2 d x^2}}{1155 c^7 \sqrt {1-c^2 x^2}}\)

Input:

Int[x^7*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]
 

Output:

(b*d*Sqrt[d - c^2*d*x^2]*(16*x + (8*c^2*x^3)/3 + (6*c^4*x^5)/5 + (5*c^6*x^ 
7)/7 - (140*c^8*x^9)/9 + (105*c^10*x^11)/11))/(1155*c^7*Sqrt[1 - c^2*x^2]) 
 - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^8*d) + (3*(d - c^2*d*x 
^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^8*d^2) - ((d - c^2*d*x^2)^(9/2)*(a + b 
*ArcSin[c*x]))/(3*c^8*d^3) + ((d - c^2*d*x^2)^(11/2)*(a + b*ArcSin[c*x]))/ 
(11*c^8*d^4)
 

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 2341

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

rule 5194

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) 
, x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin 
[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[Sim 
plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] 
&& EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 
1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.68

method	result	size
orering	\(\frac {\left (694575 x^{12} c^{12}-1619450 c^{10} x^{10}+904475 c^{8} x^{8}+27720 c^{6} x^{6}+70224 c^{4} x^{4}+517440 c^{2} x^{2}-443520\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )}{4002075 c^{8} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}-\frac {\left (33075 c^{10} x^{10}-53900 c^{8} x^{8}+2475 c^{6} x^{6}+4158 c^{4} x^{4}+9240 c^{2} x^{2}+55440\right ) \left (7 x^{6} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )-3 x^{8} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right ) c^{2} d +\frac {x^{7} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{4002075 x^{6} c^{8} \left (c x -1\right ) \left (c x +1\right )}\)	\(255\)
default	\(\text {Expression too large to display}\)	\(1781\)
parts	\(\text {Expression too large to display}\)	\(1781\)

Input:

int(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
 

Output:

1/4002075*(694575*c^12*x^12-1619450*c^10*x^10+904475*c^8*x^8+27720*c^6*x^6 
+70224*c^4*x^4+517440*c^2*x^2-443520)/c^8/(c*x-1)/(c*x+1)/(c^2*x^2-1)*(-c^ 
2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))-1/4002075/x^6*(33075*c^10*x^10-53900*c^ 
8*x^8+2475*c^6*x^6+4158*c^4*x^4+9240*c^2*x^2+55440)/c^8/(c*x-1)/(c*x+1)*(7 
*x^6*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))-3*x^8*(-c^2*d*x^2+d)^(1/2)*(a+ 
b*arcsin(c*x))*c^2*d+x^7*(-c^2*d*x^2+d)^(3/2)*b*c/(-c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.66 \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=-\frac {{\left (33075 \, b c^{11} d x^{11} - 53900 \, b c^{9} d x^{9} + 2475 \, b c^{7} d x^{7} + 4158 \, b c^{5} d x^{5} + 9240 \, b c^{3} d x^{3} + 55440 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 3465 \, {\left (105 \, a c^{12} d x^{12} - 245 \, a c^{10} d x^{10} + 145 \, a c^{8} d x^{8} + a c^{6} d x^{6} + 2 \, a c^{4} d x^{4} + 8 \, a c^{2} d x^{2} - 16 \, a d + {\left (105 \, b c^{12} d x^{12} - 245 \, b c^{10} d x^{10} + 145 \, b c^{8} d x^{8} + b c^{6} d x^{6} + 2 \, b c^{4} d x^{4} + 8 \, b c^{2} d x^{2} - 16 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{4002075 \, {\left (c^{10} x^{2} - c^{8}\right )}} \] Input:

integrate(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="fricas" 
)
 

Output:

-1/4002075*((33075*b*c^11*d*x^11 - 53900*b*c^9*d*x^9 + 2475*b*c^7*d*x^7 + 
4158*b*c^5*d*x^5 + 9240*b*c^3*d*x^3 + 55440*b*c*d*x)*sqrt(-c^2*d*x^2 + d)* 
sqrt(-c^2*x^2 + 1) + 3465*(105*a*c^12*d*x^12 - 245*a*c^10*d*x^10 + 145*a*c 
^8*d*x^8 + a*c^6*d*x^6 + 2*a*c^4*d*x^4 + 8*a*c^2*d*x^2 - 16*a*d + (105*b*c 
^12*d*x^12 - 245*b*c^10*d*x^10 + 145*b*c^8*d*x^8 + b*c^6*d*x^6 + 2*b*c^4*d 
*x^4 + 8*b*c^2*d*x^2 - 16*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^10*x^ 
2 - c^8)
 

Sympy [F(-1)]

Timed out. \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \] Input:

integrate(x**7*(-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.71 \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=-\frac {1}{1155} \, {\left (\frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{6}}{c^{2} d} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{4} d} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{6} d} + \frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{8} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{1155} \, {\left (\frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{6}}{c^{2} d} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{4} d} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{6} d} + \frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{8} d}\right )} a + \frac {{\left (33075 \, c^{10} d^{\frac {3}{2}} x^{11} - 53900 \, c^{8} d^{\frac {3}{2}} x^{9} + 2475 \, c^{6} d^{\frac {3}{2}} x^{7} + 4158 \, c^{4} d^{\frac {3}{2}} x^{5} + 9240 \, c^{2} d^{\frac {3}{2}} x^{3} + 55440 \, d^{\frac {3}{2}} x\right )} b}{4002075 \, c^{7}} \] Input:

integrate(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="maxima" 
)
 

Output:

-1/1155*(105*(-c^2*d*x^2 + d)^(5/2)*x^6/(c^2*d) + 70*(-c^2*d*x^2 + d)^(5/2 
)*x^4/(c^4*d) + 40*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^6*d) + 16*(-c^2*d*x^2 + d 
)^(5/2)/(c^8*d))*b*arcsin(c*x) - 1/1155*(105*(-c^2*d*x^2 + d)^(5/2)*x^6/(c 
^2*d) + 70*(-c^2*d*x^2 + d)^(5/2)*x^4/(c^4*d) + 40*(-c^2*d*x^2 + d)^(5/2)* 
x^2/(c^6*d) + 16*(-c^2*d*x^2 + d)^(5/2)/(c^8*d))*a + 1/4002075*(33075*c^10 
*d^(3/2)*x^11 - 53900*c^8*d^(3/2)*x^9 + 2475*c^6*d^(3/2)*x^7 + 4158*c^4*d^ 
(3/2)*x^5 + 9240*c^2*d^(3/2)*x^3 + 55440*d^(3/2)*x)*b/c^7
 

Giac [F(-2)]

Exception generated. \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int x^7\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:

int(x^7*(a + b*asin(c*x))*(d - c^2*d*x^2)^(3/2),x)
 

Output:

int(x^7*(a + b*asin(c*x))*(d - c^2*d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {d}\, d \left (-105 \sqrt {-c^{2} x^{2}+1}\, a \,c^{10} x^{10}+140 \sqrt {-c^{2} x^{2}+1}\, a \,c^{8} x^{8}-5 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}-6 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}-8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-16 \sqrt {-c^{2} x^{2}+1}\, a -1155 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{9}d x \right ) b \,c^{10}+1155 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{7}d x \right ) b \,c^{8}\right )}{1155 c^{8}} \] Input:

int(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*asin(c*x)),x)
                                                                                    
                                                                                    
 

Output:

(sqrt(d)*d*( - 105*sqrt( - c**2*x**2 + 1)*a*c**10*x**10 + 140*sqrt( - c**2 
*x**2 + 1)*a*c**8*x**8 - 5*sqrt( - c**2*x**2 + 1)*a*c**6*x**6 - 6*sqrt( - 
c**2*x**2 + 1)*a*c**4*x**4 - 8*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - 16*sqr 
t( - c**2*x**2 + 1)*a - 1155*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**9,x)* 
b*c**10 + 1155*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**7,x)*b*c**8))/(1155 
*c**8)