\(\int x^m (d-c^2 d x^2) (a+b \arcsin (c x))^2 \, dx\) [271]

Optimal result

Integrand size = 25, antiderivative size = 371 \[ \int x^m \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=\frac {2 b^2 c^2 d x^{3+m}}{(3+m)^3}-\frac {2 b c d x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m)^2}+\frac {2 d x^{1+m} (a+b \arcsin (c x))^2}{3+4 m+m^2}+\frac {d x^{1+m} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3+m}-\frac {2 b c d x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(2+m) (3+m)^2}-\frac {4 b c d x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{6+11 m+6 m^2+m^3}+\frac {2 b^2 c^2 d x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(2+m) (3+m)^3}+\frac {4 b^2 c^2 d x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(3+m)^2 \left (2+3 m+m^2\right )} \] Output:

2*b^2*c^2*d*x^(3+m)/(3+m)^3-2*b*c*d*x^(2+m)*(-c^2*x^2+1)^(1/2)*(a+b*arcsin 
(c*x))/(3+m)^2+2*d*x^(1+m)*(a+b*arcsin(c*x))^2/(m^2+4*m+3)+d*x^(1+m)*(-c^2 
*x^2+1)*(a+b*arcsin(c*x))^2/(3+m)-2*b*c*d*x^(2+m)*(a+b*arcsin(c*x))*hyperg 
eom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/(2+m)/(3+m)^2-4*b*c*d*x^(2+m)*(a+b*a 
rcsin(c*x))*hypergeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/(m^3+6*m^2+11*m+6) 
+2*b^2*c^2*d*x^(3+m)*hypergeom([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/2 
*m],c^2*x^2)/(2+m)/(3+m)^3+4*b^2*c^2*d*x^(3+m)*hypergeom([1, 3/2+1/2*m, 3/ 
2+1/2*m],[2+1/2*m, 5/2+1/2*m],c^2*x^2)/(3+m)^2/(m^2+3*m+2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.22 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.72 \[ \int x^m \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=d x^{1+m} \left (\frac {(a+b \arcsin (c x))^2}{1+m}-\frac {c^2 x^2 (a+b \arcsin (c x))^2}{3+m}+\frac {2 b c x \left (-\left ((3+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(1+m) (2+m) (3+m)}-\frac {2 b c^3 x^3 \left (-\left ((5+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {5}{2}+\frac {m}{2},\frac {5}{2}+\frac {m}{2};3+\frac {m}{2},\frac {7}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(3+m) (4+m) (5+m)}\right ) \] Input:

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

Output:

d*x^(1 + m)*((a + b*ArcSin[c*x])^2/(1 + m) - (c^2*x^2*(a + b*ArcSin[c*x])^ 
2)/(3 + m) + (2*b*c*x*(-((3 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2 
, (2 + m)/2, (4 + m)/2, c^2*x^2]) + b*c*x*HypergeometricPFQ[{1, 3/2 + m/2, 
 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2]))/((1 + m)*(2 + m)*(3 + m)) - 
(2*b*c^3*x^3*(-((5 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (4 + m) 
/2, (6 + m)/2, c^2*x^2]) + b*c*x*HypergeometricPFQ[{1, 5/2 + m/2, 5/2 + m/ 
2}, {3 + m/2, 7/2 + m/2}, c^2*x^2]))/((3 + m)*(4 + m)*(5 + m)))
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5202, 5138, 5198, 15, 5220}

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.

\(\displaystyle \int x^m \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx\)

\(\Big \downarrow \) 5202

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5198

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5220

\(\displaystyle \frac {2 d \left (\frac {x^{m+1} (a+b \arcsin (c x))^2}{m+1}-\frac {2 b c \left (\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{m+2}-\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{m^2+5 m+6}\right )}{m+1}\right )}{m+3}-\frac {2 b c d \left (\frac {\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{m+2}-\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{m^2+5 m+6}}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}\right )}{m+3}+\frac {d \left (1-c^2 x^2\right ) x^{m+1} (a+b \arcsin (c x))^2}{m+3}\)

Input:

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

Output:

(d*x^(1 + m)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3 + m) + (2*d*((x^(1 + 
m)*(a + b*ArcSin[c*x])^2)/(1 + m) - (2*b*c*((x^(2 + m)*(a + b*ArcSin[c*x]) 
*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(2 + m) - (b*c*x^( 
3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, 
c^2*x^2])/(6 + 5*m + m^2)))/(1 + m)))/(3 + m) - (2*b*c*d*(-((b*c*x^(3 + m) 
)/(3 + m)^2) + (x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3 + m) + 
 ((x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m) 
/2, c^2*x^2])/(2 + m) - (b*c*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/ 
2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(6 + 5*m + m^2))/(3 + m)))/(3 + 
m)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 5138

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 
*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 5198

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
(e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS 
in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 
 - c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x 
] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[ 
(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, 
 f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 5202

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

rule 5220

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_. 
)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2* 
x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, 
 (3 + m)/2, c^2*x^2], x] - Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*S 
imp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m 
/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] 
&& EqQ[c^2*d + e, 0] &&  !IntegerQ[m]
 

Maple [F]

\[\int x^{m} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2}d x\]

Input:

int(x^m*(-c^2*d*x^2+d)*(a+b*arcsin(c*x))^2,x)
 

Output:

int(x^m*(-c^2*d*x^2+d)*(a+b*arcsin(c*x))^2,x)
 

Fricas [F]

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

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

Output:

integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arcsin(c*x)^2 + 
 2*(a*b*c^2*d*x^2 - a*b*d)*arcsin(c*x))*x^m, x)
 

Sympy [F]

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

integrate(x**m*(-c**2*d*x**2+d)*(a+b*asin(c*x))**2,x)
 

Output:

-d*(Integral(-a**2*x**m, x) + Integral(-b**2*x**m*asin(c*x)**2, x) + Integ 
ral(-2*a*b*x**m*asin(c*x), x) + Integral(a**2*c**2*x**2*x**m, x) + Integra 
l(b**2*c**2*x**2*x**m*asin(c*x)**2, x) + Integral(2*a*b*c**2*x**2*x**m*asi 
n(c*x), x))
 

Maxima [F]

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

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

Output:

-a^2*c^2*d*x^(m + 3)/(m + 3) + a^2*d*x^(m + 1)/(m + 1) - (((b^2*c^2*d*m + 
b^2*c^2*d)*x^3 - (b^2*d*m + 3*b^2*d)*x)*x^m*arctan2(c*x, sqrt(c*x + 1)*sqr 
t(-c*x + 1))^2 + (m^2 + 4*m + 3)*integrate(2*(((b^2*c^3*d*m + b^2*c^3*d)*x 
^3 - (b^2*c*d*m + 3*b^2*c*d)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^m*arctan2(c 
*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (a*b*d*m^2 + (a*b*c^4*d*m^2 + 4*a*b*c^ 
4*d*m + 3*a*b*c^4*d)*x^4 + 4*a*b*d*m + 3*a*b*d - 2*(a*b*c^2*d*m^2 + 4*a*b* 
c^2*d*m + 3*a*b*c^2*d)*x^2)*x^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) 
)/((c^2*m^2 + 4*c^2*m + 3*c^2)*x^2 - m^2 - 4*m - 3), x))/(m^2 + 4*m + 3)
 

Giac [F]

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

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

Output:

integrate(-(c^2*d*x^2 - d)*(b*arcsin(c*x) + a)^2*x^m, x)
 

Mupad [F(-1)]

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

int(x^m*(a + b*asin(c*x))^2*(d - c^2*d*x^2),x)
 

Output:

int(x^m*(a + b*asin(c*x))^2*(d - c^2*d*x^2), x)
 

Reduce [F]

\[ \int x^m \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=\frac {d \left (-x^{m} a^{2} c^{2} m \,x^{3}-x^{m} a^{2} c^{2} x^{3}+x^{m} a^{2} m x +3 x^{m} a^{2} x -2 \left (\int x^{m} \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{2} m^{2}-8 \left (\int x^{m} \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{2} m -6 \left (\int x^{m} \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{2}+2 \left (\int x^{m} \mathit {asin} \left (c x \right )d x \right ) a b \,m^{2}+8 \left (\int x^{m} \mathit {asin} \left (c x \right )d x \right ) a b m +6 \left (\int x^{m} \mathit {asin} \left (c x \right )d x \right ) a b -\left (\int x^{m} \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{2} m^{2}-4 \left (\int x^{m} \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{2} m -3 \left (\int x^{m} \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{2}+\left (\int x^{m} \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2} m^{2}+4 \left (\int x^{m} \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2} m +3 \left (\int x^{m} \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2}\right )}{m^{2}+4 m +3} \] Input:

int(x^m*(-c^2*d*x^2+d)*(a+b*asin(c*x))^2,x)
 

Output:

(d*( - x**m*a**2*c**2*m*x**3 - x**m*a**2*c**2*x**3 + x**m*a**2*m*x + 3*x** 
m*a**2*x - 2*int(x**m*asin(c*x)*x**2,x)*a*b*c**2*m**2 - 8*int(x**m*asin(c* 
x)*x**2,x)*a*b*c**2*m - 6*int(x**m*asin(c*x)*x**2,x)*a*b*c**2 + 2*int(x**m 
*asin(c*x),x)*a*b*m**2 + 8*int(x**m*asin(c*x),x)*a*b*m + 6*int(x**m*asin(c 
*x),x)*a*b - int(x**m*asin(c*x)**2*x**2,x)*b**2*c**2*m**2 - 4*int(x**m*asi 
n(c*x)**2*x**2,x)*b**2*c**2*m - 3*int(x**m*asin(c*x)**2*x**2,x)*b**2*c**2 
+ int(x**m*asin(c*x)**2,x)*b**2*m**2 + 4*int(x**m*asin(c*x)**2,x)*b**2*m + 
 3*int(x**m*asin(c*x)**2,x)*b**2))/(m**2 + 4*m + 3)