\(\int (f x)^m (d+e x^2)^2 (a+b \arcsin (c x)) \, dx\) [478]

Optimal result

Integrand size = 23, antiderivative size = 293 \[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \sqrt {1-c^2 x^2}}{c^3 f^2 (3+m)^2 (5+m)^2}+\frac {b e^2 (f x)^{4+m} \sqrt {1-c^2 x^2}}{c f^4 (5+m)^2}+\frac {d^2 (f x)^{1+m} (a+b \arcsin (c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \arcsin (c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \arcsin (c x))}{f^5 (5+m)}-\frac {b \left (\frac {c^4 d^2 (3+m) (5+m)}{1+m}+\frac {e (2+m) \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{(3+m) (5+m)}\right ) (f x)^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{c^3 f^2 (2+m) (3+m) (5+m)} \] Output:

b*e*(2*c^2*d*(5+m)^2+e*(m^2+7*m+12))*(f*x)^(2+m)*(-c^2*x^2+1)^(1/2)/c^3/f^ 
2/(3+m)^2/(5+m)^2+b*e^2*(f*x)^(4+m)*(-c^2*x^2+1)^(1/2)/c/f^4/(5+m)^2+d^2*( 
f*x)^(1+m)*(a+b*arcsin(c*x))/f/(1+m)+2*d*e*(f*x)^(3+m)*(a+b*arcsin(c*x))/f 
^3/(3+m)+e^2*(f*x)^(5+m)*(a+b*arcsin(c*x))/f^5/(5+m)-b*(c^4*d^2*(3+m)*(5+m 
)/(1+m)+e*(2+m)*(2*c^2*d*(5+m)^2+e*(m^2+7*m+12))/(3+m)/(5+m))*(f*x)^(2+m)* 
hypergeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/c^3/f^2/(2+m)/(3+m)/(5+m)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.76 \[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=x (f x)^m \left (\frac {a d^2}{1+m}+\frac {2 a d e x^2}{3+m}+\frac {a e^2 x^4}{5+m}+\frac {b d^2 \arcsin (c x)}{1+m}+\frac {2 b d e x^2 \arcsin (c x)}{3+m}+\frac {b e^2 x^4 \arcsin (c x)}{5+m}-\frac {b c d^2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )}{2+3 m+m^2}-\frac {2 b c d e x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},2+\frac {m}{2},3+\frac {m}{2},c^2 x^2\right )}{12+7 m+m^2}-\frac {b c e^2 x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3+\frac {m}{2},4+\frac {m}{2},c^2 x^2\right )}{(5+m) (6+m)}\right ) \] Input:

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

Output:

x*(f*x)^m*((a*d^2)/(1 + m) + (2*a*d*e*x^2)/(3 + m) + (a*e^2*x^4)/(5 + m) + 
 (b*d^2*ArcSin[c*x])/(1 + m) + (2*b*d*e*x^2*ArcSin[c*x])/(3 + m) + (b*e^2* 
x^4*ArcSin[c*x])/(5 + m) - (b*c*d^2*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + 
m/2, c^2*x^2])/(2 + 3*m + m^2) - (2*b*c*d*e*x^3*Hypergeometric2F1[1/2, 2 + 
 m/2, 3 + m/2, c^2*x^2])/(12 + 7*m + m^2) - (b*c*e^2*x^5*Hypergeometric2F1 
[1/2, 3 + m/2, 4 + m/2, c^2*x^2])/((5 + m)*(6 + m)))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5230, 27, 1590, 25, 363, 278}

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.

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

\(\Big \downarrow \) 5230

\(\displaystyle -b c \int \frac {(f x)^{m+1} \left (\frac {e^2 x^4}{m+5}+\frac {2 d e x^2}{m+3}+\frac {d^2}{m+1}\right )}{f \sqrt {1-c^2 x^2}}dx+\frac {d^2 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \int \frac {(f x)^{m+1} \left (\frac {e^2 x^4}{m+5}+\frac {2 d e x^2}{m+3}+\frac {d^2}{m+1}\right )}{\sqrt {1-c^2 x^2}}dx}{f}+\frac {d^2 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 1590

\(\displaystyle -\frac {b c \left (-\frac {\int -\frac {(f x)^{m+1} \left (\frac {c^2 (m+5) d^2}{m+1}+\frac {e \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right ) x^2}{(m+3) (m+5)}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+5)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+4}}{c^2 f^3 (m+5)^2}\right )}{f}+\frac {d^2 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b c \left (\frac {\int \frac {(f x)^{m+1} \left (\frac {c^2 (m+5) d^2}{m+1}+\frac {e \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right ) x^2}{(m+3) (m+5)}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+5)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+4}}{c^2 f^3 (m+5)^2}\right )}{f}+\frac {d^2 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 363

\(\displaystyle -\frac {b c \left (\frac {\left (\frac {c^2 d^2 (m+5)}{m+1}+\frac {e (m+2) \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^2 (m+3)^2 (m+5)}\right ) \int \frac {(f x)^{m+1}}{\sqrt {1-c^2 x^2}}dx-\frac {e \sqrt {1-c^2 x^2} (f x)^{m+2} \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^2 f (m+3)^2 (m+5)}}{c^2 (m+5)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+4}}{c^2 f^3 (m+5)^2}\right )}{f}+\frac {d^2 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {d^2 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}-\frac {b c \left (\frac {\frac {(f x)^{m+2} \left (\frac {c^2 d^2 (m+5)}{m+1}+\frac {e (m+2) \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^2 (m+3)^2 (m+5)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{f (m+2)}-\frac {e \sqrt {1-c^2 x^2} (f x)^{m+2} \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^2 f (m+3)^2 (m+5)}}{c^2 (m+5)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+4}}{c^2 f^3 (m+5)^2}\right )}{f}\)

Input:

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

Output:

(d^2*(f*x)^(1 + m)*(a + b*ArcSin[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m) 
*(a + b*ArcSin[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcSin[c*x 
]))/(f^5*(5 + m)) - (b*c*(-((e^2*(f*x)^(4 + m)*Sqrt[1 - c^2*x^2])/(c^2*f^3 
*(5 + m)^2)) + (-((e*(2*c^2*d*(5 + m)^2 + e*(12 + 7*m + m^2))*(f*x)^(2 + m 
)*Sqrt[1 - c^2*x^2])/(c^2*f*(3 + m)^2*(5 + m))) + (((c^2*d^2*(5 + m))/(1 + 
 m) + (e*(2 + m)*(2*c^2*d*(5 + m)^2 + e*(12 + 7*m + m^2)))/(c^2*(3 + m)^2* 
(5 + m)))*(f*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x 
^2])/(f*(2 + m)))/(c^2*(5 + m))))/f
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 1590

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ 
(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q 
 + 1))   Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + 
b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], 
x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 
0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
 

rule 5230

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d 
^2)*arcsin(c*x))*(f*x)^m, x)
 

Sympy [F]

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

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

Output:

Integral((f*x)**m*(a + b*asin(c*x))*(d + e*x**2)**2, x)
 

Maxima [F]

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

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

Output:

a*e^2*f^m*x^5*x^m/(m + 5) + 2*a*d*e*f^m*x^3*x^m/(m + 3) + (f*x)^(m + 1)*a* 
d^2/(f*(m + 1)) + (((b*e^2*f^m*m^2 + 4*b*e^2*f^m*m + 3*b*e^2*f^m)*x^5 + 2* 
(b*d*e*f^m*m^2 + 6*b*d*e*f^m*m + 5*b*d*e*f^m)*x^3 + (b*d^2*f^m*m^2 + 8*b*d 
^2*f^m*m + 15*b*d^2*f^m)*x)*x^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) 
 + (m^3 + 9*m^2 + 23*m + 15)*integrate(-((b*c*e^2*f^m*m^2 + 4*b*c*e^2*f^m* 
m + 3*b*c*e^2*f^m)*x^5 + 2*(b*c*d*e*f^m*m^2 + 6*b*c*d*e*f^m*m + 5*b*c*d*e* 
f^m)*x^3 + (b*c*d^2*f^m*m^2 + 8*b*c*d^2*f^m*m + 15*b*c*d^2*f^m)*x)*sqrt(c* 
x + 1)*sqrt(-c*x + 1)*x^m/(m^3 - (c^2*m^3 + 9*c^2*m^2 + 23*c^2*m + 15*c^2) 
*x^2 + 9*m^2 + 23*m + 15), x))/(m^3 + 9*m^2 + 23*m + 15)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {f^{m} \left (x^{m} a \,d^{2} m^{2} x +8 x^{m} a \,d^{2} m x +15 x^{m} a \,d^{2} x +2 x^{m} a d e \,m^{2} x^{3}+12 x^{m} a d e m \,x^{3}+10 x^{m} a d e \,x^{3}+x^{m} a \,e^{2} m^{2} x^{5}+4 x^{m} a \,e^{2} m \,x^{5}+3 x^{m} a \,e^{2} x^{5}+\left (\int x^{m} \mathit {asin} \left (c x \right ) x^{4}d x \right ) b \,e^{2} m^{3}+9 \left (\int x^{m} \mathit {asin} \left (c x \right ) x^{4}d x \right ) b \,e^{2} m^{2}+23 \left (\int x^{m} \mathit {asin} \left (c x \right ) x^{4}d x \right ) b \,e^{2} m +15 \left (\int x^{m} \mathit {asin} \left (c x \right ) x^{4}d x \right ) b \,e^{2}+2 \left (\int x^{m} \mathit {asin} \left (c x \right ) x^{2}d x \right ) b d e \,m^{3}+18 \left (\int x^{m} \mathit {asin} \left (c x \right ) x^{2}d x \right ) b d e \,m^{2}+46 \left (\int x^{m} \mathit {asin} \left (c x \right ) x^{2}d x \right ) b d e m +30 \left (\int x^{m} \mathit {asin} \left (c x \right ) x^{2}d x \right ) b d e +\left (\int x^{m} \mathit {asin} \left (c x \right )d x \right ) b \,d^{2} m^{3}+9 \left (\int x^{m} \mathit {asin} \left (c x \right )d x \right ) b \,d^{2} m^{2}+23 \left (\int x^{m} \mathit {asin} \left (c x \right )d x \right ) b \,d^{2} m +15 \left (\int x^{m} \mathit {asin} \left (c x \right )d x \right ) b \,d^{2}\right )}{m^{3}+9 m^{2}+23 m +15} \] Input:

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

Output:

(f**m*(x**m*a*d**2*m**2*x + 8*x**m*a*d**2*m*x + 15*x**m*a*d**2*x + 2*x**m* 
a*d*e*m**2*x**3 + 12*x**m*a*d*e*m*x**3 + 10*x**m*a*d*e*x**3 + x**m*a*e**2* 
m**2*x**5 + 4*x**m*a*e**2*m*x**5 + 3*x**m*a*e**2*x**5 + int(x**m*asin(c*x) 
*x**4,x)*b*e**2*m**3 + 9*int(x**m*asin(c*x)*x**4,x)*b*e**2*m**2 + 23*int(x 
**m*asin(c*x)*x**4,x)*b*e**2*m + 15*int(x**m*asin(c*x)*x**4,x)*b*e**2 + 2* 
int(x**m*asin(c*x)*x**2,x)*b*d*e*m**3 + 18*int(x**m*asin(c*x)*x**2,x)*b*d* 
e*m**2 + 46*int(x**m*asin(c*x)*x**2,x)*b*d*e*m + 30*int(x**m*asin(c*x)*x** 
2,x)*b*d*e + int(x**m*asin(c*x),x)*b*d**2*m**3 + 9*int(x**m*asin(c*x),x)*b 
*d**2*m**2 + 23*int(x**m*asin(c*x),x)*b*d**2*m + 15*int(x**m*asin(c*x),x)* 
b*d**2))/(m**3 + 9*m**2 + 23*m + 15)