\(\int (f x)^m (d+e x^2)^3 (a+b \arcsin (c x)) \, dx\) [477]

Optimal result

Integrand size = 23, antiderivative size = 484 \[ \int (f x)^m \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {b e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) (f x)^{2+m} \sqrt {1-c^2 x^2}}{c^5 f^2 (3+m)^2 (5+m)^2 (7+m)^2}+\frac {b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \sqrt {1-c^2 x^2}}{c^3 f^4 (5+m)^2 (7+m)^2}+\frac {b e^3 (f x)^{6+m} \sqrt {1-c^2 x^2}}{c f^6 (7+m)^2}+\frac {d^3 (f x)^{1+m} (a+b \arcsin (c x))}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} (a+b \arcsin (c x))}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} (a+b \arcsin (c x))}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} (a+b \arcsin (c x))}{f^7 (7+m)}-\frac {b \left (\frac {c^6 d^3 (3+m) (5+m) (7+m)}{1+m}+\frac {e (2+m) \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right )}{(3+m) (5+m) (7+m)}\right ) (f x)^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{c^5 f^2 (2+m) (3+m) (5+m) (7+m)} \] Output:

b*e*(3*c^2*d*e*(7+m)^2*(m^2+7*m+12)+3*c^4*d^2*(m^2+12*m+35)^2+e^2*(m^4+18* 
m^3+119*m^2+342*m+360))*(f*x)^(2+m)*(-c^2*x^2+1)^(1/2)/c^5/f^2/(3+m)^2/(5+ 
m)^2/(7+m)^2+b*e^2*(3*c^2*d*(7+m)^2+e*(m^2+11*m+30))*(f*x)^(4+m)*(-c^2*x^2 
+1)^(1/2)/c^3/f^4/(5+m)^2/(7+m)^2+b*e^3*(f*x)^(6+m)*(-c^2*x^2+1)^(1/2)/c/f 
^6/(7+m)^2+d^3*(f*x)^(1+m)*(a+b*arcsin(c*x))/f/(1+m)+3*d^2*e*(f*x)^(3+m)*( 
a+b*arcsin(c*x))/f^3/(3+m)+3*d*e^2*(f*x)^(5+m)*(a+b*arcsin(c*x))/f^5/(5+m) 
+e^3*(f*x)^(7+m)*(a+b*arcsin(c*x))/f^7/(7+m)-b*(c^6*d^3*(3+m)*(5+m)*(7+m)/ 
(1+m)+e*(2+m)*(3*c^2*d*e*(7+m)^2*(m^2+7*m+12)+3*c^4*d^2*(m^2+12*m+35)^2+e^ 
2*(m^4+18*m^3+119*m^2+342*m+360))/(3+m)/(5+m)/(7+m))*(f*x)^(2+m)*hypergeom 
([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/c^5/f^2/(2+m)/(3+m)/(5+m)/(7+m)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.64 \[ \int (f x)^m \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=x (f x)^m \left (\frac {a d^3}{1+m}+\frac {3 a d^2 e x^2}{3+m}+\frac {3 a d e^2 x^4}{5+m}+\frac {a e^3 x^6}{7+m}+\frac {b d^3 \arcsin (c x)}{1+m}+\frac {3 b d^2 e x^2 \arcsin (c x)}{3+m}+\frac {3 b d e^2 x^4 \arcsin (c x)}{5+m}+\frac {b e^3 x^6 \arcsin (c x)}{7+m}-\frac {b c d^3 x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )}{2+3 m+m^2}-\frac {3 b c d^2 e x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},2+\frac {m}{2},3+\frac {m}{2},c^2 x^2\right )}{12+7 m+m^2}-\frac {3 b c d e^2 x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3+\frac {m}{2},4+\frac {m}{2},c^2 x^2\right )}{(5+m) (6+m)}-\frac {b c e^3 x^7 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4+\frac {m}{2},5+\frac {m}{2},c^2 x^2\right )}{(7+m) (8+m)}\right ) \] Input:

Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]
 

Output:

x*(f*x)^m*((a*d^3)/(1 + m) + (3*a*d^2*e*x^2)/(3 + m) + (3*a*d*e^2*x^4)/(5 
+ m) + (a*e^3*x^6)/(7 + m) + (b*d^3*ArcSin[c*x])/(1 + m) + (3*b*d^2*e*x^2* 
ArcSin[c*x])/(3 + m) + (3*b*d*e^2*x^4*ArcSin[c*x])/(5 + m) + (b*e^3*x^6*Ar 
cSin[c*x])/(7 + m) - (b*c*d^3*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, c 
^2*x^2])/(2 + 3*m + m^2) - (3*b*c*d^2*e*x^3*Hypergeometric2F1[1/2, 2 + m/2 
, 3 + m/2, c^2*x^2])/(12 + 7*m + m^2) - (3*b*c*d*e^2*x^5*Hypergeometric2F1 
[1/2, 3 + m/2, 4 + m/2, c^2*x^2])/((5 + m)*(6 + m)) - (b*c*e^3*x^7*Hyperge 
ometric2F1[1/2, 4 + m/2, 5 + m/2, c^2*x^2])/((7 + m)*(8 + m)))
 

Rubi [A] (verified)

Time = 2.23 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5230, 27, 2340, 25, 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.

\(\displaystyle \int \left (d+e x^2\right )^3 (f x)^m (a+b \arcsin (c x)) \, dx\)

\(\Big \downarrow \) 5230

\(\displaystyle -b c \int \frac {(f x)^{m+1} \left (\frac {e^3 x^6}{m+7}+\frac {3 d e^2 x^4}{m+5}+\frac {3 d^2 e x^2}{m+3}+\frac {d^3}{m+1}\right )}{f \sqrt {1-c^2 x^2}}dx+\frac {d^3 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \arcsin (c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \int \frac {(f x)^{m+1} \left (\frac {e^3 x^6}{m+7}+\frac {3 d e^2 x^4}{m+5}+\frac {3 d^2 e x^2}{m+3}+\frac {d^3}{m+1}\right )}{\sqrt {1-c^2 x^2}}dx}{f}+\frac {d^3 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \arcsin (c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 2340

\(\displaystyle -\frac {b c \left (-\frac {\int -\frac {(f x)^{m+1} \left (\frac {e^2 \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right ) x^4}{(m+5) (m+7)}+\frac {3 c^2 d^2 e (m+7) x^2}{m+3}+\frac {c^2 d^3 (m+7)}{m+1}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+7)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+6}}{c^2 f^5 (m+7)^2}\right )}{f}+\frac {d^3 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \arcsin (c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b c \left (\frac {\int \frac {(f x)^{m+1} \left (\frac {e^2 \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right ) x^4}{(m+5) (m+7)}+\frac {3 c^2 d^2 e (m+7) x^2}{m+3}+\frac {c^2 d^3 (m+7)}{m+1}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+7)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+6}}{c^2 f^5 (m+7)^2}\right )}{f}+\frac {d^3 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \arcsin (c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 1590

\(\displaystyle -\frac {b c \left (\frac {-\frac {\int -\frac {(f x)^{m+1} \left (\frac {d^3 (m+5) (m+7) c^4}{m+1}+\frac {e \left (3 d^2 \left (m^2+12 m+35\right )^2 c^4+3 d e (m+7)^2 \left (m^2+7 m+12\right ) c^2+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right ) x^2}{(m+3) (m+5) (m+7)}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+5)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^2 f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+6}}{c^2 f^5 (m+7)^2}\right )}{f}+\frac {d^3 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \arcsin (c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b c \left (\frac {\frac {\int \frac {(f x)^{m+1} \left (\frac {d^3 (m+5) (m+7) c^4}{m+1}+\frac {e \left (3 d^2 \left (m^2+12 m+35\right )^2 c^4+3 d e (m+7)^2 \left (m^2+7 m+12\right ) c^2+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right ) x^2}{(m+3) (m+5) (m+7)}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+5)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^2 f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+6}}{c^2 f^5 (m+7)^2}\right )}{f}+\frac {d^3 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \arcsin (c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 363

\(\displaystyle -\frac {b c \left (\frac {\frac {\left (\frac {c^4 d^3 (m+5) (m+7)}{m+1}+\frac {e (m+2) \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^2 (m+3)^2 (m+5) (m+7)}\right ) \int \frac {(f x)^{m+1}}{\sqrt {1-c^2 x^2}}dx-\frac {e \sqrt {1-c^2 x^2} (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^2 f (m+3)^2 (m+5) (m+7)}}{c^2 (m+5)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^2 f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+6}}{c^2 f^5 (m+7)^2}\right )}{f}+\frac {d^3 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \arcsin (c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {d^3 (f x)^{m+1} (a+b \arcsin (c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \arcsin (c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \arcsin (c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \arcsin (c x))}{f^7 (m+7)}-\frac {b c \left (\frac {\frac {\frac {(f x)^{m+2} \left (\frac {c^4 d^3 (m+5) (m+7)}{m+1}+\frac {e (m+2) \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^2 (m+3)^2 (m+5) (m+7)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{f (m+2)}-\frac {e \sqrt {1-c^2 x^2} (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^2 f (m+3)^2 (m+5) (m+7)}}{c^2 (m+5)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^2 f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+6}}{c^2 f^5 (m+7)^2}\right )}{f}\)

Input:

Int[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]
 

Output:

(d^3*(f*x)^(1 + m)*(a + b*ArcSin[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + 
m)*(a + b*ArcSin[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcS 
in[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcSin[c*x]))/(f^7*(7 
+ m)) - (b*c*(-((e^3*(f*x)^(6 + m)*Sqrt[1 - c^2*x^2])/(c^2*f^5*(7 + m)^2)) 
 + (-((e^2*(3*c^2*d*(7 + m)^2 + e*(30 + 11*m + m^2))*(f*x)^(4 + m)*Sqrt[1 
- c^2*x^2])/(c^2*f^3*(5 + m)^2*(7 + m))) + (-((e*(3*c^2*d*e*(7 + m)^2*(12 
+ 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 
+ 18*m^3 + m^4))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2])/(c^2*f*(3 + m)^2*(5 + m) 
*(7 + m))) + (((c^4*d^3*(5 + m)*(7 + m))/(1 + m) + (e*(2 + m)*(3*c^2*d*e*( 
7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342 
*m + 119*m^2 + 18*m^3 + m^4)))/(c^2*(3 + m)^2*(5 + m)*(7 + 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)))/(c^2*(7 + 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 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

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]

\[\int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )d x\]

Input:

int((f*x)^m*(e*x^2+d)^3*(a+b*arcsin(c*x)),x)
 

Output:

int((f*x)^m*(e*x^2+d)^3*(a+b*arcsin(c*x)),x)
 

Fricas [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="fricas")
 

Output:

integral((a*e^3*x^6 + 3*a*d*e^2*x^4 + 3*a*d^2*e*x^2 + a*d^3 + (b*e^3*x^6 + 
 3*b*d*e^2*x^4 + 3*b*d^2*e*x^2 + b*d^3)*arcsin(c*x))*(f*x)^m, x)
 

Sympy [F(-1)]

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

integrate((f*x)**m*(e*x**2+d)**3*(a+b*asin(c*x)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="maxima")
 

Output:

a*e^3*f^m*x^7*x^m/(m + 7) + 3*a*d*e^2*f^m*x^5*x^m/(m + 5) + 3*a*d^2*e*f^m* 
x^3*x^m/(m + 3) + (f*x)^(m + 1)*a*d^3/(f*(m + 1)) + (((b*e^3*f^m*m^3 + 9*b 
*e^3*f^m*m^2 + 23*b*e^3*f^m*m + 15*b*e^3*f^m)*x^7 + 3*(b*d*e^2*f^m*m^3 + 1 
1*b*d*e^2*f^m*m^2 + 31*b*d*e^2*f^m*m + 21*b*d*e^2*f^m)*x^5 + 3*(b*d^2*e*f^ 
m*m^3 + 13*b*d^2*e*f^m*m^2 + 47*b*d^2*e*f^m*m + 35*b*d^2*e*f^m)*x^3 + (b*d 
^3*f^m*m^3 + 15*b*d^3*f^m*m^2 + 71*b*d^3*f^m*m + 105*b*d^3*f^m)*x)*x^m*arc 
tan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (m^4 + 16*m^3 + 86*m^2 + 176*m + 
 105)*integrate(-((b*c*e^3*f^m*m^3 + 9*b*c*e^3*f^m*m^2 + 23*b*c*e^3*f^m*m 
+ 15*b*c*e^3*f^m)*x^7 + 3*(b*c*d*e^2*f^m*m^3 + 11*b*c*d*e^2*f^m*m^2 + 31*b 
*c*d*e^2*f^m*m + 21*b*c*d*e^2*f^m)*x^5 + 3*(b*c*d^2*e*f^m*m^3 + 13*b*c*d^2 
*e*f^m*m^2 + 47*b*c*d^2*e*f^m*m + 35*b*c*d^2*e*f^m)*x^3 + (b*c*d^3*f^m*m^3 
 + 15*b*c*d^3*f^m*m^2 + 71*b*c*d^3*f^m*m + 105*b*c*d^3*f^m)*x)*sqrt(c*x + 
1)*sqrt(-c*x + 1)*x^m/(m^4 + 16*m^3 - (c^2*m^4 + 16*c^2*m^3 + 86*c^2*m^2 + 
 176*c^2*m + 105*c^2)*x^2 + 86*m^2 + 176*m + 105), x))/(m^4 + 16*m^3 + 86* 
m^2 + 176*m + 105)
 

Giac [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="giac")
 

Output:

integrate((e*x^2 + d)^3*(b*arcsin(c*x) + a)*(f*x)^m, x)
 

Mupad [F(-1)]

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

int((a + b*asin(c*x))*(f*x)^m*(d + e*x^2)^3,x)
 

Output:

int((a + b*asin(c*x))*(f*x)^m*(d + e*x^2)^3, x)
 

Reduce [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx =\text {Too large to display} \] Input:

int((f*x)^m*(e*x^2+d)^3*(a+b*asin(c*x)),x)
 

Output:

(f**m*(x**m*a*d**3*m**3*x + 15*x**m*a*d**3*m**2*x + 71*x**m*a*d**3*m*x + 1 
05*x**m*a*d**3*x + 3*x**m*a*d**2*e*m**3*x**3 + 39*x**m*a*d**2*e*m**2*x**3 
+ 141*x**m*a*d**2*e*m*x**3 + 105*x**m*a*d**2*e*x**3 + 3*x**m*a*d*e**2*m**3 
*x**5 + 33*x**m*a*d*e**2*m**2*x**5 + 93*x**m*a*d*e**2*m*x**5 + 63*x**m*a*d 
*e**2*x**5 + x**m*a*e**3*m**3*x**7 + 9*x**m*a*e**3*m**2*x**7 + 23*x**m*a*e 
**3*m*x**7 + 15*x**m*a*e**3*x**7 + int(x**m*asin(c*x)*x**6,x)*b*e**3*m**4 
+ 16*int(x**m*asin(c*x)*x**6,x)*b*e**3*m**3 + 86*int(x**m*asin(c*x)*x**6,x 
)*b*e**3*m**2 + 176*int(x**m*asin(c*x)*x**6,x)*b*e**3*m + 105*int(x**m*asi 
n(c*x)*x**6,x)*b*e**3 + 3*int(x**m*asin(c*x)*x**4,x)*b*d*e**2*m**4 + 48*in 
t(x**m*asin(c*x)*x**4,x)*b*d*e**2*m**3 + 258*int(x**m*asin(c*x)*x**4,x)*b* 
d*e**2*m**2 + 528*int(x**m*asin(c*x)*x**4,x)*b*d*e**2*m + 315*int(x**m*asi 
n(c*x)*x**4,x)*b*d*e**2 + 3*int(x**m*asin(c*x)*x**2,x)*b*d**2*e*m**4 + 48* 
int(x**m*asin(c*x)*x**2,x)*b*d**2*e*m**3 + 258*int(x**m*asin(c*x)*x**2,x)* 
b*d**2*e*m**2 + 528*int(x**m*asin(c*x)*x**2,x)*b*d**2*e*m + 315*int(x**m*a 
sin(c*x)*x**2,x)*b*d**2*e + int(x**m*asin(c*x),x)*b*d**3*m**4 + 16*int(x** 
m*asin(c*x),x)*b*d**3*m**3 + 86*int(x**m*asin(c*x),x)*b*d**3*m**2 + 176*in 
t(x**m*asin(c*x),x)*b*d**3*m + 105*int(x**m*asin(c*x),x)*b*d**3))/(m**4 + 
16*m**3 + 86*m**2 + 176*m + 105)