\(\int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^5} \, dx\) [38]

Optimal result

Integrand size = 21, antiderivative size = 360 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{12 e \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 g-c^2 d (5 e f-d g)\right ) \sqrt {1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c^3 \left (4 e^2 (e f-4 d g)+c^2 d^2 (11 e f+d g)\right ) \sqrt {1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {(e f-d g) (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}-\frac {g (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}-\frac {b c^3 \left (4 e^4 g-c^2 d e^2 (9 e f-13 d g)-2 c^4 d^3 (3 e f+d g)\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{24 e^2 \left (c^2 d^2-e^2\right )^{7/2}} \] Output:

1/12*b*c*(-d*g+e*f)*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)/(e*x+d)^3-1/24*b*c* 
(4*e^2*g-c^2*d*(-d*g+5*e*f))*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)^2/(e*x+d)^ 
2+1/24*b*c^3*(4*e^2*(-4*d*g+e*f)+c^2*d^2*(d*g+11*e*f))*(-c^2*x^2+1)^(1/2)/ 
e/(c^2*d^2-e^2)^3/(e*x+d)-1/4*(-d*g+e*f)*(a+b*arcsin(c*x))/e^2/(e*x+d)^4-1 
/3*g*(a+b*arcsin(c*x))/e^2/(e*x+d)^3-1/24*b*c^3*(4*e^4*g-c^2*d*e^2*(-13*d* 
g+9*e*f)-2*c^4*d^3*(d*g+3*e*f))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c 
^2*x^2+1)^(1/2))/e^2/(c^2*d^2-e^2)^(7/2)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.16 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\frac {\frac {a (-6 e f+6 d g)}{(d+e x)^4}-\frac {8 a g}{(d+e x)^3}-\frac {b e \sqrt {1-c^2 x^2} \left (c^5 d^2 \left (-2 d^3 g+11 e^3 f x^2+d^2 e (18 f+g x)+d e^2 x (27 f+g x)\right )+2 c e^4 (d g+e (f+2 g x))-c^3 e^2 \left (15 d^3 g-4 e^3 f x^2+5 d^2 e (f+7 g x)+d e^2 x (-3 f+16 g x)\right )\right )}{\left (-c^2 d^2+e^2\right )^3 (d+e x)^3}-\frac {2 b (3 e f+d g+4 e g x) \arcsin (c x)}{(d+e x)^4}+\frac {b c^3 \left (4 e^4 g-2 c^4 d^3 (3 e f+d g)+c^2 d e^2 (-9 e f+13 d g)\right ) \log (d+e x)}{(-c d+e)^3 (c d+e)^3 \sqrt {-c^2 d^2+e^2}}+\frac {b c^3 \left (-4 e^4 g+c^2 d e^2 (9 e f-13 d g)+2 c^4 d^3 (3 e f+d g)\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{(-c d+e)^3 (c d+e)^3 \sqrt {-c^2 d^2+e^2}}}{24 e^2} \] Input:

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

Output:

((a*(-6*e*f + 6*d*g))/(d + e*x)^4 - (8*a*g)/(d + e*x)^3 - (b*e*Sqrt[1 - c^ 
2*x^2]*(c^5*d^2*(-2*d^3*g + 11*e^3*f*x^2 + d^2*e*(18*f + g*x) + d*e^2*x*(2 
7*f + g*x)) + 2*c*e^4*(d*g + e*(f + 2*g*x)) - c^3*e^2*(15*d^3*g - 4*e^3*f* 
x^2 + 5*d^2*e*(f + 7*g*x) + d*e^2*x*(-3*f + 16*g*x))))/((-(c^2*d^2) + e^2) 
^3*(d + e*x)^3) - (2*b*(3*e*f + d*g + 4*e*g*x)*ArcSin[c*x])/(d + e*x)^4 + 
(b*c^3*(4*e^4*g - 2*c^4*d^3*(3*e*f + d*g) + c^2*d*e^2*(-9*e*f + 13*d*g))*L 
og[d + e*x])/((-(c*d) + e)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]) + (b*c^3* 
(-4*e^4*g + c^2*d*e^2*(9*e*f - 13*d*g) + 2*c^4*d^3*(3*e*f + d*g))*Log[e + 
c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c*d) + e)^3*(c*d 
+ e)^3*Sqrt[-(c^2*d^2) + e^2]))/(24*e^2)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5252, 27, 688, 27, 688, 25, 27, 679, 488, 217}

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.

\(\displaystyle \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^5} \, dx\)

\(\Big \downarrow \) 5252

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 679

\(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (4 e^2 g-c^2 d (5 e f-d g)\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^2 \left (\frac {e \sqrt {1-c^2 x^2} \left (c^2 d^2 (d g+11 e f)+4 e^2 (e f-4 d g)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (-2 c^4 d^3 (d g+3 e f)-c^2 d e^2 (9 e f-13 d g)+4 e^4 g\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}\right )}{2 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{12 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}-\frac {g (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (4 e^2 g-c^2 d (5 e f-d g)\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^2 \left (\frac {\left (-2 c^4 d^3 (d g+3 e f)-c^2 d e^2 (9 e f-13 d g)+4 e^4 g\right ) \int \frac {1}{-c^2 d^2+e^2-\frac {\left (d x c^2+e\right )^2}{1-c^2 x^2}}d\frac {d x c^2+e}{\sqrt {1-c^2 x^2}}}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} \left (c^2 d^2 (d g+11 e f)+4 e^2 (e f-4 d g)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{12 e^2}-\frac {(e f-d g) (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}-\frac {g (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {(e f-d g) (a+b \arcsin (c x))}{4 e^2 (d+e x)^4}-\frac {g (a+b \arcsin (c x))}{3 e^2 (d+e x)^3}+\frac {b c \left (\frac {e \sqrt {1-c^2 x^2} (e f-d g)}{\left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (4 e^2 g-c^2 d (5 e f-d g)\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^2 \left (\frac {e \sqrt {1-c^2 x^2} \left (c^2 d^2 (d g+11 e f)+4 e^2 (e f-4 d g)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (-2 c^4 d^3 (d g+3 e f)-c^2 d e^2 (9 e f-13 d g)+4 e^4 g\right )}{\left (c^2 d^2-e^2\right )^{3/2}}\right )}{2 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{12 e^2}\)

Input:

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

Output:

-1/4*((e*f - d*g)*(a + b*ArcSin[c*x]))/(e^2*(d + e*x)^4) - (g*(a + b*ArcSi 
n[c*x]))/(3*e^2*(d + e*x)^3) + (b*c*((e*(e*f - d*g)*Sqrt[1 - c^2*x^2])/((c 
^2*d^2 - e^2)*(d + e*x)^3) - ((e*(4*e^2*g - c^2*d*(5*e*f - d*g))*Sqrt[1 - 
c^2*x^2])/(2*(c^2*d^2 - e^2)*(d + e*x)^2) - (c^2*((e*(4*e^2*(e*f - 4*d*g) 
+ c^2*d^2*(11*e*f + d*g))*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - 
 ((4*e^4*g - c^2*d*e^2*(9*e*f - 13*d*g) - 2*c^4*d^3*(3*e*f + d*g))*ArcTan[ 
(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3 
/2)))/(2*(c^2*d^2 - e^2)))/(c^2*d^2 - e^2)))/(12*e^2)
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 5252

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_ 
Symbol] :> With[{u = IntHide[Px*(d + e*x)^m, 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, m}, x] && PolynomialQ[Px, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1549\) vs. \(2(336)=672\).

Time = 1.06 (sec) , antiderivative size = 1550, normalized size of antiderivative = 4.31

method	result	size
parts	\(\text {Expression too large to display}\)	\(1550\)
derivativedivides	\(\text {Expression too large to display}\)	\(1554\)
default	\(\text {Expression too large to display}\)	\(1554\)

Input:

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

Output:

a*(-1/3*g/e^2/(e*x+d)^3-1/4*(-d*g+e*f)/e^2/(e*x+d)^4)+b/c*(1/4*c^5*arcsin( 
c*x)/e^2/(c*e*x+c*d)^4*d*g-1/4*c^5*arcsin(c*x)/e/(c*e*x+c*d)^4*f-1/3*c^4*a 
rcsin(c*x)*g/e^2/(c*e*x+c*d)^3+1/12*c^4/e^2*(4*g/e^3*(1/2/(c^2*d^2-e^2)*e^ 
2/(c*x+d*c/e)^2*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/ 
2)+3/2*d*c*e/(c^2*d^2-e^2)*(1/(c^2*d^2-e^2)*e^2/(c*x+d*c/e)*(-(c*x+d*c/e)^ 
2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)-d*c*e/(c^2*d^2-e^2)/(-(c^2* 
d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2* 
d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2) 
^(1/2))/(c*x+d*c/e)))+1/2/(c^2*d^2-e^2)*e^2/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln( 
(-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-( 
c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e)))-3 
*c*(d*g-e*f)/e^4*(1/3/(c^2*d^2-e^2)*e^2/(c*x+d*c/e)^3*(-(c*x+d*c/e)^2+2*d* 
c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)+5/3*d*c*e/(c^2*d^2-e^2)*(1/2/(c^2 
*d^2-e^2)*e^2/(c*x+d*c/e)^2*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e 
^2)/e^2)^(1/2)+3/2*d*c*e/(c^2*d^2-e^2)*(1/(c^2*d^2-e^2)*e^2/(c*x+d*c/e)*(- 
(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)-d*c*e/(c^2*d^2- 
e^2)/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/ 
e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d 
^2-e^2)/e^2)^(1/2))/(c*x+d*c/e)))+1/2/(c^2*d^2-e^2)*e^2/(-(c^2*d^2-e^2)/e^ 
2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1407 vs. \(2 (334) = 668\).

Time = 85.38 (sec) , antiderivative size = 2839, normalized size of antiderivative = 7.89 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/48*(16*(a*c^8*d^8*e - 4*a*c^6*d^6*e^3 + 6*a*c^4*d^4*e^5 - 4*a*c^2*d^2* 
e^7 + a*e^9)*g*x + ((3*(2*b*c^7*d^3*e^5 + 3*b*c^5*d*e^7)*f + (2*b*c^7*d^4* 
e^4 - 13*b*c^5*d^2*e^6 - 4*b*c^3*e^8)*g)*x^4 + 4*(3*(2*b*c^7*d^4*e^4 + 3*b 
*c^5*d^2*e^6)*f + (2*b*c^7*d^5*e^3 - 13*b*c^5*d^3*e^5 - 4*b*c^3*d*e^7)*g)* 
x^3 + 6*(3*(2*b*c^7*d^5*e^3 + 3*b*c^5*d^3*e^5)*f + (2*b*c^7*d^6*e^2 - 13*b 
*c^5*d^4*e^4 - 4*b*c^3*d^2*e^6)*g)*x^2 + 3*(2*b*c^7*d^7*e + 3*b*c^5*d^5*e^ 
3)*f + (2*b*c^7*d^8 - 13*b*c^5*d^6*e^2 - 4*b*c^3*d^4*e^4)*g + 4*(3*(2*b*c^ 
7*d^6*e^2 + 3*b*c^5*d^4*e^4)*f + (2*b*c^7*d^7*e - 13*b*c^5*d^5*e^3 - 4*b*c 
^3*d^3*e^5)*g)*x)*sqrt(-c^2*d^2 + e^2)*log((2*c^2*d*e*x - c^2*d^2 + (2*c^4 
*d^2 - c^2*e^2)*x^2 - 2*sqrt(-c^2*d^2 + e^2)*(c^2*d*x + e)*sqrt(-c^2*x^2 + 
 1) + 2*e^2)/(e^2*x^2 + 2*d*e*x + d^2)) + 12*(a*c^8*d^8*e - 4*a*c^6*d^6*e^ 
3 + 6*a*c^4*d^4*e^5 - 4*a*c^2*d^2*e^7 + a*e^9)*f + 4*(a*c^8*d^9 - 4*a*c^6* 
d^7*e^2 + 6*a*c^4*d^5*e^4 - 4*a*c^2*d^3*e^6 + a*d*e^8)*g + 4*(4*(b*c^8*d^8 
*e - 4*b*c^6*d^6*e^3 + 6*b*c^4*d^4*e^5 - 4*b*c^2*d^2*e^7 + b*e^9)*g*x + 3* 
(b*c^8*d^8*e - 4*b*c^6*d^6*e^3 + 6*b*c^4*d^4*e^5 - 4*b*c^2*d^2*e^7 + b*e^9 
)*f + (b*c^8*d^9 - 4*b*c^6*d^7*e^2 + 6*b*c^4*d^5*e^4 - 4*b*c^2*d^3*e^6 + b 
*d*e^8)*g)*arcsin(c*x) - 2*sqrt(-c^2*x^2 + 1)*(((11*b*c^7*d^4*e^5 - 7*b*c^ 
5*d^2*e^7 - 4*b*c^3*e^9)*f + (b*c^7*d^5*e^4 - 17*b*c^5*d^3*e^6 + 16*b*c^3* 
d*e^8)*g)*x^3 + ((38*b*c^7*d^5*e^4 - 31*b*c^5*d^3*e^6 - 7*b*c^3*d*e^8)*f + 
 (2*b*c^7*d^6*e^3 - 53*b*c^5*d^4*e^5 + 55*b*c^3*d^2*e^7 - 4*b*c*e^9)*g)...
 

Sympy [F]

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

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

Output:

Integral((a + b*asin(c*x))*(f + g*x)/(d + e*x)**5, x)
 

Maxima [F]

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

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

Output:

-1/12*(4*e*x + d)*a*g/(e^6*x^4 + 4*d*e^5*x^3 + 6*d^2*e^4*x^2 + 4*d^3*e^3*x 
 + d^4*e^2) - 1/4*a*f/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^2 + 4*d^3*e^2*x 
 + d^4*e) - 1/12*((4*b*e*g*x + 3*b*e*f + b*d*g)*arctan2(c*x, sqrt(c*x + 1) 
*sqrt(-c*x + 1)) + 12*(e^6*x^4 + 4*d*e^5*x^3 + 6*d^2*e^4*x^2 + 4*d^3*e^3*x 
 + d^4*e^2)*integrate(1/12*(4*b*c*e*g*x + 3*b*c*e*f + b*c*d*g)*e^(1/2*log( 
c*x + 1) + 1/2*log(-c*x + 1))/(c^4*e^6*x^8 + 4*c^4*d*e^5*x^7 - 4*c^2*d^3*e 
^3*x^3 - c^2*d^4*e^2*x^2 + (6*c^4*d^2*e^4 - c^2*e^6)*x^6 + 4*(c^4*d^3*e^3 
- c^2*d*e^5)*x^5 + (c^4*d^4*e^2 - 6*c^2*d^2*e^4)*x^4 + (c^2*e^6*x^6 + 4*c^ 
2*d*e^5*x^5 - 4*d^3*e^3*x - d^4*e^2 + (6*c^2*d^2*e^4 - e^6)*x^4 + 4*(c^2*d 
^3*e^3 - d*e^5)*x^3 + (c^2*d^4*e^2 - 6*d^2*e^4)*x^2)*e^(log(c*x + 1) + log 
(-c*x + 1))), x))/(e^6*x^4 + 4*d*e^5*x^3 + 6*d^2*e^4*x^2 + 4*d^3*e^3*x + d 
^4*e^2)
 

Giac [F(-2)]

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

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

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. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d+e\,x\right )}^5} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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