\(\int \frac {(f+g x+h x^2) (a+b \arcsin (c x))}{(d+e x)^5} \, dx\) [168]

Optimal result

Integrand size = 26, antiderivative size = 470 \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) (a+b \arcsin (c x))}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{3 e^3 (d+e x)^3}-\frac {h (a+b \arcsin (c x))}{2 e^3 (d+e x)^2}-\frac {b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{24 e^3 \left (c^2 d^2-e^2\right )^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.75 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.22 \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=-\frac {\frac {6 a \left (e^2 f-d e g+d^2 h\right )}{(d+e x)^4}+\frac {8 a (e g-2 d h)}{(d+e x)^3}+\frac {12 a h}{(d+e x)^2}+\frac {b c e \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (-2 d^4 h+11 e^4 f x^2+d e^3 x (27 f+g x)-d^3 e (2 g+5 h x)+d^2 e^2 (18 f+x (g-h x))\right )+2 e^4 \left (3 d^2 h+d e (g+8 h x)+e^2 (f+2 x (g+3 h x))\right )+c^2 e^2 \left (11 d^4 h+4 e^4 f x^2+d e^3 x (3 f-16 g x)+d^3 e (-15 g+19 h x)+d^2 e^2 (-5 f+x (-35 g+4 h x))\right )\right )}{\left (-c^2 d^2+e^2\right )^3 (d+e x)^3}+\frac {2 b \left (d^2 h+d e (g+4 h x)+e^2 \left (3 f+4 g x+6 h x^2\right )\right ) \arcsin (c x)}{(d+e x)^4}-\frac {b c^3 \left (-4 e^4 (e g-5 d h)+c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )+2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\right ) \log (d+e x)}{(c d-e)^3 (c d+e)^3 \sqrt {-c^2 d^2+e^2}}+\frac {b c^3 \left (-4 e^4 (e g-5 d h)+c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )+2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{(c d-e)^3 (c d+e)^3 \sqrt {-c^2 d^2+e^2}}}{24 e^3} \] Input:

Integrate[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^5,x]
 

Output:

-1/24*((6*a*(e^2*f - d*e*g + d^2*h))/(d + e*x)^4 + (8*a*(e*g - 2*d*h))/(d 
+ e*x)^3 + (12*a*h)/(d + e*x)^2 + (b*c*e*Sqrt[1 - c^2*x^2]*(c^4*d^2*(-2*d^ 
4*h + 11*e^4*f*x^2 + d*e^3*x*(27*f + g*x) - d^3*e*(2*g + 5*h*x) + d^2*e^2* 
(18*f + x*(g - h*x))) + 2*e^4*(3*d^2*h + d*e*(g + 8*h*x) + e^2*(f + 2*x*(g 
 + 3*h*x))) + c^2*e^2*(11*d^4*h + 4*e^4*f*x^2 + d*e^3*x*(3*f - 16*g*x) + d 
^3*e*(-15*g + 19*h*x) + d^2*e^2*(-5*f + x*(-35*g + 4*h*x)))))/((-(c^2*d^2) 
 + e^2)^3*(d + e*x)^3) + (2*b*(d^2*h + d*e*(g + 4*h*x) + e^2*(3*f + 4*g*x 
+ 6*h*x^2))*ArcSin[c*x])/(d + e*x)^4 - (b*c^3*(-4*e^4*(e*g - 5*d*h) + c^2* 
d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) + 2*c^4*d^3*(3*e^2*f + d*e*g + d^2*h) 
)*Log[d + e*x])/((c*d - e)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]) + (b*c^3* 
(-4*e^4*(e*g - 5*d*h) + c^2*d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) + 2*c^4*d 
^3*(3*e^2*f + d*e*g + d^2*h))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqr 
t[1 - c^2*x^2]])/((c*d - e)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]))/e^3
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5252, 27, 2182, 27, 688, 25, 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.

\(\displaystyle \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^5} \, dx\)

\(\Big \downarrow \) 5252

\(\displaystyle -b c \int -\frac {h d^2+e g d+6 e^2 h x^2+3 e^2 f+4 e (e g+d h) x}{12 e^3 (d+e x)^4 \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{3 e^3 (d+e x)^3}-\frac {h (a+b \arcsin (c x))}{2 e^3 (d+e x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c \int \frac {h d^2+e g d+6 e^2 h x^2+3 e^2 f+4 e (e g+d h) x}{(d+e x)^4 \sqrt {1-c^2 x^2}}dx}{12 e^3}-\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{3 e^3 (d+e x)^3}-\frac {h (a+b \arcsin (c x))}{2 e^3 (d+e x)^2}\)

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

\(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{\left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\frac {\int -\frac {e \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 h d^2-e g d+5 e^2 f\right )\right ) x c^2+2 \left (d^2 \left (h d^2+e g d+3 e^2 f\right ) c^4+2 e^2 \left (-h d^2-3 e g d+e^2 f\right ) c^2+6 e^4 h\right )}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 \left (c^2 d^2-e^2\right )}+\frac {e \sqrt {1-c^2 x^2} \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-d e g+5 e^2 f\right )\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}}{c^2 d^2-e^2}\right )}{12 e^3}-\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{3 e^3 (d+e x)^3}-\frac {h (a+b \arcsin (c x))}{2 e^3 (d+e x)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{\left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-d e g+5 e^2 f\right )\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\int \frac {e \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 h d^2-e g d+5 e^2 f\right )\right ) x c^2+2 \left (d^2 \left (h d^2+e g d+3 e^2 f\right ) c^4+2 e^2 \left (-h d^2-3 e g d+e^2 f\right ) c^2+6 e^4 h\right )}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{12 e^3}-\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{3 e^3 (d+e x)^3}-\frac {h (a+b \arcsin (c x))}{2 e^3 (d+e x)^2}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{\left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-d e g+5 e^2 f\right )\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (d^2 (-h)+d e g+11 e^2 f\right )+4 c^2 e^2 \left (d^2 h-4 d e g+e^2 f\right )+12 e^4 h\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {c^2 \left (-2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )-c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )+4 e^4 (e g-5 d h)\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}}{2 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{12 e^3}-\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{3 e^3 (d+e x)^3}-\frac {h (a+b \arcsin (c x))}{2 e^3 (d+e x)^2}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{\left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-d e g+5 e^2 f\right )\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\frac {c^2 \left (-2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )-c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )+4 e^4 (e g-5 d h)\right ) \int \frac {1}{-c^2 d^2+e^2-\frac {\left (d x c^2+e\right )^2}{1-c^2 x^2}}d\frac {d x c^2+e}{\sqrt {1-c^2 x^2}}}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (d^2 (-h)+d e g+11 e^2 f\right )+4 c^2 e^2 \left (d^2 h-4 d e g+e^2 f\right )+12 e^4 h\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}}{2 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{12 e^3}-\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{3 e^3 (d+e x)^3}-\frac {h (a+b \arcsin (c x))}{2 e^3 (d+e x)^2}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {(a+b \arcsin (c x)) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) (a+b \arcsin (c x))}{3 e^3 (d+e x)^3}-\frac {h (a+b \arcsin (c x))}{2 e^3 (d+e x)^2}+\frac {b c \left (\frac {e \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{\left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-d e g+5 e^2 f\right )\right )}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\frac {e \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (d^2 (-h)+d e g+11 e^2 f\right )+4 c^2 e^2 \left (d^2 h-4 d e g+e^2 f\right )+12 e^4 h\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {c^2 \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (-2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )-c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )+4 e^4 (e g-5 d h)\right )}{\left (c^2 d^2-e^2\right )^{3/2}}}{2 \left (c^2 d^2-e^2\right )}}{c^2 d^2-e^2}\right )}{12 e^3}\)

Input:

Int[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^5,x]
 

Output:

-1/4*((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(e^3*(d + e*x)^4) - ((e 
*g - 2*d*h)*(a + b*ArcSin[c*x]))/(3*e^3*(d + e*x)^3) - (h*(a + b*ArcSin[c* 
x]))/(2*e^3*(d + e*x)^2) + (b*c*((e*(e^2*f - d*e*g + d^2*h)*Sqrt[1 - c^2*x 
^2])/((c^2*d^2 - e^2)*(d + e*x)^3) - ((e*(4*e^2*(e*g - 2*d*h) - c^2*d*(5*e 
^2*f - d*e*g - 3*d^2*h))*Sqrt[1 - c^2*x^2])/(2*(c^2*d^2 - e^2)*(d + e*x)^2 
) - ((e*(12*e^4*h + c^4*d^2*(11*e^2*f + d*e*g - d^2*h) + 4*c^2*e^2*(e^2*f 
- 4*d*e*g + d^2*h))*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - (c^2* 
(4*e^4*(e*g - 5*d*h) - c^2*d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) - 2*c^4*d^ 
3*(3*e^2*f + d*e*g + d^2*h))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqr 
t[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^3)
 

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 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

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. \(1923\) vs. \(2(444)=888\).

Time = 0.28 (sec) , antiderivative size = 1924, normalized size of antiderivative = 4.09

method	result	size
parts	\(\text {Expression too large to display}\)	\(1924\)
derivativedivides	\(\text {Expression too large to display}\)	\(1935\)
default	\(\text {Expression too large to display}\)	\(1935\)

Input:

int((h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^5,x,method=_RETURNVERBOSE)
 

Output:

a*(-1/2*h/e^3/(e*x+d)^2-1/4*(d^2*h-d*e*g+e^2*f)/e^3/(e*x+d)^4-1/3*(-2*d*h+ 
e*g)/e^3/(e*x+d)^3)+b/c*(-1/4*c^5*arcsin(c*x)/e^3/(c*e*x+c*d)^4*d^2*h+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+2/3*c^4*arcsin(c*x)/e^3/(c*e*x+c*d)^3*d*h-1/3*c^4*arcsin(c*x)/e^2/(c*e*x 
+c*d)^3*g-1/2*c^3*arcsin(c*x)*h/e^3/(c*e*x+c*d)^2+1/12*c^3/e^3*(6*h/e^2*(1 
/(c^2*d^2-e^2)*e^2/(c*x+c*d/e)*(-(c*x+c*d/e)^2+2*d*c/e*(c*x+c*d/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+c*d/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+c* 
d/e)^2+2*d*c/e*(c*x+c*d/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+c*d/e)))-4*c*(2* 
d*h-e*g)/e^3*(1/2/(c^2*d^2-e^2)*e^2/(c*x+c*d/e)^2*(-(c*x+c*d/e)^2+2*d*c/e* 
(c*x+c*d/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+c*d/e)*(-(c*x+c*d/e)^2+2*d*c/e*(c*x+c*d/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+c*d/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+c*d/e)^2+2*d* 
c/e*(c*x+c*d/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+c*d/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+c*d/e 
)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+c*d/e)^2+2*d*c/e*(c*x+c*d/e)-(c^2*d^ 
2-e^2)/e^2)^(1/2))/(c*x+c*d/e)))+3*c^2*(d^2*h-d*e*g+e^2*f)/e^4*(1/3/(c^2*d 
^2-e^2)*e^2/(c*x+c*d/e)^3*(-(c*x+c*d/e)^2+2*d*c/e*(c*x+c*d/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+c*d/e)...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\text {Timed out} \] Input:

integrate((h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^5,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2}\right )}{\left (d + e x\right )^{5}}\, dx \] Input:

integrate((h*x**2+g*x+f)*(a+b*asin(c*x))/(e*x+d)**5,x)
 

Output:

Integral((a + b*asin(c*x))*(f + g*x + h*x**2)/(d + e*x)**5, x)
 

Maxima [F]

\[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\int { \frac {{\left (h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{5}} \,d x } \] Input:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((h*x^2+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. \[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^5} \,d x \] Input:

int(((a + b*asin(c*x))*(f + g*x + h*x^2))/(d + e*x)^5,x)
 

Output:

int(((a + b*asin(c*x))*(f + g*x + h*x^2))/(d + e*x)^5, x)
 

Reduce [F]

\[ \int \frac {\left (f+g x+h x^2\right ) (a+b \arcsin (c x))}{(d+e x)^5} \, dx=\int \frac {\left (h \,x^{2}+g x +f \right ) \left (\mathit {asin} \left (c x \right ) b +a \right )}{\left (e x +d \right )^{5}}d x \] Input:

int((h*x^2+g*x+f)*(a+b*asin(c*x))/(e*x+d)^5,x)
 

Output:

int((h*x^2+g*x+f)*(a+b*asin(c*x))/(e*x+d)^5,x)