\(\int (d+e x)^p (a+b \arcsin (c x)) \, dx\) [28]

Optimal result

Integrand size = 16, antiderivative size = 154 \[ \int (d+e x)^p (a+b \arcsin (c x)) \, dx=-\frac {b c (d+e x)^{2+p} \sqrt {1-\frac {c (d+e x)}{c d-e}} \sqrt {1-\frac {c (d+e x)}{c d+e}} \operatorname {AppellF1}\left (2+p,\frac {1}{2},\frac {1}{2},3+p,\frac {c (d+e x)}{c d-e},\frac {c (d+e x)}{c d+e}\right )}{e^2 (1+p) (2+p) \sqrt {1-c^2 x^2}}+\frac {(d+e x)^{1+p} (a+b \arcsin (c x))}{e (1+p)} \] Output:

-b*c*(e*x+d)^(2+p)*(1-c*(e*x+d)/(c*d-e))^(1/2)*(1-c*(e*x+d)/(c*d+e))^(1/2) 
*AppellF1(2+p,1/2,1/2,3+p,c*(e*x+d)/(c*d-e),c*(e*x+d)/(c*d+e))/e^2/(p+1)/( 
2+p)/(-c^2*x^2+1)^(1/2)+(e*x+d)^(p+1)*(a+b*arcsin(c*x))/e/(p+1)
 

Mathematica [F]

\[ \int (d+e x)^p (a+b \arcsin (c x)) \, dx=\int (d+e x)^p (a+b \arcsin (c x)) \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5242, 513, 156, 155}

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.

\(\displaystyle \int (d+e x)^p (a+b \arcsin (c x)) \, dx\)

\(\Big \downarrow \) 5242

\(\displaystyle \frac {(d+e x)^{p+1} (a+b \arcsin (c x))}{e (p+1)}-\frac {b c \int \frac {(d+e x)^{p+1}}{\sqrt {1-c^2 x^2}}dx}{e (p+1)}\)

\(\Big \downarrow \) 513

\(\displaystyle \frac {(d+e x)^{p+1} (a+b \arcsin (c x))}{e (p+1)}-\frac {b c \int \frac {(d+e x)^{p+1}}{\sqrt {1-c x} \sqrt {c x+1}}dx}{e (p+1)}\)

\(\Big \downarrow \) 156

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

\(\Big \downarrow \) 155

\(\displaystyle \frac {(d+e x)^{p+1} (a+b \arcsin (c x))}{e (p+1)}+\frac {\sqrt {2} b \sqrt {1-c x} (c d+e) (d+e x)^p \left (\frac {c (d+e x)}{c d+e}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p-1,\frac {3}{2},\frac {1}{2} (1-c x),\frac {e (1-c x)}{c d+e}\right )}{c e (p+1)}\)

Input:

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

Output:

(Sqrt[2]*b*(c*d + e)*Sqrt[1 - c*x]*(d + e*x)^p*AppellF1[1/2, 1/2, -1 - p, 
3/2, (1 - c*x)/2, (e*(1 - c*x))/(c*d + e)])/(c*e*(1 + p)*((c*(d + e*x))/(c 
*d + e))^p) + ((d + e*x)^(1 + p)*(a + b*ArcSin[c*x]))/(e*(1 + p))
 

Defintions of rubi rules used

rule 155

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* 
Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ 
(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, 
 m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[Sim 
plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] &&  !(GtQ[Simpl 
ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d 
*x, a + b*x]) &&  !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c 
- e*d)], 0] && SimplerQ[e + f*x, a + b*x])
 

rule 156

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p 
]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p])   Int[(a + b*x)^m*(c + d*x)^n*Si 
mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
& GtQ[Simplify[b/(b*c - a*d)], 0] &&  !GtQ[Simplify[b/(b*e - a*f)], 0]
 

rule 513

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
a^p   Int[(c + d*x)^n*(1 + Rt[-b/a, 2]*x)^p*(1 - Rt[-b/a, 2]*x)^p, x], x] / 
; FreeQ[{a, b, c, d, n, p}, x] && GtQ[a, 0] && NegQ[b/a]
 

rule 5242

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

Maple [F]

\[\int \left (e x +d \right )^{p} \left (a +b \arcsin \left (c x \right )\right )d x\]

Input:

int((e*x+d)^p*(a+b*arcsin(c*x)),x)
 

Output:

int((e*x+d)^p*(a+b*arcsin(c*x)),x)
 

Fricas [F]

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

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

Output:

integral((b*arcsin(c*x) + a)*(e*x + d)^p, x)
 

Sympy [F]

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

integrate((e*x+d)**p*(a+b*asin(c*x)),x)
 

Output:

Integral((a + b*asin(c*x))*(d + e*x)**p, x)
 

Maxima [F]

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

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

Output:

((e*x + d)*(e*x + d)^p*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (e*p + 
 e)*integrate((c*e*x + c*d)*sqrt(c*x + 1)*sqrt(-c*x + 1)*(e*x + d)^p/((c^2 
*e*p + c^2*e)*x^2 - e*p - e), x))*b/(e*p + e) + (e*x + d)^(p + 1)*a/(e*(p 
+ 1))
 

Giac [F]

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

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

Output:

integrate((b*arcsin(c*x) + a)*(e*x + d)^p, x)
 

Mupad [F(-1)]

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

int((a + b*asin(c*x))*(d + e*x)^p,x)
 

Output:

int((a + b*asin(c*x))*(d + e*x)^p, x)
 

Reduce [F]

\[ \int (d+e x)^p (a+b \arcsin (c x)) \, dx=\frac {\left (e x +d \right )^{p} a d +\left (e x +d \right )^{p} a e x +\left (\int \left (e x +d \right )^{p} \mathit {asin} \left (c x \right )d x \right ) b e p +\left (\int \left (e x +d \right )^{p} \mathit {asin} \left (c x \right )d x \right ) b e}{e \left (p +1\right )} \] Input:

int((e*x+d)^p*(a+b*asin(c*x)),x)
 

Output:

((d + e*x)**p*a*d + (d + e*x)**p*a*e*x + int((d + e*x)**p*asin(c*x),x)*b*e 
*p + int((d + e*x)**p*asin(c*x),x)*b*e)/(e*(p + 1))