\(\int (c e+d e x)^m (a+b \arcsin (c+d x)) \, dx\) [186]

Optimal result

Integrand size = 21, antiderivative size = 89 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x)) \, dx=\frac {(e (c+d x))^{1+m} (a+b \arcsin (c+d x))}{d e (1+m)}-\frac {b (e (c+d x))^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},(c+d x)^2\right )}{d e^2 (1+m) (2+m)} \] Output:

(e*(d*x+c))^(1+m)*(a+b*arcsin(d*x+c))/d/e/(1+m)-b*(e*(d*x+c))^(2+m)*hyperg 
eom([1/2, 1+1/2*m],[2+1/2*m],(d*x+c)^2)/d/e^2/(1+m)/(2+m)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.87 \[ \int (c e+d e x)^m (a+b \arcsin (c+d x)) \, dx=-\frac {(c+d x) (e (c+d x))^m \left (-((2+m) (a+b \arcsin (c+d x)))+b (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},(c+d x)^2\right )\right )}{d (1+m) (2+m)} \] Input:

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

Output:

-(((c + d*x)*(e*(c + d*x))^m*(-((2 + m)*(a + b*ArcSin[c + d*x])) + b*(c + 
d*x)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2]))/(d*(1 + m 
)*(2 + m)))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5304, 5138, 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.

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

\(\Big \downarrow \) 5304

\(\displaystyle \frac {\int (e (c+d x))^m (a+b \arcsin (c+d x))d(c+d x)}{d}\)

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

(((e*(c + d*x))^(1 + m)*(a + b*ArcSin[c + d*x]))/(e*(1 + m)) - (b*(e*(c + 
d*x))^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/( 
e^2*(1 + m)*(2 + m)))/d
 

Defintions of rubi rules used

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 5138

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

rule 5304

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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