\(\int \sqrt {d x} (a+b \arcsin (c x))^2 \, dx\) [74]

Optimal result

Integrand size = 18, antiderivative size = 109 \[ \int \sqrt {d x} (a+b \arcsin (c x))^2 \, dx=\frac {2 (d x)^{3/2} (a+b \arcsin (c x))^2}{3 d}-\frac {8 b c (d x)^{5/2} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )}{15 d^2}+\frac {16 b^2 c^2 (d x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 d^3} \] Output:

2/3*(d*x)^(3/2)*(a+b*arcsin(c*x))^2/d-8/15*b*c*(d*x)^(5/2)*(a+b*arcsin(c*x 
))*hypergeom([1/2, 5/4],[9/4],c^2*x^2)/d^2+16/105*b^2*c^2*(d*x)^(7/2)*hype 
rgeom([1, 7/4, 7/4],[9/4, 11/4],c^2*x^2)/d^3
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int \sqrt {d x} (a+b \arcsin (c x))^2 \, dx=\frac {2}{105} x \sqrt {d x} \left (7 (a+b \arcsin (c x)) \left (5 (a+b \arcsin (c x))-4 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )\right )+8 b^2 c^2 x^2 \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )\right ) \] Input:

Integrate[Sqrt[d*x]*(a + b*ArcSin[c*x])^2,x]
 

Output:

(2*x*Sqrt[d*x]*(7*(a + b*ArcSin[c*x])*(5*(a + b*ArcSin[c*x]) - 4*b*c*x*Hyp 
ergeometric2F1[1/2, 5/4, 9/4, c^2*x^2]) + 8*b^2*c^2*x^2*HypergeometricPFQ[ 
{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2]))/105
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5138, 5220}

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.

\(\displaystyle \int \sqrt {d x} (a+b \arcsin (c x))^2 \, dx\)

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5220

\(\displaystyle \frac {2 (d x)^{3/2} (a+b \arcsin (c x))^2}{3 d}-\frac {4 b c \left (\frac {2 (d x)^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right ) (a+b \arcsin (c x))}{5 d}-\frac {4 b c (d x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{35 d^2}\right )}{3 d}\)

Input:

Int[Sqrt[d*x]*(a + b*ArcSin[c*x])^2,x]
 

Output:

(2*(d*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*d) - (4*b*c*((2*(d*x)^(5/2)*(a + 
b*ArcSin[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2])/(5*d) - (4*b*c*( 
d*x)^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2])/(35*d^2 
)))/(3*d)
 

Defintions of rubi rules used

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 5220

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_. 
)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2* 
x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, 
 (3 + m)/2, c^2*x^2], x] - Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*S 
imp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m 
/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] 
&& EqQ[c^2*d + e, 0] &&  !IntegerQ[m]
 

Maple [F]

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

Input:

int((d*x)^(1/2)*(a+b*arcsin(c*x))^2,x)
 

Output:

int((d*x)^(1/2)*(a+b*arcsin(c*x))^2,x)
 

Fricas [F]

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

integrate((d*x)^(1/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")
 

Output:

integral((b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)*sqrt(d*x), x)
 

Sympy [F]

\[ \int \sqrt {d x} (a+b \arcsin (c x))^2 \, dx=\int \sqrt {d x} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \] Input:

integrate((d*x)**(1/2)*(a+b*asin(c*x))**2,x)
 

Output:

Integral(sqrt(d*x)*(a + b*asin(c*x))**2, x)
 

Maxima [F]

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

integrate((d*x)^(1/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")
 

Output:

2/3*b^2*sqrt(d)*x^(3/2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 1/6 
*a^2*c^2*sqrt(d)*(4*x^(3/2)/c^2 + 6*arctan(sqrt(c)*sqrt(x))/c^(7/2) + 3*lo 
g((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/c^(7/2)) + 6*a*b*c^2*sqrt(d 
)*integrate(1/3*x^(5/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*x^ 
2 - 1), x) + 4*b^2*c*sqrt(d)*integrate(1/3*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^ 
(3/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*x^2 - 1), x) - 1/2*a 
^2*sqrt(d)*(2*arctan(sqrt(c)*sqrt(x))/c^(3/2) + log((c*sqrt(x) - sqrt(c))/ 
(c*sqrt(x) + sqrt(c)))/c^(3/2)) - 6*a*b*sqrt(d)*integrate(1/3*sqrt(x)*arct 
an(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*x^2 - 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int \sqrt {d x} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((d*x)^(1/2)*(a+b*arcsin(c*x))^2,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. \[ \int \sqrt {d x} (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d\,x} \,d x \] Input:

int((a + b*asin(c*x))^2*(d*x)^(1/2),x)
 

Output:

int((a + b*asin(c*x))^2*(d*x)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {d x} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d}\, \left (2 \sqrt {x}\, a^{2} x +6 \left (\int \sqrt {x}\, \mathit {asin} \left (c x \right )d x \right ) a b +3 \left (\int \sqrt {x}\, \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2}\right )}{3} \] Input:

int((d*x)^(1/2)*(a+b*asin(c*x))^2,x)
 

Output:

(sqrt(d)*(2*sqrt(x)*a**2*x + 6*int(sqrt(x)*asin(c*x),x)*a*b + 3*int(sqrt(x 
)*asin(c*x)**2,x)*b**2))/3