\(\int \frac {(a+b \arcsin (c x))^2}{(d x)^{3/2}} \, dx\) [76]

Optimal result

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

-2*(a+b*arcsin(c*x))^2/d/(d*x)^(1/2)+8*b*c*(d*x)^(1/2)*(a+b*arcsin(c*x))*h 
ypergeom([1/4, 1/2],[5/4],c^2*x^2)/d^2-16/3*b^2*c^2*(d*x)^(3/2)*hypergeom( 
[3/4, 3/4, 1],[5/4, 7/4],c^2*x^2)/d^3
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{3/2}} \, dx=-\frac {2 x \left (3 (a+b \arcsin (c x)) \left (a+b \arcsin (c x)-4 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )\right )+8 b^2 c^2 x^2 \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )\right )}{3 (d x)^{3/2}} \] Input:

Integrate[(a + b*ArcSin[c*x])^2/(d*x)^(3/2),x]
 

Output:

(-2*x*(3*(a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] - 4*b*c*x*Hypergeometric2F 
1[1/4, 1/2, 5/4, c^2*x^2]) + 8*b^2*c^2*x^2*HypergeometricPFQ[{3/4, 3/4, 1} 
, {5/4, 7/4}, c^2*x^2]))/(3*(d*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.02, 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.

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5220

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

Input:

Int[(a + b*ArcSin[c*x])^2/(d*x)^(3/2),x]
 

Output:

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

\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (d x \right )^{\frac {3}{2}}}d x\]

Input:

int((a+b*arcsin(c*x))^2/(d*x)^(3/2),x)
 

Output:

int((a+b*arcsin(c*x))^2/(d*x)^(3/2),x)
 

Fricas [F]

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

integrate((a+b*arcsin(c*x))^2/(d*x)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*asin(c*x))**2/(d*x)**(3/2),x)
 

Output:

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

Maxima [F]

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

integrate((a+b*arcsin(c*x))^2/(d*x)^(3/2),x, algorithm="maxima")
 

Output:

-1/2*(4*b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 - (a^2*c^2*sqrt(d 
)*(2*arctan(sqrt(c)*sqrt(x))/(c^(3/2)*d^2) + log((c*sqrt(x) - sqrt(c))/(c* 
sqrt(x) + sqrt(c)))/(c^(3/2)*d^2)) + 4*a*b*c^2*sqrt(d)*integrate(x^(5/2)*a 
rctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*d^2*x^4 - d^2*x^2), x) - 8* 
b^2*c*sqrt(d)*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*x^(3/2)*arctan(c*x/(s 
qrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*d^2*x^4 - d^2*x^2), x) - a^2*sqrt(d)*(2 
*sqrt(c)*arctan(sqrt(c)*sqrt(x))/d^2 + sqrt(c)*log((c*sqrt(x) - sqrt(c))/( 
c*sqrt(x) + sqrt(c)))/d^2 + 4/(d^2*sqrt(x))) - 4*a*b*sqrt(d)*integrate(sqr 
t(x)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*d^2*x^4 - d^2*x^2), x 
))*d^(3/2)*sqrt(x))/(d^(3/2)*sqrt(x))
 

Giac [F]

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

integrate((a+b*arcsin(c*x))^2/(d*x)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*arcsin(c*x) + a)^2/(d*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d\,x\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(x)*int(asin(c*x)/(sqrt(x)*x),x)*a*b + sqrt(x)*int(asin(c*x)**2/(sq 
rt(x)*x),x)*b**2 - 2*a**2)/(sqrt(x)*sqrt(d)*d)