\(\int \frac {(a+b \arcsin (c x))^3}{\sqrt {d x}} \, dx\) [80]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {(a+b \arcsin (c x))^3}{\sqrt {d x}} \, dx=\text {Int}\left (\frac {(a+b \arcsin (c x))^3}{\sqrt {d x}},x\right ) \] Output:

Defer(Int)((a+b*arcsin(c*x))^3/(d*x)^(1/2),x)
 

Mathematica [N/A]

Not integrable

Time = 60.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \arcsin (c x))^3}{\sqrt {d x}} \, dx=\int \frac {(a+b \arcsin (c x))^3}{\sqrt {d x}} \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

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.

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

\(\Big \downarrow \) 5138

\(\displaystyle \frac {2 \sqrt {d x} (a+b \arcsin (c x))^3}{d}-\frac {6 b c \int \frac {\sqrt {d x} (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{d}\)

\(\Big \downarrow \) 5234

\(\displaystyle \frac {2 \sqrt {d x} (a+b \arcsin (c x))^3}{d}-\frac {6 b c \int \frac {\sqrt {d x} (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{d}\)

Input:

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

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

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

Input:

int((a+b*arcsin(c*x))^3/(d*x)^(1/2),x)
 

Output:

int((a+b*arcsin(c*x))^3/(d*x)^(1/2),x)
 

Fricas [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int \frac {(a+b \arcsin (c x))^3}{\sqrt {d x}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3}}{\sqrt {d x}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^3/(d*x)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

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

integrate((a+b*asin(c*x))**3/(d*x)**(1/2),x)
 

Output:

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

Maxima [N/A]

Not integrable

Time = 3.31 (sec) , antiderivative size = 438, normalized size of antiderivative = 24.33 \[ \int \frac {(a+b \arcsin (c x))^3}{\sqrt {d x}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3}}{\sqrt {d x}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^3/(d*x)^(1/2),x, algorithm="maxima")
 

Output:

1/2*(4*b^3*sqrt(x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^3 + (a^3*c^2 
*sqrt(d)*(4*sqrt(x)/(c^2*d) - 2*arctan(sqrt(c)*sqrt(x))/(c^(5/2)*d) + log( 
(c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/(c^(5/2)*d)) + 6*a*b^2*c^2*sq 
rt(d)*integrate(x^(5/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))^2/(c^2* 
d*x^3 - d*x), x) + 6*a^2*b*c^2*sqrt(d)*integrate(x^(5/2)*arctan(c*x/(sqrt( 
c*x + 1)*sqrt(-c*x + 1)))/(c^2*d*x^3 - d*x), x) + 12*b^3*c*sqrt(d)*integra 
te(sqrt(c*x + 1)*sqrt(-c*x + 1)*x^(3/2)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c* 
x + 1)))^2/(c^2*d*x^3 - d*x), x) + a^3*sqrt(d)*(2*arctan(sqrt(c)*sqrt(x))/ 
(sqrt(c)*d) - log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/(sqrt(c)*d) 
) - 6*a*b^2*sqrt(d)*integrate(sqrt(x)*arctan(c*x/(sqrt(c*x + 1)*sqrt(-c*x 
+ 1)))^2/(c^2*d*x^3 - d*x), x) - 6*a^2*b*sqrt(d)*integrate(sqrt(x)*arctan( 
c*x/(sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^2*d*x^3 - d*x), x))*sqrt(d))/sqrt(d 
)
 

Giac [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c x))^3}{\sqrt {d x}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3}}{\sqrt {d x}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^3/(d*x)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arcsin (c x))^3}{\sqrt {d x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3}{\sqrt {d\,x}} \,d x \] Input:

int((a + b*asin(c*x))^3/(d*x)^(1/2),x)
 

Output:

int((a + b*asin(c*x))^3/(d*x)^(1/2), x)
 

Reduce [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.67 \[ \int \frac {(a+b \arcsin (c x))^3}{\sqrt {d x}} \, dx=\frac {2 \sqrt {x}\, a^{3}+3 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {x}}d x \right ) a^{2} b +\left (\int \frac {\mathit {asin} \left (c x \right )^{3}}{\sqrt {x}}d x \right ) b^{3}+3 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {x}}d x \right ) a \,b^{2}}{\sqrt {d}} \] Input:

int((a+b*asin(c*x))^3/(d*x)^(1/2),x)
 

Output:

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