\(\int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx\) [97]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx=\text {Int}\left (\frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}},x\right ) \] Output:

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

Mathematica [N/A]

Not integrable

Time = 16.42 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx=\int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx \] Input:

Integrate[(Sqrt[f + g*x]*(a + b*ArcSin[c*x]))/Sqrt[d + e*x],x]
 

Output:

Integrate[(Sqrt[f + g*x]*(a + b*ArcSin[c*x]))/Sqrt[d + e*x], x]
 

Rubi [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, 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.

\(\displaystyle \int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx\)

\(\Big \downarrow \) 5300

\(\displaystyle \int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}}dx\)

Input:

Int[(Sqrt[f + g*x]*(a + b*ArcSin[c*x]))/Sqrt[d + e*x],x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

\[\int \frac {\sqrt {g x +f}\, \left (a +b \arcsin \left (c x \right )\right )}{\sqrt {e x +d}}d x\]

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx=\int { \frac {\sqrt {g x + f} {\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt {e x + d}} \,d x } \] Input:

integrate((g*x+f)^(1/2)*(a+b*arcsin(c*x))/(e*x+d)^(1/2),x, algorithm="fric 
as")
 

Output:

integral(sqrt(g*x + f)*(b*arcsin(c*x) + a)/sqrt(e*x + d), x)
 

Sympy [N/A]

Not integrable

Time = 3.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \sqrt {f + g x}}{\sqrt {d + e x}}\, dx \] Input:

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

Output:

Integral((a + b*asin(c*x))*sqrt(f + g*x)/sqrt(d + e*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x+f)^(1/2)*(a+b*arcsin(c*x))/(e*x+d)^(1/2),x, algorithm="maxi 
ma")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e*(d*g-e*f)>0)', see `assume?` f 
or more de
 

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx=\int { \frac {\sqrt {g x + f} {\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt {e x + d}} \,d x } \] Input:

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

Output:

integrate(sqrt(g*x + f)*(b*arcsin(c*x) + a)/sqrt(e*x + d), x)
 

Mupad [N/A]

Not integrable

Time = 5.40 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx=\int \frac {\sqrt {f+g\,x}\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{\sqrt {d+e\,x}} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.89 \[ \int \frac {\sqrt {f+g x} (a+b \arcsin (c x))}{\sqrt {d+e x}} \, dx=\frac {\sqrt {g x +f}\, \sqrt {e x +d}\, a e g -\sqrt {g}\, \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {e x +d}+\sqrt {e}\, \sqrt {g x +f}}{\sqrt {d g -e f}}\right ) a d g +\sqrt {g}\, \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {e x +d}+\sqrt {e}\, \sqrt {g x +f}}{\sqrt {d g -e f}}\right ) a e f +\left (\int \frac {\sqrt {g x +f}\, \mathit {asin} \left (c x \right )}{\sqrt {e x +d}}d x \right ) b \,e^{2} g}{e^{2} g} \] Input:

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

Output:

(sqrt(f + g*x)*sqrt(d + e*x)*a*e*g - sqrt(g)*sqrt(e)*log((sqrt(g)*sqrt(d + 
 e*x) + sqrt(e)*sqrt(f + g*x))/sqrt(d*g - e*f))*a*d*g + sqrt(g)*sqrt(e)*lo 
g((sqrt(g)*sqrt(d + e*x) + sqrt(e)*sqrt(f + g*x))/sqrt(d*g - e*f))*a*e*f + 
 int((sqrt(f + g*x)*asin(c*x))/sqrt(d + e*x),x)*b*e**2*g)/(e**2*g)