\(\int \frac {a+b \arcsin (c x)}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx\) [100]

Optimal result

Integrand size = 27, antiderivative size = 769 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=-\frac {2 \sqrt {f+g x} (a+b \arcsin (c x))}{3 (e f-d g) (d+e x)^{3/2}}+\frac {4 g \sqrt {f+g x} (a+b \arcsin (c x))}{3 (e f-d g)^2 \sqrt {d+e x}}-\frac {4 \sqrt {2} b c^{3/2} \sqrt {-c-c^2 x} \sqrt {\frac {(c d-e) (1-c x)}{c (d+e x)}} \sqrt {f+g x} E\left (\arcsin \left (\frac {\sqrt {-e f+d g} \sqrt {-c-c^2 x}}{\sqrt {c} \sqrt {c f-g} \sqrt {d+e x}}\right )|\frac {(c d+e) (c f-g)}{2 c (e f-d g)}\right )}{3 \left (c^2 d^2-e^2\right ) \sqrt {c f-g} \sqrt {-e f+d g} \sqrt {\frac {(c d-e) (f+g x)}{(c f-g) (d+e x)}} \sqrt {1-c^2 x^2}}+\frac {2 \sqrt {2} b \sqrt {c} (c f+g) \sqrt {-c-c^2 x} \sqrt {\frac {(c d-e) (1-c x)}{c (d+e x)}} \sqrt {f+g x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-e f+d g} \sqrt {-c-c^2 x}}{\sqrt {c} \sqrt {c f-g} \sqrt {d+e x}}\right ),\frac {(c d+e) (c f-g)}{2 c (e f-d g)}\right )}{3 (c d+e) \sqrt {c f-g} (-e f+d g)^{3/2} \sqrt {\frac {(c d-e) (f+g x)}{(c f-g) (d+e x)}} \sqrt {1-c^2 x^2}}-\frac {8 b c \sqrt {-c (c d+e)} g \sqrt {\frac {(e f-d g) (1-c x)}{(c d+e) (f+g x)}} \sqrt {-\frac {(e f-d g) (1+c x)}{(c d-e) (f+g x)}} (f+g x) \operatorname {EllipticPi}\left (\frac {(c d+e) g}{e (c f+g)},\arcsin \left (\frac {\sqrt {-c (c f+g)} \sqrt {d+e x}}{\sqrt {-c (c d+e)} \sqrt {f+g x}}\right ),\frac {(c d+e) (c f-g)}{(c d-e) (c f+g)}\right )}{3 e \sqrt {-c (c f+g)} (e f-d g)^2 \sqrt {1-c^2 x^2}} \] Output:

-2/3*(g*x+f)^(1/2)*(a+b*arcsin(c*x))/(-d*g+e*f)/(e*x+d)^(3/2)+4/3*g*(g*x+f 
)^(1/2)*(a+b*arcsin(c*x))/(-d*g+e*f)^2/(e*x+d)^(1/2)-4/3*2^(1/2)*b*c^(3/2) 
*(-c^2*x-c)^(1/2)*((c*d-e)*(-c*x+1)/c/(e*x+d))^(1/2)*(g*x+f)^(1/2)*Ellipti 
cE((d*g-e*f)^(1/2)*(-c^2*x-c)^(1/2)/c^(1/2)/(c*f-g)^(1/2)/(e*x+d)^(1/2),1/ 
2*2^(1/2)*((c*d+e)*(c*f-g)/c/(-d*g+e*f))^(1/2))/(c^2*d^2-e^2)/(c*f-g)^(1/2 
)/(d*g-e*f)^(1/2)/((c*d-e)*(g*x+f)/(c*f-g)/(e*x+d))^(1/2)/(-c^2*x^2+1)^(1/ 
2)+2/3*2^(1/2)*b*c^(1/2)*(c*f+g)*(-c^2*x-c)^(1/2)*((c*d-e)*(-c*x+1)/c/(e*x 
+d))^(1/2)*(g*x+f)^(1/2)*EllipticF((d*g-e*f)^(1/2)*(-c^2*x-c)^(1/2)/c^(1/2 
)/(c*f-g)^(1/2)/(e*x+d)^(1/2),1/2*2^(1/2)*((c*d+e)*(c*f-g)/c/(-d*g+e*f))^( 
1/2))/(c*d+e)/(c*f-g)^(1/2)/(d*g-e*f)^(3/2)/((c*d-e)*(g*x+f)/(c*f-g)/(e*x+ 
d))^(1/2)/(-c^2*x^2+1)^(1/2)-8/3*b*c*(-c*(c*d+e))^(1/2)*g*((-d*g+e*f)*(-c* 
x+1)/(c*d+e)/(g*x+f))^(1/2)*(-(-d*g+e*f)*(c*x+1)/(c*d-e)/(g*x+f))^(1/2)*(g 
*x+f)*EllipticPi((-c*(c*f+g))^(1/2)*(e*x+d)^(1/2)/(-c*(c*d+e))^(1/2)/(g*x+ 
f)^(1/2),(c*d+e)*g/e/(c*f+g),((c*d+e)*(c*f-g)/(c*d-e)/(c*f+g))^(1/2))/e/(- 
c*(c*f+g))^(1/2)/(-d*g+e*f)^2/(-c^2*x^2+1)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1998\) vs. \(2(769)=1538\).

Time = 22.48 (sec) , antiderivative size = 1998, normalized size of antiderivative = 2.60 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcSin[c*x])/((d + e*x)^(5/2)*Sqrt[f + g*x]),x]
 

Output:

(-4*b*c*e*Sqrt[f + g*x]*Sqrt[1 - c^2*x^2])/(3*(-(c^2*d^2) + e^2)*(e*f - d* 
g)*Sqrt[d + e*x]) + Sqrt[d + e*x]*Sqrt[f + g*x]*((2*a)/(3*(-(e*f) + d*g)*( 
d + e*x)^2) + (4*a*g)/(3*(e*f - d*g)^2*(d + e*x))) + (2*b*Sqrt[f + g*x]*(- 
(e*f) + 3*d*g + 2*e*g*x)*ArcSin[c*x])/(3*(-(e*f) + d*g)^2*(d + e*x)^(3/2)) 
 - (4*b*c*Sqrt[d + e*x]*Sqrt[1 - (c^2*(d + e*x)^2*(-1 + d/(d + e*x))^2)/e^ 
2]*((e*f - d*g)*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x))*(e^2/(d + e*x)^2 - 
 c^2*(-1 + d/(d + e*x))^2) + ((c^2*d^2 - e^2)*Sqrt[((c*d + e)*(g + (e*f)/( 
d + e*x) - (d*g)/(d + e*x)))/(e*(c*f + g))]*(-(e^2*f^2*Sqrt[((e*f - d*g)*( 
c - (c*d)/(d + e*x) + e/(d + e*x)))/(e*(c*f - g))]*(c - (c*d + e)/(d + e*x 
))*Sqrt[1 - e/(c*(d + e*x)) + (c*d*(-1 + d/(d + e*x)))/e]*(e*(c*f - g)*Ell 
ipticE[ArcSin[Sqrt[((c*d + e)*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x)))/(e* 
(c*f + g))]], ((c*d - e)*(c*f + g))/((c*d + e)*(c*f - g))] + c*(-(e*f) + d 
*g)*EllipticF[ArcSin[Sqrt[((c*d + e)*(g + (e*f)/(d + e*x) - (d*g)/(d + e*x 
)))/(e*(c*f + g))]], ((c*d - e)*(c*f + g))/((c*d + e)*(c*f - g))])) + 2*d* 
e*f*g*Sqrt[((e*f - d*g)*(c - (c*d)/(d + e*x) + e/(d + e*x)))/(e*(c*f - g)) 
]*(c - (c*d + e)/(d + e*x))*Sqrt[1 - e/(c*(d + e*x)) + (c*d*(-1 + d/(d + e 
*x)))/e]*(e*(c*f - g)*EllipticE[ArcSin[Sqrt[((c*d + e)*(g + (e*f)/(d + e*x 
) - (d*g)/(d + e*x)))/(e*(c*f + g))]], ((c*d - e)*(c*f + g))/((c*d + e)*(c 
*f - g))] + c*(-(e*f) + d*g)*EllipticF[ArcSin[Sqrt[((c*d + e)*(g + (e*f)/( 
d + e*x) - (d*g)/(d + e*x)))/(e*(c*f + g))]], ((c*d - e)*(c*f + g))/((c...
 

Rubi [F]

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.

\(\displaystyle \int \frac {a+b \arcsin (c x)}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx\)

\(\Big \downarrow \) 5286

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 726

\(\displaystyle \frac {2 b c \left ((e f-d g) \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx-\frac {4 g \sqrt {-c (c d+e)} (f+g x) \sqrt {\frac {(1-c x) (e f-d g)}{(c d+e) (f+g x)}} \sqrt {-\frac {(c x+1) (e f-d g)}{(c d-e) (f+g x)}} \operatorname {EllipticPi}\left (\frac {(c d+e) g}{e (c f+g)},\arcsin \left (\frac {\sqrt {-c (c f+g)} \sqrt {d+e x}}{\sqrt {-c (c d+e)} \sqrt {f+g x}}\right ),\frac {(c d+e) (c f-g)}{(c d-e) (c f+g)}\right )}{e \sqrt {1-c^2 x^2} \sqrt {-c (c f+g)}}\right )}{3 (e f-d g)^2}+\frac {4 g \sqrt {f+g x} (a+b \arcsin (c x))}{3 \sqrt {d+e x} (e f-d g)^2}-\frac {2 \sqrt {f+g x} (a+b \arcsin (c x))}{3 (d+e x)^{3/2} (e f-d g)}\)

\(\Big \downarrow \) 744

\(\displaystyle \frac {2 b c \left ((e f-d g) \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx-\frac {4 g \sqrt {-c (c d+e)} (f+g x) \sqrt {\frac {(1-c x) (e f-d g)}{(c d+e) (f+g x)}} \sqrt {-\frac {(c x+1) (e f-d g)}{(c d-e) (f+g x)}} \operatorname {EllipticPi}\left (\frac {(c d+e) g}{e (c f+g)},\arcsin \left (\frac {\sqrt {-c (c f+g)} \sqrt {d+e x}}{\sqrt {-c (c d+e)} \sqrt {f+g x}}\right ),\frac {(c d+e) (c f-g)}{(c d-e) (c f+g)}\right )}{e \sqrt {1-c^2 x^2} \sqrt {-c (c f+g)}}\right )}{3 (e f-d g)^2}+\frac {4 g \sqrt {f+g x} (a+b \arcsin (c x))}{3 \sqrt {d+e x} (e f-d g)^2}-\frac {2 \sqrt {f+g x} (a+b \arcsin (c x))}{3 (d+e x)^{3/2} (e f-d g)}\)

Input:

Int[(a + b*ArcSin[c*x])/((d + e*x)^(5/2)*Sqrt[f + g*x]),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {a +b \arcsin \left (c x \right )}{\left (e x +d \right )^{\frac {5}{2}} \sqrt {g x +f}}d x\]

Input:

int((a+b*arcsin(c*x))/(e*x+d)^(5/2)/(g*x+f)^(1/2),x)
 

Output:

int((a+b*arcsin(c*x))/(e*x+d)^(5/2)/(g*x+f)^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\text {Timed out} \] Input:

integrate((a+b*arcsin(c*x))/(e*x+d)^(5/2)/(g*x+f)^(1/2),x, algorithm="fric 
as")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {a+b \arcsin (c x)}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (d + e x\right )^{\frac {5}{2}} \sqrt {f + g x}}\, dx \] Input:

integrate((a+b*asin(c*x))/(e*x+d)**(5/2)/(g*x+f)**(1/2),x)
 

Output:

Integral((a + b*asin(c*x))/((d + e*x)**(5/2)*sqrt(f + g*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arcsin(c*x))/(e*x+d)^(5/2)/(g*x+f)^(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 [F]

\[ \int \frac {a+b \arcsin (c x)}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}} \sqrt {g x + f}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))/(e*x+d)^(5/2)/(g*x+f)^(1/2),x, algorithm="giac 
")
 

Output:

integrate((b*arcsin(c*x) + a)/((e*x + d)^(5/2)*sqrt(g*x + f)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:

int((a + b*asin(c*x))/((f + g*x)^(1/2)*(d + e*x)^(5/2)),x)
 

Output:

int((a + b*asin(c*x))/((f + g*x)^(1/2)*(d + e*x)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \arcsin (c x)}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx=\int \frac {\mathit {asin} \left (c x \right ) b +a}{\left (e x +d \right )^{\frac {5}{2}} \sqrt {g x +f}}d x \] Input:

int((a+b*asin(c*x))/(e*x+d)^(5/2)/(g*x+f)^(1/2),x)
 

Output:

int((a+b*asin(c*x))/(e*x+d)^(5/2)/(g*x+f)^(1/2),x)