\(\int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{9/2}} \, dx\) [169]

Optimal result

Integrand size = 23, antiderivative size = 172 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{9/2}} \, dx=-\frac {4 b \sqrt {1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac {12 b \sqrt {1-(c+d x)^2}}{35 d e^4 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{7 d e (e (c+d x))^{7/2}}-\frac {12 b E\left (\left .\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{35 d e^{9/2}}+\frac {12 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{35 d e^{9/2}} \] Output:

-4/35*b*(1-(d*x+c)^2)^(1/2)/d/e^2/(e*(d*x+c))^(5/2)-12/35*b*(1-(d*x+c)^2)^ 
(1/2)/d/e^4/(e*(d*x+c))^(1/2)-2/7*(a+b*arcsin(d*x+c))/d/e/(e*(d*x+c))^(7/2 
)-12/35*b*EllipticE((e*(d*x+c))^(1/2)/e^(1/2),I)/d/e^(9/2)+12/35*b*Ellipti 
cF((e*(d*x+c))^(1/2)/e^(1/2),I)/d/e^(9/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.38 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{9/2}} \, dx=-\frac {2 \sqrt {e (c+d x)} \left (5 (a+b \arcsin (c+d x))+2 b (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},-\frac {1}{4},(c+d x)^2\right )\right )}{35 d e^5 (c+d x)^4} \] Input:

Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(9/2),x]
 

Output:

(-2*Sqrt[e*(c + d*x)]*(5*(a + b*ArcSin[c + d*x]) + 2*b*(c + d*x)*Hypergeom 
etric2F1[-5/4, 1/2, -1/4, (c + d*x)^2]))/(35*d*e^5*(c + d*x)^4)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5304, 5138, 264, 264, 261, 259, 327}

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.

\(\displaystyle \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{9/2}} \, dx\)

\(\Big \downarrow \) 5304

\(\displaystyle \frac {\int \frac {a+b \arcsin (c+d x)}{(e (c+d x))^{9/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {\frac {2 b \int \frac {1}{(e (c+d x))^{7/2} \sqrt {1-(c+d x)^2}}d(c+d x)}{7 e}-\frac {2 (a+b \arcsin (c+d x))}{7 e (e (c+d x))^{7/2}}}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {\frac {2 b \left (\frac {3 \int \frac {1}{(e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}d(c+d x)}{5 e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{5 e (e (c+d x))^{5/2}}\right )}{7 e}-\frac {2 (a+b \arcsin (c+d x))}{7 e (e (c+d x))^{7/2}}}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {\frac {2 b \left (\frac {3 \left (-\frac {\int \frac {\sqrt {e (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)}{e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{e \sqrt {e (c+d x)}}\right )}{5 e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{5 e (e (c+d x))^{5/2}}\right )}{7 e}-\frac {2 (a+b \arcsin (c+d x))}{7 e (e (c+d x))^{7/2}}}{d}\)

\(\Big \downarrow \) 261

\(\displaystyle \frac {\frac {2 b \left (\frac {3 \left (-\frac {\sqrt {e (c+d x)} \int \frac {\sqrt {c+d x}}{\sqrt {1-(c+d x)^2}}d(c+d x)}{e^2 \sqrt {c+d x}}-\frac {2 \sqrt {1-(c+d x)^2}}{e \sqrt {e (c+d x)}}\right )}{5 e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{5 e (e (c+d x))^{5/2}}\right )}{7 e}-\frac {2 (a+b \arcsin (c+d x))}{7 e (e (c+d x))^{7/2}}}{d}\)

\(\Big \downarrow \) 259

\(\displaystyle \frac {\frac {2 b \left (\frac {3 \left (\frac {2 \sqrt {e (c+d x)} \int \frac {\sqrt {c+d x}}{\sqrt {\frac {1}{2} (c+d x-1)+1}}d\frac {\sqrt {-c-d x+1}}{\sqrt {2}}}{e^2 \sqrt {c+d x}}-\frac {2 \sqrt {1-(c+d x)^2}}{e \sqrt {e (c+d x)}}\right )}{5 e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{5 e (e (c+d x))^{5/2}}\right )}{7 e}-\frac {2 (a+b \arcsin (c+d x))}{7 e (e (c+d x))^{7/2}}}{d}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {2 b \left (\frac {3 \left (\frac {2 \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{e^2 \sqrt {c+d x}}-\frac {2 \sqrt {1-(c+d x)^2}}{e \sqrt {e (c+d x)}}\right )}{5 e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{5 e (e (c+d x))^{5/2}}\right )}{7 e}-\frac {2 (a+b \arcsin (c+d x))}{7 e (e (c+d x))^{7/2}}}{d}\)

Input:

Int[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(9/2),x]
 

Output:

((-2*(a + b*ArcSin[c + d*x]))/(7*e*(e*(c + d*x))^(7/2)) + (2*b*((-2*Sqrt[1 
 - (c + d*x)^2])/(5*e*(e*(c + d*x))^(5/2)) + (3*((-2*Sqrt[1 - (c + d*x)^2] 
)/(e*Sqrt[e*(c + d*x)]) + (2*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 - c 
 - d*x]/Sqrt[2]], 2])/(e^2*Sqrt[c + d*x])))/(5*e^2)))/(7*e))/d
 

Defintions of rubi rules used

rule 259

Int[Sqrt[x_]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[-2/(Sqrt[a]*(-b/a 
)^(3/4))   Subst[Int[Sqrt[1 - 2*x^2]/Sqrt[1 - x^2], x], x, Sqrt[1 - Sqrt[-b 
/a]*x]/Sqrt[2]], x] /; FreeQ[{a, b}, x] && GtQ[-b/a, 0] && GtQ[a, 0]
 

rule 261

Int[Sqrt[(c_)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[c*x]/ 
Sqrt[x]   Int[Sqrt[x]/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ 
[-b/a, 0]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

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 5304

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{7 \left (d e x +c e \right )^{\frac {7}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{7 \left (d e x +c e \right )^{\frac {7}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{35 \left (d e x +c e \right )^{\frac {5}{2}}}-\frac {6 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{35 e^{2} \sqrt {d e x +c e}}+\frac {6 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{35 e^{3} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(225\)
default	\(\frac {-\frac {2 a}{7 \left (d e x +c e \right )^{\frac {7}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{7 \left (d e x +c e \right )^{\frac {7}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{35 \left (d e x +c e \right )^{\frac {5}{2}}}-\frac {6 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{35 e^{2} \sqrt {d e x +c e}}+\frac {6 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{35 e^{3} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(225\)
parts	\(-\frac {2 a}{7 \left (d e x +c e \right )^{\frac {7}{2}} d e}+\frac {2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{7 \left (d e x +c e \right )^{\frac {7}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{35 \left (d e x +c e \right )^{\frac {5}{2}}}-\frac {6 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{35 e^{2} \sqrt {d e x +c e}}+\frac {6 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{35 e^{3} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(230\)

Input:

int((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

2/d/e*(-1/7*a/(d*e*x+c*e)^(7/2)+b*(-1/7/(d*e*x+c*e)^(7/2)*arcsin((d*e*x+c* 
e)/e)+2/7/e*(-1/5*(-(d*e*x+c*e)^2/e^2+1)^(1/2)/(d*e*x+c*e)^(5/2)-3/5/e^2*( 
-(d*e*x+c*e)^2/e^2+1)^(1/2)/(d*e*x+c*e)^(1/2)+3/5/e^3/(1/e)^(1/2)*(1-(d*e* 
x+c*e)/e)^(1/2)*(1+(d*e*x+c*e)/e)^(1/2)/(-(d*e*x+c*e)^2/e^2+1)^(1/2)*(Elli 
pticF((d*e*x+c*e)^(1/2)*(1/e)^(1/2),I)-EllipticE((d*e*x+c*e)^(1/2)*(1/e)^( 
1/2),I)))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.37 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{9/2}} \, dx=-\frac {2 \, {\left (6 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \sqrt {-d^{3} e} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + \sqrt {d e x + c e} {\left (5 \, b d \arcsin \left (d x + c\right ) + 5 \, a d + 2 \, {\left (3 \, b d^{4} x^{3} + 9 \, b c d^{3} x^{2} + {\left (9 \, b c^{2} + b\right )} d^{2} x + {\left (3 \, b c^{3} + b c\right )} d\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )}\right )}}{35 \, {\left (d^{6} e^{5} x^{4} + 4 \, c d^{5} e^{5} x^{3} + 6 \, c^{2} d^{4} e^{5} x^{2} + 4 \, c^{3} d^{3} e^{5} x + c^{4} d^{2} e^{5}\right )}} \] Input:

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(9/2),x, algorithm="fricas")
 

Output:

-2/35*(6*(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^ 
4)*sqrt(-d^3*e)*weierstrassZeta(4/d^2, 0, weierstrassPInverse(4/d^2, 0, (d 
*x + c)/d)) + sqrt(d*e*x + c*e)*(5*b*d*arcsin(d*x + c) + 5*a*d + 2*(3*b*d^ 
4*x^3 + 9*b*c*d^3*x^2 + (9*b*c^2 + b)*d^2*x + (3*b*c^3 + b*c)*d)*sqrt(-d^2 
*x^2 - 2*c*d*x - c^2 + 1)))/(d^6*e^5*x^4 + 4*c*d^5*e^5*x^3 + 6*c^2*d^4*e^5 
*x^2 + 4*c^3*d^3*e^5*x + c^4*d^2*e^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*asin(d*x+c))/(d*e*x+c*e)**(9/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{9/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(9/2),x, algorithm="maxima")
 

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>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{9/2}} \, dx=\int { \frac {b \arcsin \left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(9/2),x, algorithm="giac")
 

Output:

integrate((b*arcsin(d*x + c) + a)/(d*e*x + c*e)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{9/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{9/2}} \,d x \] Input:

int((a + b*asin(c + d*x))/(c*e + d*e*x)^(9/2),x)
 

Output:

int((a + b*asin(c + d*x))/(c*e + d*e*x)^(9/2), x)
 

Reduce [F]

\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{9/2}} \, dx=\frac {7 \sqrt {d x +c}\, \left (\int \frac {\mathit {asin} \left (d x +c \right )}{\sqrt {d x +c}\, c^{4}+4 \sqrt {d x +c}\, c^{3} d x +6 \sqrt {d x +c}\, c^{2} d^{2} x^{2}+4 \sqrt {d x +c}\, c \,d^{3} x^{3}+\sqrt {d x +c}\, d^{4} x^{4}}d x \right ) b \,c^{3} d +21 \sqrt {d x +c}\, \left (\int \frac {\mathit {asin} \left (d x +c \right )}{\sqrt {d x +c}\, c^{4}+4 \sqrt {d x +c}\, c^{3} d x +6 \sqrt {d x +c}\, c^{2} d^{2} x^{2}+4 \sqrt {d x +c}\, c \,d^{3} x^{3}+\sqrt {d x +c}\, d^{4} x^{4}}d x \right ) b \,c^{2} d^{2} x +21 \sqrt {d x +c}\, \left (\int \frac {\mathit {asin} \left (d x +c \right )}{\sqrt {d x +c}\, c^{4}+4 \sqrt {d x +c}\, c^{3} d x +6 \sqrt {d x +c}\, c^{2} d^{2} x^{2}+4 \sqrt {d x +c}\, c \,d^{3} x^{3}+\sqrt {d x +c}\, d^{4} x^{4}}d x \right ) b c \,d^{3} x^{2}+7 \sqrt {d x +c}\, \left (\int \frac {\mathit {asin} \left (d x +c \right )}{\sqrt {d x +c}\, c^{4}+4 \sqrt {d x +c}\, c^{3} d x +6 \sqrt {d x +c}\, c^{2} d^{2} x^{2}+4 \sqrt {d x +c}\, c \,d^{3} x^{3}+\sqrt {d x +c}\, d^{4} x^{4}}d x \right ) b \,d^{4} x^{3}-2 a}{7 \sqrt {e}\, \sqrt {d x +c}\, d \,e^{4} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )} \] Input:

int((a+b*asin(d*x+c))/(d*e*x+c*e)^(9/2),x)
 

Output:

(7*sqrt(c + d*x)*int(asin(c + d*x)/(sqrt(c + d*x)*c**4 + 4*sqrt(c + d*x)*c 
**3*d*x + 6*sqrt(c + d*x)*c**2*d**2*x**2 + 4*sqrt(c + d*x)*c*d**3*x**3 + s 
qrt(c + d*x)*d**4*x**4),x)*b*c**3*d + 21*sqrt(c + d*x)*int(asin(c + d*x)/( 
sqrt(c + d*x)*c**4 + 4*sqrt(c + d*x)*c**3*d*x + 6*sqrt(c + d*x)*c**2*d**2* 
x**2 + 4*sqrt(c + d*x)*c*d**3*x**3 + sqrt(c + d*x)*d**4*x**4),x)*b*c**2*d* 
*2*x + 21*sqrt(c + d*x)*int(asin(c + d*x)/(sqrt(c + d*x)*c**4 + 4*sqrt(c + 
 d*x)*c**3*d*x + 6*sqrt(c + d*x)*c**2*d**2*x**2 + 4*sqrt(c + d*x)*c*d**3*x 
**3 + sqrt(c + d*x)*d**4*x**4),x)*b*c*d**3*x**2 + 7*sqrt(c + d*x)*int(asin 
(c + d*x)/(sqrt(c + d*x)*c**4 + 4*sqrt(c + d*x)*c**3*d*x + 6*sqrt(c + d*x) 
*c**2*d**2*x**2 + 4*sqrt(c + d*x)*c*d**3*x**3 + sqrt(c + d*x)*d**4*x**4),x 
)*b*d**4*x**3 - 2*a)/(7*sqrt(e)*sqrt(c + d*x)*d*e**4*(c**3 + 3*c**2*d*x + 
3*c*d**2*x**2 + d**3*x**3))