\(\int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{11/2}} \, dx\) [164]

Optimal result

Integrand size = 23, antiderivative size = 139 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{11/2}} \, dx=-\frac {4 b \sqrt {1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}-\frac {20 b \sqrt {1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac {2 (a+b \arcsin (c+d x))}{9 d e (e (c+d x))^{9/2}}+\frac {20 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{189 d e^{11/2}} \] Output:

-4/63*b*(1-(d*x+c)^2)^(1/2)/d/e^2/(e*(d*x+c))^(7/2)-20/189*b*(1-(d*x+c)^2) 
^(1/2)/d/e^4/(e*(d*x+c))^(3/2)-2/9*(a+b*arcsin(d*x+c))/d/e/(e*(d*x+c))^(9/ 
2)+20/189*b*EllipticF((e*(d*x+c))^(1/2)/e^(1/2),I)/d/e^(11/2)
 

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 = 0.47 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{11/2}} \, dx=-\frac {2 \sqrt {e (c+d x)} \left (7 (a+b \arcsin (c+d x))+2 b (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {1}{2},-\frac {3}{4},(c+d x)^2\right )\right )}{63 d e^6 (c+d x)^5} \] Input:

Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(11/2),x]
 

Output:

(-2*Sqrt[e*(c + d*x)]*(7*(a + b*ArcSin[c + d*x]) + 2*b*(c + d*x)*Hypergeom 
etric2F1[-7/4, 1/2, -3/4, (c + d*x)^2]))/(63*d*e^6*(c + d*x)^5)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5304, 5138, 264, 264, 266, 762}

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.

\(\displaystyle \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{11/2}} \, dx\)

\(\Big \downarrow \) 5304

\(\displaystyle \frac {\int \frac {a+b \arcsin (c+d x)}{(e (c+d x))^{11/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {\frac {2 b \int \frac {1}{(e (c+d x))^{9/2} \sqrt {1-(c+d x)^2}}d(c+d x)}{9 e}-\frac {2 (a+b \arcsin (c+d x))}{9 e (e (c+d x))^{9/2}}}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {\frac {2 b \left (\frac {5 \int \frac {1}{(e (c+d x))^{5/2} \sqrt {1-(c+d x)^2}}d(c+d x)}{7 e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{7 e (e (c+d x))^{7/2}}\right )}{9 e}-\frac {2 (a+b \arcsin (c+d x))}{9 e (e (c+d x))^{9/2}}}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {\frac {2 b \left (\frac {5 \left (\frac {\int \frac {1}{\sqrt {e (c+d x)} \sqrt {1-(c+d x)^2}}d(c+d x)}{3 e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{3 e (e (c+d x))^{3/2}}\right )}{7 e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{7 e (e (c+d x))^{7/2}}\right )}{9 e}-\frac {2 (a+b \arcsin (c+d x))}{9 e (e (c+d x))^{9/2}}}{d}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\frac {2 b \left (\frac {5 \left (\frac {2 \int \frac {1}{\sqrt {1-(c+d x)^2}}d\sqrt {e (c+d x)}}{3 e^3}-\frac {2 \sqrt {1-(c+d x)^2}}{3 e (e (c+d x))^{3/2}}\right )}{7 e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{7 e (e (c+d x))^{7/2}}\right )}{9 e}-\frac {2 (a+b \arcsin (c+d x))}{9 e (e (c+d x))^{9/2}}}{d}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\frac {2 b \left (\frac {5 \left (\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{3 e^{5/2}}-\frac {2 \sqrt {1-(c+d x)^2}}{3 e (e (c+d x))^{3/2}}\right )}{7 e^2}-\frac {2 \sqrt {1-(c+d x)^2}}{7 e (e (c+d x))^{7/2}}\right )}{9 e}-\frac {2 (a+b \arcsin (c+d x))}{9 e (e (c+d x))^{9/2}}}{d}\)

Input:

Int[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(11/2),x]
 

Output:

((-2*(a + b*ArcSin[c + d*x]))/(9*e*(e*(c + d*x))^(9/2)) + (2*b*((-2*Sqrt[1 
 - (c + d*x)^2])/(7*e*(e*(c + d*x))^(7/2)) + (5*((-2*Sqrt[1 - (c + d*x)^2] 
)/(3*e*(e*(c + d*x))^(3/2)) + (2*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e 
]], -1])/(3*e^(5/2))))/(7*e^2)))/(9*e))/d
 

Defintions of rubi rules used

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 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && 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 = 1.67 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{9 \left (d e x +c e \right )^{\frac {9}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{9 \left (d e x +c e \right )^{\frac {9}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{63 \left (d e x +c e \right )^{\frac {7}{2}}}-\frac {10 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{189 e^{2} \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {10 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{189 e^{4} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(203\)
default	\(\frac {-\frac {2 a}{9 \left (d e x +c e \right )^{\frac {9}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{9 \left (d e x +c e \right )^{\frac {9}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{63 \left (d e x +c e \right )^{\frac {7}{2}}}-\frac {10 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{189 e^{2} \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {10 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{189 e^{4} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(203\)
parts	\(-\frac {2 a}{9 \left (d e x +c e \right )^{\frac {9}{2}} d e}+\frac {2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{9 \left (d e x +c e \right )^{\frac {9}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{63 \left (d e x +c e \right )^{\frac {7}{2}}}-\frac {10 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{189 e^{2} \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {10 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{189 e^{4} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(208\)

Input:

int((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

2/d/e*(-1/9*a/(d*e*x+c*e)^(9/2)+b*(-1/9/(d*e*x+c*e)^(9/2)*arcsin((d*e*x+c* 
e)/e)+2/9/e*(-1/7*(-(d*e*x+c*e)^2/e^2+1)^(1/2)/(d*e*x+c*e)^(7/2)-5/21/e^2* 
(-(d*e*x+c*e)^2/e^2+1)^(1/2)/(d*e*x+c*e)^(3/2)+5/21/e^4/(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)*Ell 
ipticF((d*e*x+c*e)^(1/2)*(1/e)^(1/2),I))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (114) = 228\).

Time = 0.11 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.89 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{11/2}} \, dx=-\frac {2 \, {\left (10 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (21 \, b d^{2} \arcsin \left (d x + c\right ) + 21 \, a d^{2} + 2 \, {\left (5 \, b d^{5} x^{3} + 15 \, b c d^{4} x^{2} + 3 \, {\left (5 \, b c^{2} + b\right )} d^{3} x + {\left (5 \, b c^{3} + 3 \, b c\right )} d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )} \sqrt {d e x + c e}\right )}}{189 \, {\left (d^{8} e^{6} x^{5} + 5 \, c d^{7} e^{6} x^{4} + 10 \, c^{2} d^{6} e^{6} x^{3} + 10 \, c^{3} d^{5} e^{6} x^{2} + 5 \, c^{4} d^{4} e^{6} x + c^{5} d^{3} e^{6}\right )}} \] Input:

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(11/2),x, algorithm="fricas")
 

Output:

-2/189*(10*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 + 10*b*c^3*d^2*x^ 
2 + 5*b*c^4*d*x + b*c^5)*sqrt(-d^3*e)*weierstrassPInverse(4/d^2, 0, (d*x + 
 c)/d) + (21*b*d^2*arcsin(d*x + c) + 21*a*d^2 + 2*(5*b*d^5*x^3 + 15*b*c*d^ 
4*x^2 + 3*(5*b*c^2 + b)*d^3*x + (5*b*c^3 + 3*b*c)*d^2)*sqrt(-d^2*x^2 - 2*c 
*d*x - c^2 + 1))*sqrt(d*e*x + c*e))/(d^8*e^6*x^5 + 5*c*d^7*e^6*x^4 + 10*c^ 
2*d^6*e^6*x^3 + 10*c^3*d^5*e^6*x^2 + 5*c^4*d^4*e^6*x + c^5*d^3*e^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*asin(d*x+c))/(d*e*x+c*e)**(11/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{11/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(11/2),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]

\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{11/2}} \, dx=\int { \frac {b \arcsin \left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(11/2),x, algorithm="giac")
 

Output:

integrate((b*arcsin(d*x + c) + a)/(d*e*x + c*e)^(11/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{11/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{11/2}} \,d x \] Input:

int((a + b*asin(c + d*x))/(c*e + d*e*x)^(11/2),x)
 

Output:

int((a + b*asin(c + d*x))/(c*e + d*e*x)^(11/2), x)
 

Reduce [F]

\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{11/2}} \, dx=\frac {9 \sqrt {d x +c}\, \left (\int \frac {\mathit {asin} \left (d x +c \right )}{\sqrt {d x +c}\, c^{5}+5 \sqrt {d x +c}\, c^{4} d x +10 \sqrt {d x +c}\, c^{3} d^{2} x^{2}+10 \sqrt {d x +c}\, c^{2} d^{3} x^{3}+5 \sqrt {d x +c}\, c \,d^{4} x^{4}+\sqrt {d x +c}\, d^{5} x^{5}}d x \right ) b \,c^{4} d +36 \sqrt {d x +c}\, \left (\int \frac {\mathit {asin} \left (d x +c \right )}{\sqrt {d x +c}\, c^{5}+5 \sqrt {d x +c}\, c^{4} d x +10 \sqrt {d x +c}\, c^{3} d^{2} x^{2}+10 \sqrt {d x +c}\, c^{2} d^{3} x^{3}+5 \sqrt {d x +c}\, c \,d^{4} x^{4}+\sqrt {d x +c}\, d^{5} x^{5}}d x \right ) b \,c^{3} d^{2} x +54 \sqrt {d x +c}\, \left (\int \frac {\mathit {asin} \left (d x +c \right )}{\sqrt {d x +c}\, c^{5}+5 \sqrt {d x +c}\, c^{4} d x +10 \sqrt {d x +c}\, c^{3} d^{2} x^{2}+10 \sqrt {d x +c}\, c^{2} d^{3} x^{3}+5 \sqrt {d x +c}\, c \,d^{4} x^{4}+\sqrt {d x +c}\, d^{5} x^{5}}d x \right ) b \,c^{2} d^{3} x^{2}+36 \sqrt {d x +c}\, \left (\int \frac {\mathit {asin} \left (d x +c \right )}{\sqrt {d x +c}\, c^{5}+5 \sqrt {d x +c}\, c^{4} d x +10 \sqrt {d x +c}\, c^{3} d^{2} x^{2}+10 \sqrt {d x +c}\, c^{2} d^{3} x^{3}+5 \sqrt {d x +c}\, c \,d^{4} x^{4}+\sqrt {d x +c}\, d^{5} x^{5}}d x \right ) b c \,d^{4} x^{3}+9 \sqrt {d x +c}\, \left (\int \frac {\mathit {asin} \left (d x +c \right )}{\sqrt {d x +c}\, c^{5}+5 \sqrt {d x +c}\, c^{4} d x +10 \sqrt {d x +c}\, c^{3} d^{2} x^{2}+10 \sqrt {d x +c}\, c^{2} d^{3} x^{3}+5 \sqrt {d x +c}\, c \,d^{4} x^{4}+\sqrt {d x +c}\, d^{5} x^{5}}d x \right ) b \,d^{5} x^{4}-2 a}{9 \sqrt {e}\, \sqrt {d x +c}\, d \,e^{5} \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}\right )} \] Input:

int((a+b*asin(d*x+c))/(d*e*x+c*e)^(11/2),x)
 

Output:

(9*sqrt(c + d*x)*int(asin(c + d*x)/(sqrt(c + d*x)*c**5 + 5*sqrt(c + d*x)*c 
**4*d*x + 10*sqrt(c + d*x)*c**3*d**2*x**2 + 10*sqrt(c + d*x)*c**2*d**3*x** 
3 + 5*sqrt(c + d*x)*c*d**4*x**4 + sqrt(c + d*x)*d**5*x**5),x)*b*c**4*d + 3 
6*sqrt(c + d*x)*int(asin(c + d*x)/(sqrt(c + d*x)*c**5 + 5*sqrt(c + d*x)*c* 
*4*d*x + 10*sqrt(c + d*x)*c**3*d**2*x**2 + 10*sqrt(c + d*x)*c**2*d**3*x**3 
 + 5*sqrt(c + d*x)*c*d**4*x**4 + sqrt(c + d*x)*d**5*x**5),x)*b*c**3*d**2*x 
 + 54*sqrt(c + d*x)*int(asin(c + d*x)/(sqrt(c + d*x)*c**5 + 5*sqrt(c + d*x 
)*c**4*d*x + 10*sqrt(c + d*x)*c**3*d**2*x**2 + 10*sqrt(c + d*x)*c**2*d**3* 
x**3 + 5*sqrt(c + d*x)*c*d**4*x**4 + sqrt(c + d*x)*d**5*x**5),x)*b*c**2*d* 
*3*x**2 + 36*sqrt(c + d*x)*int(asin(c + d*x)/(sqrt(c + d*x)*c**5 + 5*sqrt( 
c + d*x)*c**4*d*x + 10*sqrt(c + d*x)*c**3*d**2*x**2 + 10*sqrt(c + d*x)*c** 
2*d**3*x**3 + 5*sqrt(c + d*x)*c*d**4*x**4 + sqrt(c + d*x)*d**5*x**5),x)*b* 
c*d**4*x**3 + 9*sqrt(c + d*x)*int(asin(c + d*x)/(sqrt(c + d*x)*c**5 + 5*sq 
rt(c + d*x)*c**4*d*x + 10*sqrt(c + d*x)*c**3*d**2*x**2 + 10*sqrt(c + d*x)* 
c**2*d**3*x**3 + 5*sqrt(c + d*x)*c*d**4*x**4 + sqrt(c + d*x)*d**5*x**5),x) 
*b*d**5*x**4 - 2*a)/(9*sqrt(e)*sqrt(c + d*x)*d*e**5*(c**4 + 4*c**3*d*x + 6 
*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4))