\(\int (c e+d e x)^{7/2} (a+b \arcsin (c+d x)) \, dx\) [165]

Optimal result

Integrand size = 23, antiderivative size = 169 \[ \int (c e+d e x)^{7/2} (a+b \arcsin (c+d x)) \, dx=\frac {28 b e^2 (e (c+d x))^{3/2} \sqrt {1-(c+d x)^2}}{405 d}+\frac {4 b (e (c+d x))^{7/2} \sqrt {1-(c+d x)^2}}{81 d}+\frac {2 (e (c+d x))^{9/2} (a+b \arcsin (c+d x))}{9 d e}-\frac {28 b e^{7/2} E\left (\left .\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{135 d}+\frac {28 b e^{7/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{135 d} \] Output:

28/405*b*e^2*(e*(d*x+c))^(3/2)*(1-(d*x+c)^2)^(1/2)/d+4/81*b*(e*(d*x+c))^(7 
/2)*(1-(d*x+c)^2)^(1/2)/d+2/9*(e*(d*x+c))^(9/2)*(a+b*arcsin(d*x+c))/d/e-28 
/135*b*e^(7/2)*EllipticE((e*(d*x+c))^(1/2)/e^(1/2),I)/d+28/135*b*e^(7/2)*E 
llipticF((e*(d*x+c))^(1/2)/e^(1/2),I)/d
 

Mathematica [C] (verified)

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

Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.68 \[ \int (c e+d e x)^{7/2} (a+b \arcsin (c+d x)) \, dx=\frac {2 (e (c+d x))^{7/2} \left (45 a (c+d x)^3+14 b \sqrt {1-(c+d x)^2}+10 b (c+d x)^2 \sqrt {1-(c+d x)^2}+45 b (c+d x)^3 \arcsin (c+d x)-14 b \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},(c+d x)^2\right )\right )}{405 d (c+d x)^2} \] Input:

Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSin[c + d*x]),x]
 

Output:

(2*(e*(c + d*x))^(7/2)*(45*a*(c + d*x)^3 + 14*b*Sqrt[1 - (c + d*x)^2] + 10 
*b*(c + d*x)^2*Sqrt[1 - (c + d*x)^2] + 45*b*(c + d*x)^3*ArcSin[c + d*x] - 
14*b*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(405*d*(c + d*x)^2)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5304, 5138, 262, 262, 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.

\(\displaystyle \int (c e+d e x)^{7/2} (a+b \arcsin (c+d x)) \, dx\)

\(\Big \downarrow \) 5304

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 261

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

\(\Big \downarrow \) 259

\(\displaystyle \frac {\frac {2 (e (c+d x))^{9/2} (a+b \arcsin (c+d x))}{9 e}-\frac {2 b \left (\frac {7}{9} e^2 \left (-\frac {6 e^2 \sqrt {e (c+d x)} \int \frac {\sqrt {c+d x}}{\sqrt {\frac {1}{2} (c+d x-1)+1}}d\frac {\sqrt {-c-d x+1}}{\sqrt {2}}}{5 \sqrt {c+d x}}-\frac {2}{5} e \sqrt {1-(c+d x)^2} (e (c+d x))^{3/2}\right )-\frac {2}{9} e \sqrt {1-(c+d x)^2} (e (c+d x))^{7/2}\right )}{9 e}}{d}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {2 (e (c+d x))^{9/2} (a+b \arcsin (c+d x))}{9 e}-\frac {2 b \left (\frac {7}{9} e^2 \left (-\frac {6 e^2 \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{5 \sqrt {c+d x}}-\frac {2}{5} e \sqrt {1-(c+d x)^2} (e (c+d x))^{3/2}\right )-\frac {2}{9} e \sqrt {1-(c+d x)^2} (e (c+d x))^{7/2}\right )}{9 e}}{d}\)

Input:

Int[(c*e + d*e*x)^(7/2)*(a + b*ArcSin[c + d*x]),x]
 

Output:

((2*(e*(c + d*x))^(9/2)*(a + b*ArcSin[c + d*x]))/(9*e) - (2*b*((-2*e*(e*(c 
 + d*x))^(7/2)*Sqrt[1 - (c + d*x)^2])/9 + (7*e^2*((-2*e*(e*(c + d*x))^(3/2 
)*Sqrt[1 - (c + d*x)^2])/5 - (6*e^2*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqr 
t[1 - c - d*x]/Sqrt[2]], 2])/(5*Sqrt[c + d*x])))/9))/(9*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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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 = 2.40 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.35

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (d e x +c e \right )^{\frac {9}{2}}}{9}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {9}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{9}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {7}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{9}-\frac {7 e^{4} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{45}-\frac {7 e^{5} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{15 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{9 e}\right )}{d e}\)	\(228\)
default	\(\frac {\frac {2 a \left (d e x +c e \right )^{\frac {9}{2}}}{9}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {9}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{9}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {7}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{9}-\frac {7 e^{4} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{45}-\frac {7 e^{5} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{15 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{9 e}\right )}{d e}\)	\(228\)
parts	\(\frac {2 a \left (d e x +c e \right )^{\frac {9}{2}}}{9 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {9}{2}} \arcsin \left (\frac {d e x +c e}{e}\right )}{9}-\frac {2 \left (-\frac {e^{2} \left (d e x +c e \right )^{\frac {7}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{9}-\frac {7 e^{4} \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{45}-\frac {7 e^{5} \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{15 \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{9 e}\right )}{d e}\)	\(233\)

Input:

int((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c)),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*(-1/9*e^2*(d*e*x+c*e)^(7/2)*(-(d*e*x+c*e)^2/e^2+1)^(1/2)-7/45*e^ 
4*(d*e*x+c*e)^(3/2)*(-(d*e*x+c*e)^2/e^2+1)^(1/2)-7/15*e^5/(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)*( 
EllipticF((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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (135) = 270\).

Time = 0.11 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.62 \[ \int (c e+d e x)^{7/2} (a+b \arcsin (c+d x)) \, dx=-\frac {2 \, {\left (42 \, \sqrt {-d^{3} e} b e^{3} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) - {\left (45 \, a d^{5} e^{3} x^{4} + 180 \, a c d^{4} e^{3} x^{3} + 270 \, a c^{2} d^{3} e^{3} x^{2} + 180 \, a c^{3} d^{2} e^{3} x + 45 \, a c^{4} d e^{3} + 45 \, {\left (b d^{5} e^{3} x^{4} + 4 \, b c d^{4} e^{3} x^{3} + 6 \, b c^{2} d^{3} e^{3} x^{2} + 4 \, b c^{3} d^{2} e^{3} x + b c^{4} d e^{3}\right )} \arcsin \left (d x + c\right ) + 2 \, {\left (5 \, b d^{4} e^{3} x^{3} + 15 \, b c d^{3} e^{3} x^{2} + {\left (15 \, b c^{2} + 7 \, b\right )} d^{2} e^{3} x + {\left (5 \, b c^{3} + 7 \, b c\right )} d e^{3}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )} \sqrt {d e x + c e}\right )}}{405 \, d^{2}} \] Input:

integrate((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c)),x, algorithm="fricas")
 

Output:

-2/405*(42*sqrt(-d^3*e)*b*e^3*weierstrassZeta(4/d^2, 0, weierstrassPInvers 
e(4/d^2, 0, (d*x + c)/d)) - (45*a*d^5*e^3*x^4 + 180*a*c*d^4*e^3*x^3 + 270* 
a*c^2*d^3*e^3*x^2 + 180*a*c^3*d^2*e^3*x + 45*a*c^4*d*e^3 + 45*(b*d^5*e^3*x 
^4 + 4*b*c*d^4*e^3*x^3 + 6*b*c^2*d^3*e^3*x^2 + 4*b*c^3*d^2*e^3*x + b*c^4*d 
*e^3)*arcsin(d*x + c) + 2*(5*b*d^4*e^3*x^3 + 15*b*c*d^3*e^3*x^2 + (15*b*c^ 
2 + 7*b)*d^2*e^3*x + (5*b*c^3 + 7*b*c)*d*e^3)*sqrt(-d^2*x^2 - 2*c*d*x - c^ 
2 + 1))*sqrt(d*e*x + c*e))/d^2
 

Sympy [F(-1)]

Timed out. \[ \int (c e+d e x)^{7/2} (a+b \arcsin (c+d x)) \, dx=\text {Timed out} \] Input:

integrate((d*e*x+c*e)**(7/2)*(a+b*asin(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int (c e+d e x)^{7/2} (a+b \arcsin (c+d x)) \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c)),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]

\[ \int (c e+d e x)^{7/2} (a+b \arcsin (c+d x)) \, dx=\int { {\left (d e x + c e\right )}^{\frac {7}{2}} {\left (b \arcsin \left (d x + c\right ) + a\right )} \,d x } \] Input:

integrate((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c)),x, algorithm="giac")
 

Output:

integrate((d*e*x + c*e)^(7/2)*(b*arcsin(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^{7/2} (a+b \arcsin (c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^{7/2}\,\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right ) \,d x \] Input:

int((c*e + d*e*x)^(7/2)*(a + b*asin(c + d*x)),x)
 

Output:

int((c*e + d*e*x)^(7/2)*(a + b*asin(c + d*x)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int (c e+d e x)^{7/2} (a+b \arcsin (c+d x)) \, dx=\frac {\sqrt {e}\, e^{3} \left (2 \sqrt {d x +c}\, a \,c^{4}+8 \sqrt {d x +c}\, a \,c^{3} d x +12 \sqrt {d x +c}\, a \,c^{2} d^{2} x^{2}+8 \sqrt {d x +c}\, a c \,d^{3} x^{3}+2 \sqrt {d x +c}\, a \,d^{4} x^{4}+9 \left (\int \sqrt {d x +c}\, \mathit {asin} \left (d x +c \right ) x^{3}d x \right ) b \,d^{4}+27 \left (\int \sqrt {d x +c}\, \mathit {asin} \left (d x +c \right ) x^{2}d x \right ) b c \,d^{3}+27 \left (\int \sqrt {d x +c}\, \mathit {asin} \left (d x +c \right ) x d x \right ) b \,c^{2} d^{2}+9 \left (\int \sqrt {d x +c}\, \mathit {asin} \left (d x +c \right )d x \right ) b \,c^{3} d \right )}{9 d} \] Input:

int((d*e*x+c*e)^(7/2)*(a+b*asin(d*x+c)),x)
 

Output:

(sqrt(e)*e**3*(2*sqrt(c + d*x)*a*c**4 + 8*sqrt(c + d*x)*a*c**3*d*x + 12*sq 
rt(c + d*x)*a*c**2*d**2*x**2 + 8*sqrt(c + d*x)*a*c*d**3*x**3 + 2*sqrt(c + 
d*x)*a*d**4*x**4 + 9*int(sqrt(c + d*x)*asin(c + d*x)*x**3,x)*b*d**4 + 27*i 
nt(sqrt(c + d*x)*asin(c + d*x)*x**2,x)*b*c*d**3 + 27*int(sqrt(c + d*x)*asi 
n(c + d*x)*x,x)*b*c**2*d**2 + 9*int(sqrt(c + d*x)*asin(c + d*x),x)*b*c**3* 
d))/(9*d)