3.2.54 \(\int \frac {x^m (a+b \arcsin (c x))}{(d-c^2 d x^2)^{5/2}} \, dx\) [154]

3.2.54.1 Optimal result

Integrand size = 27, antiderivative size = 408 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {x^{1+m} (a+b \arcsin (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(2-m) x^{1+m} (a+b \arcsin (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(2-m) m x^{1+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{3 d^2 (1+m) \sqrt {d-c^2 d x^2}}-\frac {b c (2-m) x^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{3 d^2 (2+m) \sqrt {d-c^2 d x^2}}-\frac {b c x^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (2,\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{3 d^2 (2+m) \sqrt {d-c^2 d x^2}}+\frac {b c (2-m) m x^{2+m} \sqrt {1-c^2 x^2} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{3 d^2 \left (2+3 m+m^2\right ) \sqrt {d-c^2 d x^2}} \]

output

1/3*x^(1+m)*(a+b*arcsin(c*x))/d/(-c^2*d*x^2+d)^(3/2)+1/3*(2-m)*x^(1+m)*(a+ 
b*arcsin(c*x))/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*(2-m)*m*x^(1+m)*(a+b*arcsin(c* 
x))*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)/d^2 
/(1+m)/(-c^2*d*x^2+d)^(1/2)-1/3*b*c*(2-m)*x^(2+m)*hypergeom([1, 1+1/2*m],[ 
2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)/d^2/(2+m)/(-c^2*d*x^2+d)^(1/2)-1/3*b* 
c*x^(2+m)*hypergeom([2, 1+1/2*m],[2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)/d^2 
/(2+m)/(-c^2*d*x^2+d)^(1/2)+1/3*b*c*(2-m)*m*x^(2+m)*hypergeom([1, 1+1/2*m, 
 1+1/2*m],[2+1/2*m, 3/2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)/d^2/(m^2+3*m+2) 
/(-c^2*d*x^2+d)^(1/2)
 

3.2.54.2 Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.68 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {x^{1+m} \left (d (1+m) (2+m) (a+b \arcsin (c x))-b c d (1+m) x \left (1-c^2 x^2\right )^{3/2} \operatorname {Hypergeometric2F1}\left (2,1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )+(2-m) \left (d-c^2 d x^2\right ) \left ((1+m) (2+m) (a+b \arcsin (c x))-b c (1+m) x \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )-m \sqrt {1-c^2 x^2} \left ((2+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )-b c x \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )\right )\right )\right )}{3 d^2 (1+m) (2+m) \left (d-c^2 d x^2\right )^{3/2}} \]

input

Integrate[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]
 

output

(x^(1 + m)*(d*(1 + m)*(2 + m)*(a + b*ArcSin[c*x]) - b*c*d*(1 + m)*x*(1 - c 
^2*x^2)^(3/2)*Hypergeometric2F1[2, 1 + m/2, 2 + m/2, c^2*x^2] + (2 - m)*(d 
 - c^2*d*x^2)*((1 + m)*(2 + m)*(a + b*ArcSin[c*x]) - b*c*(1 + m)*x*Sqrt[1 
- c^2*x^2]*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, c^2*x^2] - m*Sqrt[1 - c^ 
2*x^2]*((2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + 
 m)/2, c^2*x^2] - b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/ 
2, 2 + m/2}, c^2*x^2]))))/(3*d^2*(1 + m)*(2 + m)*(d - c^2*d*x^2)^(3/2))
 

3.2.54.3 Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5208, 278, 5208, 278, 5220}

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.

\(\displaystyle \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5208

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

\(\Big \downarrow \) 278

\(\displaystyle \frac {(2-m) \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 d}+\frac {x^{m+1} (a+b \arcsin (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} x^{m+2} \operatorname {Hypergeometric2F1}\left (2,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{3 d^2 (m+2) \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5208

\(\displaystyle \frac {(2-m) \left (-\frac {m \int \frac {x^m (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}}dx}{d}-\frac {b c \sqrt {1-c^2 x^2} \int \frac {x^{m+1}}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x^{m+1} (a+b \arcsin (c x))}{d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x^{m+1} (a+b \arcsin (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} x^{m+2} \operatorname {Hypergeometric2F1}\left (2,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{3 d^2 (m+2) \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {(2-m) \left (-\frac {m \int \frac {x^m (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}}dx}{d}+\frac {x^{m+1} (a+b \arcsin (c x))}{d \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} x^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{d (m+2) \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x^{m+1} (a+b \arcsin (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} x^{m+2} \operatorname {Hypergeometric2F1}\left (2,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{3 d^2 (m+2) \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5220

\(\displaystyle \frac {(2-m) \left (-\frac {m \left (\frac {\sqrt {1-c^2 x^2} x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{(m+1) \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} x^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt {d-c^2 d x^2}}\right )}{d}+\frac {x^{m+1} (a+b \arcsin (c x))}{d \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} x^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{d (m+2) \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x^{m+1} (a+b \arcsin (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} x^{m+2} \operatorname {Hypergeometric2F1}\left (2,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{3 d^2 (m+2) \sqrt {d-c^2 d x^2}}\)

input

Int[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]
 

output

(x^(1 + m)*(a + b*ArcSin[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) - (b*c*x^(2 + 
m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, c^2*x^2])/ 
(3*d^2*(2 + m)*Sqrt[d - c^2*d*x^2]) + ((2 - m)*((x^(1 + m)*(a + b*ArcSin[c 
*x]))/(d*Sqrt[d - c^2*d*x^2]) - (b*c*x^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeom 
etric2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d*(2 + m)*Sqrt[d - c^2*d*x^2] 
) - (m*((x^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1 
[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((1 + m)*Sqrt[d - c^2*d*x^2]) - (b*c 
*x^(2 + m)*Sqrt[1 - c^2*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 
 + m/2, 2 + m/2}, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[d - c^2*d*x^2])))/d))/(3 
*d)
 

3.2.54.3.1 Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 5208

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) 
   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c 
*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)* 
(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b 
, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

rule 5220

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_. 
)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2* 
x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, 
 (3 + m)/2, c^2*x^2], x] - Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*S 
imp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m 
/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] 
&& EqQ[c^2*d + e, 0] &&  !IntegerQ[m]
 

3.2.54.4 Maple [F]

\[\int \frac {x^{m} \left (a +b \arcsin \left (c x \right )\right )}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

input

int(x^m*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x)
 

output

int(x^m*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x)
 

3.2.54.5 Fricas [F]

\[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(x^m*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas" 
)
 

output

integral(-sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)*x^m/(c^6*d^3*x^6 - 3*c^ 
4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
 

3.2.54.6 Sympy [F]

\[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{m} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

input

integrate(x**m*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(5/2),x)
 

output

Integral(x**m*(a + b*asin(c*x))/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
 

3.2.54.7 Maxima [F]

\[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(x^m*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima" 
)
 

output

integrate((b*arcsin(c*x) + a)*x^m/(-c^2*d*x^2 + d)^(5/2), x)
 

3.2.54.8 Giac [F(-2)]

Exception generated. \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate(x^m*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

3.2.54.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^m\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

input

int((x^m*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2),x)
 

output

int((x^m*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2), x)