\(\int \frac {x^m (a+b \arcsin (c x))^2}{(d-c^2 d x^2)^3} \, dx\) [281]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {b c x^{2+m} (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (1-m) x^{2+m} (a+b \arcsin (c x))}{6 d^3 \sqrt {1-c^2 x^2}}-\frac {b c (3-m) x^{2+m} (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}+\frac {x^{1+m} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {b c (1-m) (1+m) x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{6 d^3 (2+m)}+\frac {b c (3-m) (1+m) x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{4 d^3 (2+m)}+\frac {b^2 c^2 (1-m) x^{3+m} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{6 d^3 (3+m)}+\frac {b^2 c^2 (3-m) x^{3+m} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{4 d^3 (3+m)}+\frac {b^2 c^2 x^{3+m} \operatorname {Hypergeometric2F1}\left (2,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{6 d^3 (3+m)}-\frac {b^2 c^2 (1-m) (1+m) x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{6 d^3 \left (6+5 m+m^2\right )}-\frac {b^2 c^2 (3-m) (1+m) x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{4 d^3 \left (6+5 m+m^2\right )}+\frac {(1-m) (3-m) \text {Int}\left (\frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2},x\right )}{8 d^2} \]

[Out]

-1/6*b*c*x^(2+m)*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^(3/2)+1/4*x^(1+m)*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)^2+1
/8*(3-m)*x^(1+m)*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)+1/6*b*c*(1-m)*(1+m)*x^(2+m)*(a+b*arcsin(c*x))*hypergeom(
[1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/d^3/(2+m)+1/4*b*c*(3-m)*(1+m)*x^(2+m)*(a+b*arcsin(c*x))*hypergeom([1/2, 1+1/
2*m],[2+1/2*m],c^2*x^2)/d^3/(2+m)+1/6*b^2*c^2*(1-m)*x^(3+m)*hypergeom([1, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)/d^3/
(3+m)+1/4*b^2*c^2*(3-m)*x^(3+m)*hypergeom([1, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)/d^3/(3+m)+1/6*b^2*c^2*x^(3+m)*hy
pergeom([2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)/d^3/(3+m)-1/6*b^2*c^2*(1-m)*(1+m)*x^(3+m)*hypergeom([1, 3/2+1/2*m,
3/2+1/2*m],[2+1/2*m, 5/2+1/2*m],c^2*x^2)/d^3/(m^2+5*m+6)-1/4*b^2*c^2*(3-m)*(1+m)*x^(3+m)*hypergeom([1, 3/2+1/
2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/2*m],c^2*x^2)/d^3/(m^2+5*m+6)-1/6*b*c*(1-m)*x^(2+m)*(a+b*arcsin(c*x))/d^3/(-c^
2*x^2+1)^(1/2)-1/4*b*c*(3-m)*x^(2+m)*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^(1/2)+1/8*(1-m)*(3-m)*Unintegrable(x^m
*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d),x)/d^2

Rubi [N/A]

Not integrable

Time = 0.62 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx \]

[In]

Int[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]

[Out]

-1/6*(b*c*x^(2 + m)*(a + b*ArcSin[c*x]))/(d^3*(1 - c^2*x^2)^(3/2)) - (b*c*(1 - m)*x^(2 + m)*(a + b*ArcSin[c*x]
))/(6*d^3*Sqrt[1 - c^2*x^2]) - (b*c*(3 - m)*x^(2 + m)*(a + b*ArcSin[c*x]))/(4*d^3*Sqrt[1 - c^2*x^2]) + (x^(1 +
m)*(a + b*ArcSin[c*x])^2)/(4*d^3*(1 - c^2*x^2)^2) + ((3 - m)*x^(1 + m)*(a + b*ArcSin[c*x])^2)/(8*d^3*(1 - c^2
*x^2)) + (b*c*(1 - m)*(1 + m)*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x
^2])/(6*d^3*(2 + m)) + (b*c*(3 - m)*(1 + m)*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4
+ m)/2, c^2*x^2])/(4*d^3*(2 + m)) + (b^2*c^2*(1 - m)*x^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, c^2
*x^2])/(6*d^3*(3 + m)) + (b^2*c^2*(3 - m)*x^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, c^2*x^2])/(4*d^
3*(3 + m)) + (b^2*c^2*x^(3 + m)*Hypergeometric2F1[2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(6*d^3*(3 + m)) - (b^2*c^
2*(1 - m)*(1 + m)*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(6*d^
3*(6 + 5*m + m^2)) - (b^2*c^2*(3 - m)*(1 + m)*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2,
5/2 + m/2}, c^2*x^2])/(4*d^3*(6 + 5*m + m^2)) + ((1 - m)*(3 - m)*Defer[Int][(x^m*(a + b*ArcSin[c*x])^2)/(d -
c^2*d*x^2), x])/(8*d^2)

Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 d^3}+\frac {(3-m) \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d} \\ & = -\frac {b c x^{2+m} (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {x^{1+m} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {\left (b^2 c^2\right ) \int \frac {x^{2+m}}{\left (1-c^2 x^2\right )^2} \, dx}{6 d^3}-\frac {(b c (1-m)) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{6 d^3}-\frac {(b c (3-m)) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{4 d^3}+\frac {((1-m) (3-m)) \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx}{8 d^2} \\ & = -\frac {b c x^{2+m} (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (1-m) x^{2+m} (a+b \arcsin (c x))}{6 d^3 \sqrt {1-c^2 x^2}}-\frac {b c (3-m) x^{2+m} (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}+\frac {x^{1+m} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {b^2 c^2 x^{3+m} \operatorname {Hypergeometric2F1}\left (2,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{6 d^3 (3+m)}+\frac {\left (b^2 c^2 (1-m)\right ) \int \frac {x^{2+m}}{1-c^2 x^2} \, dx}{6 d^3}+\frac {\left (b^2 c^2 (3-m)\right ) \int \frac {x^{2+m}}{1-c^2 x^2} \, dx}{4 d^3}+\frac {((1-m) (3-m)) \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx}{8 d^2}+\frac {(b c (1-m) (1+m)) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{6 d^3}+\frac {(b c (3-m) (1+m)) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{4 d^3} \\ & = -\frac {b c x^{2+m} (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (1-m) x^{2+m} (a+b \arcsin (c x))}{6 d^3 \sqrt {1-c^2 x^2}}-\frac {b c (3-m) x^{2+m} (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}+\frac {x^{1+m} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {b c (1-m) (1+m) x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{6 d^3 (2+m)}+\frac {b c (3-m) (1+m) x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{4 d^3 (2+m)}+\frac {b^2 c^2 (1-m) x^{3+m} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{6 d^3 (3+m)}+\frac {b^2 c^2 (3-m) x^{3+m} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{4 d^3 (3+m)}+\frac {b^2 c^2 x^{3+m} \operatorname {Hypergeometric2F1}\left (2,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{6 d^3 (3+m)}-\frac {b^2 c^2 (1-m) (1+m) x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{6 d^3 \left (6+5 m+m^2\right )}-\frac {b^2 c^2 (3-m) (1+m) x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{4 d^3 \left (6+5 m+m^2\right )}+\frac {((1-m) (3-m)) \int \frac {x^m (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx}{8 d^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 9.81 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx \]

[In]

Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]

[Out]

Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3, x]

Maple [N/A] (verified)

Not integrable

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

\[\int \frac {x^{m} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (-c^{2} d \,x^{2}+d \right )^{3}}d x\]

[In]

int(x^m*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x)

[Out]

int(x^m*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.56 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]

[In]

integrate(x^m*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="fricas")

[Out]

integral(-(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)*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)

Sympy [N/A]

Not integrable

Time = 117.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.41 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2} x^{m}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{m} \operatorname {asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]

[In]

integrate(x**m*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**3,x)

[Out]

-(Integral(a**2*x**m/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(b**2*x**m*asin(c*x)**2/(c**6*x
**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(2*a*b*x**m*asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**
2 - 1), x))/d**3

Maxima [N/A]

Not integrable

Time = 0.90 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]

[In]

integrate(x^m*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

-integrate((b*arcsin(c*x) + a)^2*x^m/(c^2*d*x^2 - d)^3, x)

Giac [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]

[In]

integrate(x^m*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="giac")

[Out]

integrate(-(b*arcsin(c*x) + a)^2*x^m/(c^2*d*x^2 - d)^3, x)

Mupad [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]

[In]

int((x^m*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^3,x)

[Out]

int((x^m*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^3, x)