3.282 \(\int x^m (d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=958 \[ \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 x^{m+1}}{m+6}+\frac {5 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 x^{m+1}}{(m+4) (m+6)}+\frac {15 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^{m+1}}{(m+6) \left (m^2+6 m+8\right )}-\frac {30 b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{(m+2)^2 (m+4) (m+6) \sqrt {1-c^2 x^2}}-\frac {10 b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{(m+6) \left (m^2+6 m+8\right ) \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{\left (m^2+8 m+12\right ) \sqrt {1-c^2 x^2}}+\frac {10 b^2 c^2 d^2 (3 m+10) \sqrt {d-c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};c^2 x^2\right ) x^{m+3}}{(m+2) (m+3) (m+4)^3 (m+6) \sqrt {1-c^2 x^2}}+\frac {2 b^2 c^2 d^2 \left (15 m^2+130 m+264\right ) \sqrt {d-c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};c^2 x^2\right ) x^{m+3}}{(m+2) (m+3) (m+4)^2 (m+6)^3 \sqrt {1-c^2 x^2}}+\frac {30 b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};c^2 x^2\right ) x^{m+3}}{(m+2)^2 (m+3) (m+4) (m+6) \sqrt {1-c^2 x^2}}+\frac {2 b^2 c^2 d^2 \left (m^2+15 m+52\right ) \sqrt {d-c^2 d x^2} x^{m+3}}{(m+4)^2 (m+6)^3}+\frac {10 b^2 c^2 d^2 \sqrt {d-c^2 d x^2} x^{m+3}}{(m+4)^3 (m+6)}+\frac {4 b c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+4}}{(m+4) (m+6) \sqrt {1-c^2 x^2}}+\frac {10 b c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+4}}{(m+4)^2 (m+6) \sqrt {1-c^2 x^2}}-\frac {2 b^2 c^4 d^2 \sqrt {d-c^2 d x^2} x^{m+5}}{(m+6)^3}-\frac {2 b c^5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+6}}{(m+6)^2 \sqrt {1-c^2 x^2}}+\frac {15 d^3 \text {Int}\left (\frac {x^m \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}},x\right )}{(m+6) \left (m^2+6 m+8\right )} \]

[Out]

5*d*x^(1+m)*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/(4+m)/(6+m)+x^(1+m)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)
)^2/(6+m)+10*b^2*c^2*d^2*x^(3+m)*(-c^2*d*x^2+d)^(1/2)/(4+m)^3/(6+m)+2*b^2*c^2*d^2*(m^2+15*m+52)*x^(3+m)*(-c^2*
d*x^2+d)^(1/2)/(4+m)^2/(6+m)^3-2*b^2*c^4*d^2*x^(5+m)*(-c^2*d*x^2+d)^(1/2)/(6+m)^3+15*d^2*x^(1+m)*(a+b*arcsin(c
*x))^2*(-c^2*d*x^2+d)^(1/2)/(6+m)/(m^2+6*m+8)-30*b*c*d^2*x^(2+m)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(2+m)^
2/(4+m)/(6+m)/(-c^2*x^2+1)^(1/2)-10*b*c*d^2*x^(2+m)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(6+m)/(m^2+6*m+8)/(
-c^2*x^2+1)^(1/2)-2*b*c*d^2*x^(2+m)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(m^2+8*m+12)/(-c^2*x^2+1)^(1/2)+10*
b*c^3*d^2*x^(4+m)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(4+m)^2/(6+m)/(-c^2*x^2+1)^(1/2)+4*b*c^3*d^2*x^(4+m)*
(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(4+m)/(6+m)/(-c^2*x^2+1)^(1/2)-2*b*c^5*d^2*x^(6+m)*(a+b*arcsin(c*x))*(-
c^2*d*x^2+d)^(1/2)/(6+m)^2/(-c^2*x^2+1)^(1/2)+10*b^2*c^2*d^2*(10+3*m)*x^(3+m)*hypergeom([1/2, 3/2+1/2*m],[5/2+
1/2*m],c^2*x^2)*(-c^2*d*x^2+d)^(1/2)/(4+m)^3/(6+m)/(m^2+5*m+6)/(-c^2*x^2+1)^(1/2)+30*b^2*c^2*d^2*x^(3+m)*hyper
geom([1/2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)*(-c^2*d*x^2+d)^(1/2)/(2+m)^2/(6+m)/(m^2+7*m+12)/(-c^2*x^2+1)^(1/2)+
2*b^2*c^2*d^2*(15*m^2+130*m+264)*x^(3+m)*hypergeom([1/2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)*(-c^2*d*x^2+d)^(1/2)/
(4+m)^2/(6+m)^3/(m^2+5*m+6)/(-c^2*x^2+1)^(1/2)+15*d^3*Unintegrable(x^m*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2
),x)/(6+m)/(m^2+6*m+8)

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Rubi [A]  time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int x^m \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int x^m \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ \end {align*}

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Mathematica [A]  time = 6.43, size = 0, normalized size = 0.00 \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcs
in(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)*x^m, x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 12.28, size = 0, normalized size = 0.00 \[ \int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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