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

Optimal. Leaf size=371 \[ \frac {4 b^2 c^2 d x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+3)^2 \left (m^2+3 m+2\right )}+\frac {2 b^2 c^2 d x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+2) (m+3)^3}-\frac {4 b c d x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{m^3+6 m^2+11 m+6}-\frac {2 b c d x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(m+2) (m+3)^2}+\frac {d \left (1-c^2 x^2\right ) x^{m+1} \left (a+b \sin ^{-1}(c x)\right )^2}{m+3}-\frac {2 b c d \sqrt {1-c^2 x^2} x^{m+2} \left (a+b \sin ^{-1}(c x)\right )}{(m+3)^2}+\frac {2 d x^{m+1} \left (a+b \sin ^{-1}(c x)\right )^2}{m^2+4 m+3}+\frac {2 b^2 c^2 d x^{m+3}}{(m+3)^3} \]

[Out]

2*b^2*c^2*d*x^(3+m)/(3+m)^3+2*d*x^(1+m)*(a+b*arcsin(c*x))^2/(m^2+4*m+3)+d*x^(1+m)*(-c^2*x^2+1)*(a+b*arcsin(c*x
))^2/(3+m)-2*b*c*d*x^(2+m)*(a+b*arcsin(c*x))*hypergeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/(2+m)/(3+m)^2-4*b*c*d
*x^(2+m)*(a+b*arcsin(c*x))*hypergeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/(m^3+6*m^2+11*m+6)+2*b^2*c^2*d*x^(3+m)*
HypergeometricPFQ([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/2*m],c^2*x^2)/(2+m)/(3+m)^3+4*b^2*c^2*d*x^(3+m)*Hy
pergeometricPFQ([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/2*m],c^2*x^2)/(3+m)^2/(m^2+3*m+2)-2*b*c*d*x^(2+m)*(a
+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(3+m)^2

________________________________________________________________________________________

Rubi [F]  time = 0.05, 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 = {} \[ \int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )\right )} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 4.48, size = 0, normalized size = 0.00 \[ \int x^{m} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{2} c^{2} d x^{m + 3}}{m + 3} + \frac {a^{2} d x^{m + 1}}{m + 1} - \frac {{\left ({\left (b^{2} c^{2} d m + b^{2} c^{2} d\right )} x^{3} - {\left (b^{2} d m + 3 \, b^{2} d\right )} x\right )} x^{m} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (m^{2} + 4 \, m + 3\right )} \int \frac {{\left ({\left (b^{2} c^{3} d m + b^{2} c^{3} d\right )} x^{3} - {\left (b^{2} c d m + 3 \, b^{2} c d\right )} x\right )} \sqrt {c x + 1} \sqrt {-c x + 1} x^{m} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left (a b d m^{2} + {\left (a b c^{4} d m^{2} + 4 \, a b c^{4} d m + 3 \, a b c^{4} d\right )} x^{4} + 4 \, a b d m + 3 \, a b d - 2 \, {\left (a b c^{2} d m^{2} + 4 \, a b c^{2} d m + 3 \, a b c^{2} d\right )} x^{2}\right )} x^{m} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{{\left (c^{2} m^{2} + 4 \, c^{2} m + 3 \, c^{2}\right )} x^{2} - m^{2} - 4 \, m - 3}\,{d x}}{m^{2} + 4 \, m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2*c^2*d*x^(m + 3)/(m + 3) + a^2*d*x^(m + 1)/(m + 1) - (((b^2*c^2*d*m + b^2*c^2*d)*x^3 - (b^2*d*m + 3*b^2*d)
*x)*x^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + (m^2 + 4*m + 3)*integrate(2*(((b^2*c^3*d*m + b^2*c^3*d)
*x^3 - (b^2*c*d*m + 3*b^2*c*d)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))
+ (a*b*d*m^2 + (a*b*c^4*d*m^2 + 4*a*b*c^4*d*m + 3*a*b*c^4*d)*x^4 + 4*a*b*d*m + 3*a*b*d - 2*(a*b*c^2*d*m^2 + 4*
a*b*c^2*d*m + 3*a*b*c^2*d)*x^2)*x^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/((c^2*m^2 + 4*c^2*m + 3*c^2)*x
^2 - m^2 - 4*m - 3), x))/(m^2 + 4*m + 3)

________________________________________________________________________________________

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

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),x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - d \left (\int \left (- a^{2} x^{m}\right )\, dx + \int \left (- b^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int \left (- 2 a b x^{m} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int a^{2} c^{2} x^{2} x^{m}\, dx + \int b^{2} c^{2} x^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{2} x^{m} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*(Integral(-a**2*x**m, x) + Integral(-b**2*x**m*asin(c*x)**2, x) + Integral(-2*a*b*x**m*asin(c*x), x) + Inte
gral(a**2*c**2*x**2*x**m, x) + Integral(b**2*c**2*x**2*x**m*asin(c*x)**2, x) + Integral(2*a*b*c**2*x**2*x**m*a
sin(c*x), x))

________________________________________________________________________________________