3.276 \(\int x^m (d-c^2 d x^2)^3 (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=1312 \[ \frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 x^{m+1}}{m+7}+\frac {6 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 x^{m+1}}{(m+5) (m+7)}+\frac {24 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 x^{m+1}}{(m+7) \left (m^2+8 m+15\right )}+\frac {48 d^3 \left (a+b \sin ^{-1}(c x)\right )^2 x^{m+1}}{(m+5) (m+7) \left (m^2+4 m+3\right )}-\frac {2 b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{(m+7)^2}-\frac {12 b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{(m+5)^2 (m+7)}-\frac {10 b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{(m+5) (m+7)^2}-\frac {48 b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{(m+3)^2 (m+5) (m+7)}-\frac {36 b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{(m+3) (m+5)^2 (m+7)}-\frac {30 b c d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{(m+7)^2 \left (m^2+8 m+15\right )}-\frac {48 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) x^{m+2}}{(m+2) (m+3)^2 (m+5) (m+7)}-\frac {36 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) x^{m+2}}{(m+5)^2 (m+7) \left (m^2+5 m+6\right )}-\frac {30 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) x^{m+2}}{(m+5) (m+7)^2 \left (m^2+5 m+6\right )}-\frac {96 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) x^{m+2}}{(m+5) (m+7) \left (m^3+6 m^2+11 m+6\right )}+\frac {48 b^2 c^2 d^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 ) x^{m+3}}{(m+2) (m+3)^3 (m+5) (m+7)}+\frac {36 b^2 c^2 d^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 ) x^{m+3}}{(m+2) (m+3)^2 (m+5)^2 (m+7)}+\frac {96 b^2 c^2 d^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 ) x^{m+3}}{(m+3)^2 (m+5) (m+7) \left (m^2+3 m+2\right )}+\frac {30 b^2 c^2 d^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 ) x^{m+3}}{(m+2) (m+3)^2 (m+5) (m+7)^2}+\frac {48 b^2 c^2 d^3 x^{m+3}}{(m+3)^3 (m+5) (m+7)}+\frac {12 b^2 c^2 d^3 x^{m+3}}{(m+3) (m+5)^2 (m+7)}+\frac {36 b^2 c^2 d^3 x^{m+3}}{(m+3)^2 (m+5)^2 (m+7)}+\frac {10 b^2 c^2 d^3 x^{m+3}}{(m+7)^2 \left (m^2+8 m+15\right )}+\frac {2 b^2 c^2 d^3 x^{m+3}}{(m+3) (m+7)^2}+\frac {30 b^2 c^2 d^3 x^{m+3}}{(m+3)^2 (m+5) (m+7)^2}-\frac {12 b^2 c^4 d^3 x^{m+5}}{(m+5)^3 (m+7)}-\frac {4 b^2 c^4 d^3 x^{m+5}}{(m+5) (m+7)^2}-\frac {10 b^2 c^4 d^3 x^{m+5}}{(m+5)^2 (m+7)^2}+\frac {2 b^2 c^6 d^3 x^{m+7}}{(m+7)^3} \]

[Out]

d^3*x^(1+m)*(-c^2*x^2+1)^3*(a+b*arcsin(c*x))^2/(7+m)-30*b*c*d^3*x^(2+m)*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(
7+m)^2/(m^2+8*m+15)-36*b*c*d^3*x^(2+m)*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(3+m)/(5+m)^2/(7+m)-48*b*c*d^3*x^(
2+m)*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(3+m)^2/(5+m)/(7+m)+30*b^2*c^2*d^3*x^(3+m)/(3+m)^2/(5+m)/(7+m)^2+12*
b^2*c^2*d^3*x^(3+m)/(3+m)/(5+m)^2/(7+m)+48*b^2*c^2*d^3*x^(3+m)/(3+m)^3/(5+m)/(7+m)-2*b*c*d^3*x^(2+m)*(-c^2*x^2
+1)^(5/2)*(a+b*arcsin(c*x))/(7+m)^2+36*b^2*c^2*d^3*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)/(m^2+8*m+15)^2/(m^2+9*m+14)+2*b^2*c^6*d^3*x^(7+m)/(7+m)^3+2*b^2*c^2*d^3*x^(3+m)/(3+m)/(
7+m)^2+36*b^2*c^2*d^3*x^(3+m)/(7+m)/(m^2+8*m+15)^2+10*b^2*c^2*d^3*x^(3+m)/(7+m)^2/(m^2+8*m+15)-10*b^2*c^4*d^3*
x^(5+m)/(5+m)^2/(7+m)^2-4*b^2*c^4*d^3*x^(5+m)/(5+m)/(7+m)^2-12*b^2*c^4*d^3*x^(5+m)/(5+m)^3/(7+m)+48*d^3*x^(1+m
)*(a+b*arcsin(c*x))^2/(5+m)/(7+m)/(m^2+4*m+3)+24*d^3*x^(1+m)*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/(7+m)/(m^2+8*m+1
5)+6*d^3*x^(1+m)*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/(5+m)/(7+m)-30*b*c*d^3*x^(2+m)*(a+b*arcsin(c*x))*hypergeom
([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/(5+m)/(7+m)^2/(m^2+5*m+6)-36*b*c*d^3*x^(2+m)*(a+b*arcsin(c*x))*hypergeom([1
/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/(5+m)^2/(7+m)/(m^2+5*m+6)-48*b*c*d^3*x^(2+m)*(a+b*arcsin(c*x))*hypergeom([1/2,
1+1/2*m],[2+1/2*m],c^2*x^2)/(3+m)^2/(7+m)/(m^2+7*m+10)-96*b*c*d^3*x^(2+m)*(a+b*arcsin(c*x))*hypergeom([1/2, 1
+1/2*m],[2+1/2*m],c^2*x^2)/(5+m)/(7+m)/(m^3+6*m^2+11*m+6)+96*b^2*c^2*d^3*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)/(3+m)^2/(5+m)/(7+m)/(m^2+3*m+2)+48*b^2*c^2*d^3*x^(3+m)*Hypergeome
tricPFQ([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)^3/(7+m)/(m^2+7*m+10)-10*b*c*d^3*x^(2+m)*
(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/(5+m)/(7+m)^2-12*b*c*d^3*x^(2+m)*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/(5+
m)^2/(7+m)+30*b^2*c^2*d^3*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)/(3
+m)^2/(7+m)^2/(m^2+7*m+10)

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Rubi [F]  time = 0.08, 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 = {} \[ \int x^m \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int x^m \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int x^m \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ \end {align*}

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Mathematica [F]  time = 3.67, size = 0, normalized size = 0.00 \[ \int x^m \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{6} d^{3} x^{6} - 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} - a^{2} d^{3} + {\left (b^{2} c^{6} d^{3} x^{6} - 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{6} d^{3} x^{6} - 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} - a b d^{3}\right )} \arcsin \left (c x\right )\right )} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 22.23, size = 0, normalized size = 0.00 \[ \int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{2} c^{6} d^{3} x^{m + 7}}{m + 7} + \frac {3 \, a^{2} c^{4} d^{3} x^{m + 5}}{m + 5} - \frac {3 \, a^{2} c^{2} d^{3} x^{m + 3}}{m + 3} + \frac {a^{2} d^{3} x^{m + 1}}{m + 1} - \frac {{\left ({\left (b^{2} c^{6} d^{3} m^{3} + 9 \, b^{2} c^{6} d^{3} m^{2} + 23 \, b^{2} c^{6} d^{3} m + 15 \, b^{2} c^{6} d^{3}\right )} x^{7} - 3 \, {\left (b^{2} c^{4} d^{3} m^{3} + 11 \, b^{2} c^{4} d^{3} m^{2} + 31 \, b^{2} c^{4} d^{3} m + 21 \, b^{2} c^{4} d^{3}\right )} x^{5} + 3 \, {\left (b^{2} c^{2} d^{3} m^{3} + 13 \, b^{2} c^{2} d^{3} m^{2} + 47 \, b^{2} c^{2} d^{3} m + 35 \, b^{2} c^{2} d^{3}\right )} x^{3} - {\left (b^{2} d^{3} m^{3} + 15 \, b^{2} d^{3} m^{2} + 71 \, b^{2} d^{3} m + 105 \, b^{2} d^{3}\right )} x\right )} x^{m} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} - 2 \, {\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} \int \frac {{\left ({\left (b^{2} c^{7} d^{3} m^{3} + 9 \, b^{2} c^{7} d^{3} m^{2} + 23 \, b^{2} c^{7} d^{3} m + 15 \, b^{2} c^{7} d^{3}\right )} x^{7} - 3 \, {\left (b^{2} c^{5} d^{3} m^{3} + 11 \, b^{2} c^{5} d^{3} m^{2} + 31 \, b^{2} c^{5} d^{3} m + 21 \, b^{2} c^{5} d^{3}\right )} x^{5} + 3 \, {\left (b^{2} c^{3} d^{3} m^{3} + 13 \, b^{2} c^{3} d^{3} m^{2} + 47 \, b^{2} c^{3} d^{3} m + 35 \, b^{2} c^{3} d^{3}\right )} x^{3} - {\left (b^{2} c d^{3} m^{3} + 15 \, b^{2} c d^{3} m^{2} + 71 \, b^{2} c d^{3} m + 105 \, b^{2} c d^{3}\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^{3} m^{4} + {\left (a b c^{8} d^{3} m^{4} + 16 \, a b c^{8} d^{3} m^{3} + 86 \, a b c^{8} d^{3} m^{2} + 176 \, a b c^{8} d^{3} m + 105 \, a b c^{8} d^{3}\right )} x^{8} + 16 \, a b d^{3} m^{3} + 86 \, a b d^{3} m^{2} - 4 \, {\left (a b c^{6} d^{3} m^{4} + 16 \, a b c^{6} d^{3} m^{3} + 86 \, a b c^{6} d^{3} m^{2} + 176 \, a b c^{6} d^{3} m + 105 \, a b c^{6} d^{3}\right )} x^{6} + 176 \, a b d^{3} m + 105 \, a b d^{3} + 6 \, {\left (a b c^{4} d^{3} m^{4} + 16 \, a b c^{4} d^{3} m^{3} + 86 \, a b c^{4} d^{3} m^{2} + 176 \, a b c^{4} d^{3} m + 105 \, a b c^{4} d^{3}\right )} x^{4} - 4 \, {\left (a b c^{2} d^{3} m^{4} + 16 \, a b c^{2} d^{3} m^{3} + 86 \, a b c^{2} d^{3} m^{2} + 176 \, a b c^{2} d^{3} m + 105 \, a b c^{2} d^{3}\right )} x^{2}\right )} x^{m} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{m^{4} + 16 \, m^{3} - {\left (c^{2} m^{4} + 16 \, c^{2} m^{3} + 86 \, c^{2} m^{2} + 176 \, c^{2} m + 105 \, c^{2}\right )} x^{2} + 86 \, m^{2} + 176 \, m + 105}\,{d x}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2*c^6*d^3*x^(m + 7)/(m + 7) + 3*a^2*c^4*d^3*x^(m + 5)/(m + 5) - 3*a^2*c^2*d^3*x^(m + 3)/(m + 3) + a^2*d^3*x
^(m + 1)/(m + 1) - (((b^2*c^6*d^3*m^3 + 9*b^2*c^6*d^3*m^2 + 23*b^2*c^6*d^3*m + 15*b^2*c^6*d^3)*x^7 - 3*(b^2*c^
4*d^3*m^3 + 11*b^2*c^4*d^3*m^2 + 31*b^2*c^4*d^3*m + 21*b^2*c^4*d^3)*x^5 + 3*(b^2*c^2*d^3*m^3 + 13*b^2*c^2*d^3*
m^2 + 47*b^2*c^2*d^3*m + 35*b^2*c^2*d^3)*x^3 - (b^2*d^3*m^3 + 15*b^2*d^3*m^2 + 71*b^2*d^3*m + 105*b^2*d^3)*x)*
x^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + (m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*integrate(-2*(((b^2*c
^7*d^3*m^3 + 9*b^2*c^7*d^3*m^2 + 23*b^2*c^7*d^3*m + 15*b^2*c^7*d^3)*x^7 - 3*(b^2*c^5*d^3*m^3 + 11*b^2*c^5*d^3*
m^2 + 31*b^2*c^5*d^3*m + 21*b^2*c^5*d^3)*x^5 + 3*(b^2*c^3*d^3*m^3 + 13*b^2*c^3*d^3*m^2 + 47*b^2*c^3*d^3*m + 35
*b^2*c^3*d^3)*x^3 - (b^2*c*d^3*m^3 + 15*b^2*c*d^3*m^2 + 71*b^2*c*d^3*m + 105*b^2*c*d^3)*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^3*m^4 + (a*b*c^8*d^3*m^4 + 16*a*b*c^8*d^3*m^
3 + 86*a*b*c^8*d^3*m^2 + 176*a*b*c^8*d^3*m + 105*a*b*c^8*d^3)*x^8 + 16*a*b*d^3*m^3 + 86*a*b*d^3*m^2 - 4*(a*b*c
^6*d^3*m^4 + 16*a*b*c^6*d^3*m^3 + 86*a*b*c^6*d^3*m^2 + 176*a*b*c^6*d^3*m + 105*a*b*c^6*d^3)*x^6 + 176*a*b*d^3*
m + 105*a*b*d^3 + 6*(a*b*c^4*d^3*m^4 + 16*a*b*c^4*d^3*m^3 + 86*a*b*c^4*d^3*m^2 + 176*a*b*c^4*d^3*m + 105*a*b*c
^4*d^3)*x^4 - 4*(a*b*c^2*d^3*m^4 + 16*a*b*c^2*d^3*m^3 + 86*a*b*c^2*d^3*m^2 + 176*a*b*c^2*d^3*m + 105*a*b*c^2*d
^3)*x^2)*x^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(m^4 + 16*m^3 - (c^2*m^4 + 16*c^2*m^3 + 86*c^2*m^2 +
176*c^2*m + 105*c^2)*x^2 + 86*m^2 + 176*m + 105), x))/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]

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)^3,x)

[Out]

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

[Out]

Timed out

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