3.3.80 \(\int \frac {x^m (a+b \text {ArcSin}(c x))^2}{(d-c^2 d x^2)^2} \, dx\) [280]

Optimal. Leaf size=280 \[ -\frac {b c x^{2+m} (a+b \text {ArcSin}(c x))}{d^2 \sqrt {1-c^2 x^2}}+\frac {x^{1+m} (a+b \text {ArcSin}(c x))^2}{2 d^2 \left (1-c^2 x^2\right )}+\frac {b c (1+m) x^{2+m} (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{d^2 (2+m)}+\frac {b^2 c^2 x^{3+m} \text {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{d^2 (3+m)}-\frac {b^2 c^2 (1+m) x^{3+m} \text {HypergeometricPFQ}\left (\left \{1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2}\right \},\left \{2+\frac {m}{2},\frac {5}{2}+\frac {m}{2}\right \},c^2 x^2\right )}{d^2 \left (6+5 m+m^2\right )}+\frac {(1-m) \text {Int}\left (\frac {x^m (a+b \text {ArcSin}(c x))^2}{d-c^2 d x^2},x\right )}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.30, 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 = {} \begin {gather*} \int \frac {x^m (a+b \text {ArcSin}(c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x^{1+m} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(b c) \int \frac {x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac {(1-m) \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{2 d}\\ &=-\frac {b c x^{2+m} \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt {1-c^2 x^2}}+\frac {x^{1+m} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (b^2 c^2\right ) \int \frac {x^{2+m}}{1-c^2 x^2} \, dx}{d^2}+\frac {(1-m) \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{2 d}+\frac {(b c (1+m)) \int \frac {x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{d^2}\\ &=-\frac {b c x^{2+m} \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt {1-c^2 x^2}}+\frac {x^{1+m} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \left (1-c^2 x^2\right )}+\frac {b c (1+m) x^{2+m} \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{d^2 (2+m)}+\frac {b^2 c^2 x^{3+m} \, _2F_1\left (1,\frac {3+m}{2};\frac {5+m}{2};c^2 x^2\right )}{d^2 (3+m)}-\frac {b^2 c^2 (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 )}{d^2 \left (6+5 m+m^2\right )}+\frac {(1-m) \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{2 d}\\ \end {align*}

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Mathematica [A]
time = 4.77, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m (a+b \text {ArcSin}(c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.39, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (-c^{2} d \,x^{2}+d \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2} x^{m}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{m} \operatorname {asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \end {gather*}

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)^2,x)

[Out]

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

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