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

Optimal. Leaf size=957 \[ \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 x^{m+1}}{m+6}+\frac{5 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 x^{m+1}}{(m+4) (m+6)}+\frac{15 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^{m+1}}{(m+6) \left (m^2+6 m+8\right )}-\frac{30 b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{(m+2)^2 (m+4) (m+6) \sqrt{1-c^2 x^2}}-\frac{10 b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{(m+6) \left (m^2+6 m+8\right ) \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+2}}{\left (m^2+8 m+12\right ) \sqrt{1-c^2 x^2}}+\frac{10 b^2 c^2 d^2 (3 m+10) \sqrt{d-c^2 d x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},c^2 x^2\right ) x^{m+3}}{(m+2) (m+3) (m+4)^3 (m+6) \sqrt{1-c^2 x^2}}+\frac{2 b^2 c^2 d^2 \left (15 m^2+130 m+264\right ) \sqrt{d-c^2 d x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},c^2 x^2\right ) x^{m+3}}{(m+2) (m+3) (m+4)^2 (m+6)^3 \sqrt{1-c^2 x^2}}+\frac{30 b^2 c^2 d^2 \sqrt{d-c^2 d x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},c^2 x^2\right ) x^{m+3}}{(m+2)^2 (m+3) (m+4) (m+6) \sqrt{1-c^2 x^2}}+\frac{2 b^2 c^2 d^2 \left (m^2+15 m+52\right ) \sqrt{d-c^2 d x^2} x^{m+3}}{(m+4)^2 (m+6)^3}+\frac{10 b^2 c^2 d^2 \sqrt{d-c^2 d x^2} x^{m+3}}{(m+4)^3 (m+6)}+\frac{4 b c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+4}}{(m+4) (m+6) \sqrt{1-c^2 x^2}}+\frac{10 b c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+4}}{(m+4)^2 (m+6) \sqrt{1-c^2 x^2}}-\frac{2 b^2 c^4 d^2 \sqrt{d-c^2 d x^2} x^{m+5}}{(m+6)^3}-\frac{2 b c^5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^{m+6}}{(m+6)^2 \sqrt{1-c^2 x^2}}+\frac{15 d^3 \text{Unintegrable}\left (\frac{x^m \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}},x\right )}{(m+6) \left (m^2+6 m+8\right )} \]

[Out]

(10*b^2*c^2*d^2*x^(3 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)^3*(6 + m)) + (2*b^2*c^2*d^2*(52 + 15*m + m^2)*x^(3 + m
)*Sqrt[d - c^2*d*x^2])/((4 + m)^2*(6 + m)^3) - (2*b^2*c^4*d^2*x^(5 + m)*Sqrt[d - c^2*d*x^2])/(6 + m)^3 - (30*b
*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 - c^2*x^2]) - (10*
b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((6 + m)*(8 + 6*m + m^2)*Sqrt[1 - c^2*x^2]) - (2*b*
c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((12 + 8*m + m^2)*Sqrt[1 - c^2*x^2]) + (10*b*c^3*d^2*
x^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((4 + m)^2*(6 + m)*Sqrt[1 - c^2*x^2]) + (4*b*c^3*d^2*x^(4 +
m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((4 + m)*(6 + m)*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*x^(6 + m)*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((6 + m)^2*Sqrt[1 - c^2*x^2]) + (15*d^2*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a
+ b*ArcSin[c*x])^2)/((6 + m)*(8 + 6*m + m^2)) + (5*d*x^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/((
4 + m)*(6 + m)) + (x^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(6 + m) + (30*b^2*c^2*d^2*x^(3 + m)*
Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((2 + m)^2*(3 + m)*(4 + m)*(6 + m)*
Sqrt[1 - c^2*x^2]) + (10*b^2*c^2*d^2*(10 + 3*m)*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2
, (5 + m)/2, c^2*x^2])/((2 + m)*(3 + m)*(4 + m)^3*(6 + m)*Sqrt[1 - c^2*x^2]) + (2*b^2*c^2*d^2*(264 + 130*m + 1
5*m^2)*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((2 + m)*(3 + m)*(
4 + m)^2*(6 + m)^3*Sqrt[1 - c^2*x^2]) + (15*d^3*Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],
x])/((6 + m)*(8 + 6*m + m^2))

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Rubi [A]  time = 0.157333, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 4.44079, size = 0, normalized size = 0. \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

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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Maple [A]  time = 7.355, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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]

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