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

Optimal. Leaf size=756 \[ \frac{16 b^2 c^2 d^2 x^{m+3} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},c^2 x^2\right )}{(m+3)^2 (m+5) \left (m^2+3 m+2\right )}+\frac{8 b^2 c^2 d^2 x^{m+3} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},c^2 x^2\right )}{(m+2) (m+3)^3 (m+5)}+\frac{6 b^2 c^2 d^2 x^{m+3} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},c^2 x^2\right )}{(m+2) (m+3)^2 (m+5)^2}-\frac{6 b c d^2 x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(m+5)^2 \left (m^2+5 m+6\right )}-\frac{16 b c d^2 x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(m+5) \left (m^3+6 m^2+11 m+6\right )}-\frac{8 b c d^2 x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(m+2) (m+3)^2 (m+5)}+\frac{4 d^2 \left (1-c^2 x^2\right ) x^{m+1} \left (a+b \sin ^{-1}(c x)\right )^2}{m^2+8 m+15}+\frac{d^2 \left (1-c^2 x^2\right )^2 x^{m+1} \left (a+b \sin ^{-1}(c x)\right )^2}{m+5}-\frac{2 b c d^2 \left (1-c^2 x^2\right )^{3/2} x^{m+2} \left (a+b \sin ^{-1}(c x)\right )}{(m+5)^2}-\frac{8 b c d^2 \sqrt{1-c^2 x^2} x^{m+2} \left (a+b \sin ^{-1}(c x)\right )}{(m+3)^2 (m+5)}-\frac{6 b c d^2 \sqrt{1-c^2 x^2} x^{m+2} \left (a+b \sin ^{-1}(c x)\right )}{(m+3) (m+5)^2}+\frac{8 d^2 x^{m+1} \left (a+b \sin ^{-1}(c x)\right )^2}{(m+5) \left (m^2+4 m+3\right )}+\frac{8 b^2 c^2 d^2 x^{m+3}}{(m+3)^3 (m+5)}+\frac{2 b^2 c^2 d^2 x^{m+3}}{(m+3) (m+5)^2}+\frac{6 b^2 c^2 d^2 x^{m+3}}{(m+3)^2 (m+5)^2}-\frac{2 b^2 c^4 d^2 x^{m+5}}{(m+5)^3} \]

[Out]

(6*b^2*c^2*d^2*x^(3 + m))/((3 + m)^2*(5 + m)^2) + (2*b^2*c^2*d^2*x^(3 + m))/((3 + m)*(5 + m)^2) + (8*b^2*c^2*d
^2*x^(3 + m))/((3 + m)^3*(5 + m)) - (2*b^2*c^4*d^2*x^(5 + m))/(5 + m)^3 - (6*b*c*d^2*x^(2 + m)*Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x]))/((3 + m)*(5 + m)^2) - (8*b*c*d^2*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3
+ m)^2*(5 + m)) - (2*b*c*d^2*x^(2 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(5 + m)^2 + (8*d^2*x^(1 + m)*(
a + b*ArcSin[c*x])^2)/((5 + m)*(3 + 4*m + m^2)) + (4*d^2*x^(1 + m)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(15 +
8*m + m^2) + (d^2*x^(1 + m)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(5 + m) - (8*b*c*d^2*x^(2 + m)*(a + b*ArcSi
n[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)) - (6*b*c*d^2*x^(2 +
m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)^2*(6 + 5*m + m^2)) - (
16*b*c*d^2*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)*(6 +
11*m + 6*m^2 + m^3)) + (6*b^2*c^2*d^2*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m
/2}, c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)^2) + (8*b^2*c^2*d^2*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 +
m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(3 + m)^3*(5 + m)) + (16*b^2*c^2*d^2*x^(3 + m)*HypergeometricP
FQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((3 + m)^2*(5 + m)*(2 + 3*m + m^2))

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Rubi [F]  time = 0.0806745, 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 = {} \[ \int x^m \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.142369, size = 0, normalized size = 0. \[ \int x^m \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 7.016, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{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)^2*(a+b*arcsin(c*x))^2,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  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 )} 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)^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))*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)**2*(a+b*asin(c*x))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c^{2} d x^{2} - d\right )}^{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)^2*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

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