3.381 \(\int x (a+b \sin ^{-1}(c x^n)) \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{2} x^2 \left (a+b \sin ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{n+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+2}{2 n},\frac{1}{2} \left (\frac{2}{n}+3\right ),c^2 x^{2 n}\right )}{2 (n+2)} \]

[Out]

(x^2*(a + b*ArcSin[c*x^n]))/2 - (b*c*n*x^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/(2*n), (3 + 2/n)/2, c^2*x^(2*n
)])/(2*(2 + n))

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Rubi [A]  time = 0.0333579, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4842, 12, 364} \[ \frac{1}{2} x^2 \left (a+b \sin ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2 n};\frac{1}{2} \left (3+\frac{2}{n}\right );c^2 x^{2 n}\right )}{2 (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSin[c*x^n]),x]

[Out]

(x^2*(a + b*ArcSin[c*x^n]))/2 - (b*c*n*x^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/(2*n), (3 + 2/n)/2, c^2*x^(2*n
)])/(2*(2 + n))

Rule 4842

Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.0560275, size = 75, normalized size = 1.09 \[ -\frac{b c n x^{n+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+2}{2 n},\frac{n+2}{2 n}+1,c^2 x^{2 n}\right )}{2 (n+2)}+\frac{a x^2}{2}+\frac{1}{2} b x^2 \sin ^{-1}\left (c x^n\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcSin[c*x^n]),x]

[Out]

(a*x^2)/2 + (b*x^2*ArcSin[c*x^n])/2 - (b*c*n*x^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/(2*n), 1 + (2 + n)/(2*n)
, c^2*x^(2*n)])/(2*(2 + n))

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Maple [F]  time = 0.015, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\arcsin \left ( c{x}^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsin(c*x^n)),x)

[Out]

int(x*(a+b*arcsin(c*x^n)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{2} \,{\left (x^{2} \arctan \left (c x^{n}, \sqrt{c x^{n} + 1} \sqrt{-c x^{n} + 1}\right ) + 2 \, c n \int \frac{\sqrt{c x^{n} + 1} \sqrt{-c x^{n} + 1} x x^{n}}{2 \,{\left (c^{2} x^{2 \, n} - 1\right )}}\,{d x}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsin(c*x^n)),x, algorithm="maxima")

[Out]

1/2*a*x^2 + 1/2*(x^2*arctan2(c*x^n, sqrt(c*x^n + 1)*sqrt(-c*x^n + 1)) + 2*c*n*integrate(1/2*sqrt(c*x^n + 1)*sq
rt(-c*x^n + 1)*x*x^n/(c^2*x^(2*n) - 1), x))*b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsin(c*x^n)),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 7.21011, size = 60, normalized size = 0.87 \begin{align*} \frac{a x^{2}}{2} + \frac{b x^{2} \operatorname{asin}{\left (c x^{n} \right )}}{2} + \frac{i b x^{2} \Gamma \left (\frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - \frac{1}{n} \\ 1 - \frac{1}{n} \end{matrix}\middle |{\frac{x^{- 2 n}}{c^{2}}} \right )}}{4 \Gamma \left (1 + \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*asin(c*x**n)),x)

[Out]

a*x**2/2 + b*x**2*asin(c*x**n)/2 + I*b*x**2*gamma(1/n)*hyper((1/2, -1/n), (1 - 1/n,), x**(-2*n)/c**2)/(4*gamma
(1 + 1/n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsin(c*x^n)),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x^n) + a)*x, x)