3.4.79 \(\int x^m (a+b \text {ArcSin}(c x^n)) \, dx\) [379]

Optimal. Leaf size=81 \[ \frac {x^{1+m} \left (a+b \text {ArcSin}\left (c x^n\right )\right )}{1+m}-\frac {b c n x^{1+m+n} \, _2F_1\left (\frac {1}{2},\frac {1+m+n}{2 n};\frac {1+m+3 n}{2 n};c^2 x^{2 n}\right )}{(1+m) (1+m+n)} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4926, 12, 371} \begin {gather*} \frac {x^{m+1} \left (a+b \text {ArcSin}\left (c x^n\right )\right )}{m+1}-\frac {b c n x^{m+n+1} \, _2F_1\left (\frac {1}{2},\frac {m+n+1}{2 n};\frac {m+3 n+1}{2 n};c^2 x^{2 n}\right )}{(m+1) (m+n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*ArcSin[c*x^n]),x]

[Out]

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

Rule 12

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4926

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]

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 78, normalized size = 0.96 \begin {gather*} \frac {x^{1+m} \left ((1+m+n) \left (a+b \text {ArcSin}\left (c x^n\right )\right )-b c n x^n \, _2F_1\left (\frac {1}{2},\frac {1+m+n}{2 n};\frac {1+m+3 n}{2 n};c^2 x^{2 n}\right )\right )}{(1+m) (1+m+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*ArcSin[c*x^n]),x]

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{m} \left (a +b \arcsin \left (c \,x^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
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^n)),x, algorithm="maxima")

[Out]

(x*x^m*arctan2(c*x^n, sqrt(c*x^n + 1)*sqrt(-c*x^n + 1)) + (c*m + c)*n*integrate(sqrt(c*x^n + 1)*sqrt(-c*x^n +
1)*e^(m*log(x) + n*log(x))/((c^2*m + c^2)*x^(2*n) - m - 1), x))*b/(m + 1) + a*x^(m + 1)/(m + 1)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \left (a + b \operatorname {asin}{\left (c x^{n} \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*(a + b*asin(c*x**n)), x)

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Giac [F]
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^n)),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^m\,\left (a+b\,\mathrm {asin}\left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*asin(c*x^n)),x)

[Out]

int(x^m*(a + b*asin(c*x^n)), x)

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