3.370 \(\int x^3 (a+b \sin ^{-1}(\frac {c}{x})) \, dx\)

Optimal. Leaf size=64 \[ \frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{12} b c x^3 \sqrt {1-\frac {c^2}{x^2}}+\frac {1}{6} b c^3 x \sqrt {1-\frac {c^2}{x^2}} \]

[Out]

1/4*x^4*(a+b*arcsin(c/x))+1/6*b*c^3*x*(1-c^2/x^2)^(1/2)+1/12*b*c*x^3*(1-c^2/x^2)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4842, 12, 271, 191} \[ \frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{12} b c x^3 \sqrt {1-\frac {c^2}{x^2}}+\frac {1}{6} b c^3 x \sqrt {1-\frac {c^2}{x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*ArcSin[c/x]),x]

[Out]

(b*c^3*Sqrt[1 - c^2/x^2]*x)/6 + (b*c*Sqrt[1 - c^2/x^2]*x^3)/12 + (x^4*(a + b*ArcSin[c/x]))/4

Rule 12

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

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]

Rubi steps

\begin {align*} \int x^3 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{4} b \int \frac {c x^2}{\sqrt {1-\frac {c^2}{x^2}}} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{4} (b c) \int \frac {x^2}{\sqrt {1-\frac {c^2}{x^2}}} \, dx\\ &=\frac {1}{12} b c \sqrt {1-\frac {c^2}{x^2}} x^3+\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{6} \left (b c^3\right ) \int \frac {1}{\sqrt {1-\frac {c^2}{x^2}}} \, dx\\ &=\frac {1}{6} b c^3 \sqrt {1-\frac {c^2}{x^2}} x+\frac {1}{12} b c \sqrt {1-\frac {c^2}{x^2}} x^3+\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 59, normalized size = 0.92 \[ \frac {a x^4}{4}+b \sqrt {\frac {x^2-c^2}{x^2}} \left (\frac {c^3 x}{6}+\frac {c x^3}{12}\right )+\frac {1}{4} b x^4 \sin ^{-1}\left (\frac {c}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*ArcSin[c/x]),x]

[Out]

(a*x^4)/4 + b*Sqrt[(-c^2 + x^2)/x^2]*((c^3*x)/6 + (c*x^3)/12) + (b*x^4*ArcSin[c/x])/4

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fricas [A]  time = 1.47, size = 51, normalized size = 0.80 \[ \frac {1}{4} \, b x^{4} \arcsin \left (\frac {c}{x}\right ) + \frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (2 \, b c^{3} x + b c x^{3}\right )} \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsin(c/x)),x, algorithm="fricas")

[Out]

1/4*b*x^4*arcsin(c/x) + 1/4*a*x^4 + 1/12*(2*b*c^3*x + b*c*x^3)*sqrt(-(c^2 - x^2)/x^2)

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giac [B]  time = 0.31, size = 340, normalized size = 5.31 \[ \frac {3 \, b c x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {c}{x}\right ) + 3 \, a c x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4} + 2 \, b c^{2} x^{3} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{3} + 12 \, b c^{3} x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {c}{x}\right ) + 12 \, a c^{3} x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} + 18 \, b c^{4} x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )} + 18 \, b c^{5} \arcsin \left (\frac {c}{x}\right ) + 18 \, a c^{5} - \frac {18 \, b c^{6}}{x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}} + \frac {12 \, b c^{7} \arcsin \left (\frac {c}{x}\right )}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}} + \frac {12 \, a c^{7}}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}} - \frac {2 \, b c^{8}}{x^{3} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{3}} + \frac {3 \, b c^{9} \arcsin \left (\frac {c}{x}\right )}{x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4}} + \frac {3 \, a c^{9}}{x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4}}}{192 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsin(c/x)),x, algorithm="giac")

[Out]

1/192*(3*b*c*x^4*(sqrt(-c^2/x^2 + 1) + 1)^4*arcsin(c/x) + 3*a*c*x^4*(sqrt(-c^2/x^2 + 1) + 1)^4 + 2*b*c^2*x^3*(
sqrt(-c^2/x^2 + 1) + 1)^3 + 12*b*c^3*x^2*(sqrt(-c^2/x^2 + 1) + 1)^2*arcsin(c/x) + 12*a*c^3*x^2*(sqrt(-c^2/x^2
+ 1) + 1)^2 + 18*b*c^4*x*(sqrt(-c^2/x^2 + 1) + 1) + 18*b*c^5*arcsin(c/x) + 18*a*c^5 - 18*b*c^6/(x*(sqrt(-c^2/x
^2 + 1) + 1)) + 12*b*c^7*arcsin(c/x)/(x^2*(sqrt(-c^2/x^2 + 1) + 1)^2) + 12*a*c^7/(x^2*(sqrt(-c^2/x^2 + 1) + 1)
^2) - 2*b*c^8/(x^3*(sqrt(-c^2/x^2 + 1) + 1)^3) + 3*b*c^9*arcsin(c/x)/(x^4*(sqrt(-c^2/x^2 + 1) + 1)^4) + 3*a*c^
9/(x^4*(sqrt(-c^2/x^2 + 1) + 1)^4))/c

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maple [A]  time = 0.01, size = 71, normalized size = 1.11 \[ -c^{4} \left (-\frac {a \,x^{4}}{4 c^{4}}+b \left (-\frac {x^{4} \arcsin \left (\frac {c}{x}\right )}{4 c^{4}}-\frac {x^{3} \sqrt {1-\frac {c^{2}}{x^{2}}}}{12 c^{3}}-\frac {x \sqrt {1-\frac {c^{2}}{x^{2}}}}{6 c}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arcsin(c/x)),x)

[Out]

-c^4*(-1/4*a/c^4*x^4+b*(-1/4/c^4*x^4*arcsin(c/x)-1/12/c^3*x^3*(1-c^2/x^2)^(1/2)-1/6/c*x*(1-c^2/x^2)^(1/2)))

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maxima [A]  time = 1.09, size = 59, normalized size = 0.92 \[ \frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arcsin \left (\frac {c}{x}\right ) + {\left (x^{3} {\left (-\frac {c^{2}}{x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, c^{2} x \sqrt {-\frac {c^{2}}{x^{2}} + 1}\right )} c\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsin(c/x)),x, algorithm="maxima")

[Out]

1/4*a*x^4 + 1/12*(3*x^4*arcsin(c/x) + (x^3*(-c^2/x^2 + 1)^(3/2) + 3*c^2*x*sqrt(-c^2/x^2 + 1))*c)*b

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^3\,\left (a+b\,\mathrm {asin}\left (\frac {c}{x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*asin(c/x)),x)

[Out]

int(x^3*(a + b*asin(c/x)), x)

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sympy [A]  time = 2.69, size = 107, normalized size = 1.67 \[ \frac {a x^{4}}{4} + \frac {b c \left (\begin {cases} \frac {2 c^{3} \sqrt {-1 + \frac {x^{2}}{c^{2}}}}{3} + \frac {c x^{2} \sqrt {-1 + \frac {x^{2}}{c^{2}}}}{3} & \text {for}\: \left |{\frac {x^{2}}{c^{2}}}\right | > 1 \\\frac {2 i c^{3} \sqrt {1 - \frac {x^{2}}{c^{2}}}}{3} + \frac {i c x^{2} \sqrt {1 - \frac {x^{2}}{c^{2}}}}{3} & \text {otherwise} \end {cases}\right )}{4} + \frac {b x^{4} \operatorname {asin}{\left (\frac {c}{x} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*asin(c/x)),x)

[Out]

a*x**4/4 + b*c*Piecewise((2*c**3*sqrt(-1 + x**2/c**2)/3 + c*x**2*sqrt(-1 + x**2/c**2)/3, Abs(x**2/c**2) > 1),
(2*I*c**3*sqrt(1 - x**2/c**2)/3 + I*c*x**2*sqrt(1 - x**2/c**2)/3, True))/4 + b*x**4*asin(c/x)/4

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