3.2.41 \(\int x^2 (a+b \text {ArcCos}(c x)) \, dx\) [141]

Optimal. Leaf size=60 \[ -\frac {b \sqrt {1-c^2 x^2}}{3 c^3}+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}+\frac {1}{3} x^3 (a+b \text {ArcCos}(c x)) \]

[Out]

1/9*b*(-c^2*x^2+1)^(3/2)/c^3+1/3*x^3*(a+b*arccos(c*x))-1/3*b*(-c^2*x^2+1)^(1/2)/c^3

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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4724, 272, 45} \begin {gather*} \frac {1}{3} x^3 (a+b \text {ArcCos}(c x))+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac {b \sqrt {1-c^2 x^2}}{3 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCos[c*x]),x]

[Out]

-1/3*(b*Sqrt[1 - c^2*x^2])/c^3 + (b*(1 - c^2*x^2)^(3/2))/(9*c^3) + (x^3*(a + b*ArcCos[c*x]))/3

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCo
s[c*x])^n/(d*(m + 1))), x] + Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int x^2 \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{3} (b c) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} (b c) \text {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} (b c) \text {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \sqrt {1-c^2 x^2}}{3 c^3}+\frac {b \left (1-c^2 x^2\right )^{3/2}}{9 c^3}+\frac {1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 55, normalized size = 0.92 \begin {gather*} \frac {a x^3}{3}+b \left (-\frac {2}{9 c^3}-\frac {x^2}{9 c}\right ) \sqrt {1-c^2 x^2}+\frac {1}{3} b x^3 \text {ArcCos}(c x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*ArcCos[c*x]),x]

[Out]

(a*x^3)/3 + b*(-2/(9*c^3) - x^2/(9*c))*Sqrt[1 - c^2*x^2] + (b*x^3*ArcCos[c*x])/3

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Maple [A]
time = 0.01, size = 64, normalized size = 1.07

method	result	size
derivativedivides	\(\frac {\frac {c^{3} x^{3} a}{3}+b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\)	\(64\)
default	\(\frac {\frac {c^{3} x^{3} a}{3}+b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c^3*(1/3*c^3*x^3*a+b*(1/3*c^3*x^3*arccos(c*x)-1/9*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/9*(-c^2*x^2+1)^(1/2)))

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Maxima [A]
time = 0.47, size = 60, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, a x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccos(c*x)),x, algorithm="maxima")

[Out]

1/3*a*x^3 + 1/9*(3*x^3*arccos(c*x) - c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b

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Fricas [A]
time = 3.19, size = 54, normalized size = 0.90 \begin {gather*} \frac {3 \, b c^{3} x^{3} \arccos \left (c x\right ) + 3 \, a c^{3} x^{3} - {\left (b c^{2} x^{2} + 2 \, b\right )} \sqrt {-c^{2} x^{2} + 1}}{9 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccos(c*x)),x, algorithm="fricas")

[Out]

1/9*(3*b*c^3*x^3*arccos(c*x) + 3*a*c^3*x^3 - (b*c^2*x^2 + 2*b)*sqrt(-c^2*x^2 + 1))/c^3

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Sympy [A]
time = 0.14, size = 70, normalized size = 1.17 \begin {gather*} \begin {cases} \frac {a x^{3}}{3} + \frac {b x^{3} \operatorname {acos}{\left (c x \right )}}{3} - \frac {b x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {2 b \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\\frac {x^{3} \left (a + \frac {\pi b}{2}\right )}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*acos(c*x)),x)

[Out]

Piecewise((a*x**3/3 + b*x**3*acos(c*x)/3 - b*x**2*sqrt(-c**2*x**2 + 1)/(9*c) - 2*b*sqrt(-c**2*x**2 + 1)/(9*c**
3), Ne(c, 0)), (x**3*(a + pi*b/2)/3, True))

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Giac [A]
time = 0.42, size = 56, normalized size = 0.93 \begin {gather*} \frac {1}{3} \, b x^{3} \arccos \left (c x\right ) + \frac {1}{3} \, a x^{3} - \frac {\sqrt {-c^{2} x^{2} + 1} b x^{2}}{9 \, c} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b}{9 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccos(c*x)),x, algorithm="giac")

[Out]

1/3*b*x^3*arccos(c*x) + 1/3*a*x^3 - 1/9*sqrt(-c^2*x^2 + 1)*b*x^2/c - 2/9*sqrt(-c^2*x^2 + 1)*b/c^3

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \left \{\begin {array}{cl} \frac {a\,x^3}{3}-b\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}-\frac {x^3\,\mathrm {acos}\left (c\,x\right )}{3}\right ) & \text {\ if\ \ }0<c\\ \int x^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right ) \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*acos(c*x)),x)

[Out]

piecewise(0 < c, - b*(((1/c^2 - x^2)^(1/2)*(2/c^2 + x^2))/9 - (x^3*acos(c*x))/3) + (a*x^3)/3, ~0 < c, int(x^2*
(a + b*acos(c*x)), x))

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