3.19 \(\int (d+e x^2) (a+b \cos ^{-1}(c x)) \, dx\)

Optimal. Leaf size=81 \[ d x \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac{b \sqrt{1-c^2 x^2} \left (3 c^2 d+e\right )}{3 c^3}+\frac{b e \left (1-c^2 x^2\right )^{3/2}}{9 c^3} \]

[Out]

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

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Rubi [A]  time = 0.0674421, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {4666, 444, 43} \[ d x \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \cos ^{-1}(c x)\right )-\frac{b \sqrt{1-c^2 x^2} \left (3 c^2 d+e\right )}{3 c^3}+\frac{b e \left (1-c^2 x^2\right )^{3/2}}{9 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*(a + b*ArcCos[c*x]),x]

[Out]

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

Rule 4666

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

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])

Rubi steps

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

Mathematica [A]  time = 0.07248, size = 91, normalized size = 1.12 \[ a d x+\frac{1}{3} a e x^3-\frac{b d \sqrt{1-c^2 x^2}}{c}+b e \left (-\frac{2}{9 c^3}-\frac{x^2}{9 c}\right ) \sqrt{1-c^2 x^2}+b d x \cos ^{-1}(c x)+\frac{1}{3} b e x^3 \cos ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)*(a + b*ArcCos[c*x]),x]

[Out]

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

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Maple [A]  time = 0.005, size = 112, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{2}} \left ({\frac{{c}^{3}{x}^{3}e}{3}}+d{c}^{3}x \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{\arccos \left ( cx \right ){c}^{3}{x}^{3}e}{3}}+\arccos \left ( cx \right ) d{c}^{3}x+{\frac{e}{3} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{c}^{2}d\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(a+b*arccos(c*x)),x)

[Out]

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

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Maxima [A]  time = 1.65647, size = 127, normalized size = 1.57 \begin{align*} \frac{1}{3} \, a e 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 e + a d x + \frac{{\left (c x \arccos \left (c x\right ) - \sqrt{-c^{2} x^{2} + 1}\right )} b d}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.1665, size = 186, normalized size = 2.3 \begin{align*} \frac{3 \, a c^{3} e x^{3} + 9 \, a c^{3} d x + 3 \,{\left (b c^{3} e x^{3} + 3 \, b c^{3} d x\right )} \arccos \left (c x\right ) -{\left (b c^{2} e x^{2} + 9 \, b c^{2} d + 2 \, b e\right )} \sqrt{-c^{2} x^{2} + 1}}{9 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.678229, size = 114, normalized size = 1.41 \begin{align*} \begin{cases} a d x + \frac{a e x^{3}}{3} + b d x \operatorname{acos}{\left (c x \right )} + \frac{b e x^{3} \operatorname{acos}{\left (c x \right )}}{3} - \frac{b d \sqrt{- c^{2} x^{2} + 1}}{c} - \frac{b e x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} - \frac{2 b e \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} & \text{for}\: c \neq 0 \\\left (a + \frac{\pi b}{2}\right ) \left (d x + \frac{e x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16234, size = 128, normalized size = 1.58 \begin{align*} \frac{1}{3} \, b x^{3} \arccos \left (c x\right ) e + \frac{1}{3} \, a x^{3} e + b d x \arccos \left (c x\right ) - \frac{\sqrt{-c^{2} x^{2} + 1} b x^{2} e}{9 \, c} + a d x - \frac{\sqrt{-c^{2} x^{2} + 1} b d}{c} - \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b e}{9 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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