\(\int (b x)^m \arccos (a x)^2 \, dx\) [121]

Optimal result

Integrand size = 12, antiderivative size = 150 \[ \int (b x)^m \arccos (a x)^2 \, dx=\frac {(b x)^{1+m} \arccos (a x)^2}{b (1+m)}+\frac {2 a (b x)^{2+m} \arccos (a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{b^2 (1+m) (2+m)}+\frac {2 a^2 (b x)^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};a^2 x^2\right )}{b^3 (1+m) (2+m) (3+m)} \]

[Out]

(b*x)^(1+m)*arccos(a*x)^2/b/(1+m)+2*a*(b*x)^(2+m)*arccos(a*x)*hypergeom([1/2, 1+1/2*m],[2+1/2*m],a^2*x^2)/b^2/
(1+m)/(2+m)+2*a^2*(b*x)^(3+m)*hypergeom([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/2*m],a^2*x^2)/b^3/(3+m)/(m^2
+3*m+2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4724, 4806} \[ \int (b x)^m \arccos (a x)^2 \, dx=\frac {2 a^2 (b x)^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};a^2 x^2\right )}{b^3 (m+1) (m+2) (m+3)}+\frac {2 a \arccos (a x) (b x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{b^2 (m+1) (m+2)}+\frac {\arccos (a x)^2 (b x)^{m+1}}{b (m+1)} \]

[In]

Int[(b*x)^m*ArcCos[a*x]^2,x]

[Out]

((b*x)^(1 + m)*ArcCos[a*x]^2)/(b*(1 + m)) + (2*a*(b*x)^(2 + m)*ArcCos[a*x]*Hypergeometric2F1[1/2, (2 + m)/2, (
4 + m)/2, a^2*x^2])/(b^2*(1 + m)*(2 + m)) + (2*a^2*(b*x)^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2},
{2 + m/2, 5/2 + m/2}, a^2*x^2])/(b^3*(1 + m)*(2 + m)*(3 + m))

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]

Rule 4806

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)
^(m + 1)/(f*(m + 1)))*(a + b*ArcCos[c*x])*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*Hypergeometric2F1[1/2, (1 +
m)/2, (3 + m)/2, c^2*x^2], x] + Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d +
e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d, e,
f, m}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {(b x)^{1+m} \arccos (a x)^2}{b (1+m)}+\frac {(2 a) \int \frac {(b x)^{1+m} \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{b (1+m)} \\ & = \frac {(b x)^{1+m} \arccos (a x)^2}{b (1+m)}+\frac {2 a (b x)^{2+m} \arccos (a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{b^2 (1+m) (2+m)}+\frac {2 a^2 (b x)^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};a^2 x^2\right )}{b^3 (1+m) (2+m) (3+m)} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 1.41 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.88 \[ \int (b x)^m \arccos (a x)^2 \, dx=\frac {x (b x)^m \left (4 \arccos (a x)^2+a x \left (\frac {8 \sqrt {1-a^2 x^2} \arccos (a x) \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {4+m}{2},a^2 x^2\right )}{2+m}+2^{-m} a \sqrt {\pi } x \operatorname {Gamma}(2+m) \, _3\tilde {F}_2\left (1,\frac {3+m}{2},\frac {3+m}{2};\frac {4+m}{2},\frac {5+m}{2};a^2 x^2\right )\right )\right )}{4 (1+m)} \]

[In]

Integrate[(b*x)^m*ArcCos[a*x]^2,x]

[Out]

(x*(b*x)^m*(4*ArcCos[a*x]^2 + a*x*((8*Sqrt[1 - a^2*x^2]*ArcCos[a*x]*Hypergeometric2F1[1, (3 + m)/2, (4 + m)/2,
a^2*x^2])/(2 + m) + (a*Sqrt[Pi]*x*Gamma[2 + m]*HypergeometricPFQRegularized[{1, (3 + m)/2, (3 + m)/2}, {(4 +
m)/2, (5 + m)/2}, a^2*x^2])/2^m)))/(4*(1 + m))

Maple [F]

\[\int \left (b x \right )^{m} \arccos \left (a x \right )^{2}d x\]

[In]

int((b*x)^m*arccos(a*x)^2,x)

[Out]

int((b*x)^m*arccos(a*x)^2,x)

Fricas [F]

\[ \int (b x)^m \arccos (a x)^2 \, dx=\int { \left (b x\right )^{m} \arccos \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

integral((b*x)^m*arccos(a*x)^2, x)

Sympy [F]

\[ \int (b x)^m \arccos (a x)^2 \, dx=\int \left (b x\right )^{m} \operatorname {acos}^{2}{\left (a x \right )}\, dx \]

[In]

integrate((b*x)**m*acos(a*x)**2,x)

[Out]

Integral((b*x)**m*acos(a*x)**2, x)

Maxima [F]

\[ \int (b x)^m \arccos (a x)^2 \, dx=\int { \left (b x\right )^{m} \arccos \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (b x)^m \arccos (a x)^2 \, dx=\int { \left (b x\right )^{m} \arccos \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

integrate((b*x)^m*arccos(a*x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (b x)^m \arccos (a x)^2 \, dx=\int {\mathrm {acos}\left (a\,x\right )}^2\,{\left (b\,x\right )}^m \,d x \]

[In]

int(acos(a*x)^2*(b*x)^m,x)

[Out]

int(acos(a*x)^2*(b*x)^m, x)