3.183 \(\int x^2 (a+b \cos ^{-1}(c x))^{5/2} \, dx\)

Optimal. Leaf size=358 \[ \frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{144 c^3}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{144 c^3}-\frac{5 b^2 x \sqrt{a+b \cos ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}-\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2} \]

[Out]

(-5*b^2*x*Sqrt[a + b*ArcCos[c*x]])/(6*c^2) - (5*b^2*x^3*Sqrt[a + b*ArcCos[c*x]])/36 - (5*b*Sqrt[1 - c^2*x^2]*(
a + b*ArcCos[c*x])^(3/2))/(9*c^3) - (5*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(3/2))/(18*c) + (x^3*(a + b
*ArcCos[c*x])^(5/2))/3 + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]
])/(16*c^3) + (5*b^(5/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(144*
c^3) + (15*b^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(16*c^3) + (5*b
^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*c^3)

________________________________________________________________________________________

Rubi [A]  time = 1.31637, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {4630, 4708, 4678, 4620, 4724, 3306, 3305, 3351, 3304, 3352, 3312} \[ \frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{144 c^3}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{144 c^3}-\frac{5 b^2 x \sqrt{a+b \cos ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}-\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCos[c*x])^(5/2),x]

[Out]

(-5*b^2*x*Sqrt[a + b*ArcCos[c*x]])/(6*c^2) - (5*b^2*x^3*Sqrt[a + b*ArcCos[c*x]])/36 - (5*b*Sqrt[1 - c^2*x^2]*(
a + b*ArcCos[c*x])^(3/2))/(9*c^3) - (5*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(3/2))/(18*c) + (x^3*(a + b
*ArcCos[c*x])^(5/2))/3 + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]
])/(16*c^3) + (5*b^(5/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(144*
c^3) + (15*b^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(16*c^3) + (5*b
^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*c^3)

Rule 4630

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

Rule 4708

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

Rule 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4620

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

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int x^2 \left (a+b \cos ^{-1}(c x)\right )^{5/2} \, dx &=\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac{1}{6} (5 b c) \int \frac{x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}-\frac{1}{12} \left (5 b^2\right ) \int x^2 \sqrt{a+b \cos ^{-1}(c x)} \, dx+\frac{(5 b) \int \frac{x \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{\sqrt{1-c^2 x^2}} \, dx}{9 c}\\ &=-\frac{5}{36} b^2 x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^2\right ) \int \sqrt{a+b \cos ^{-1}(c x)} \, dx}{6 c^2}-\frac{1}{72} \left (5 b^3 c\right ) \int \frac{x^3}{\sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}} \, dx\\ &=-\frac{5 b^2 x \sqrt{a+b \cos ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\cos ^3(x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{72 c^3}-\frac{\left (5 b^3\right ) \int \frac{x}{\sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}} \, dx}{12 c}\\ &=-\frac{5 b^2 x \sqrt{a+b \cos ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{4 \sqrt{a+b x}}+\frac{\cos (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{72 c^3}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \cos ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{288 c^3}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{96 c^3}+\frac{\left (5 b^3 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{12 c^3}+\frac{\left (5 b^3 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \cos ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac{\left (5 b^2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{6 c^3}+\frac{\left (5 b^3 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{96 c^3}+\frac{\left (5 b^3 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{288 c^3}+\frac{\left (5 b^2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{6 c^3}+\frac{\left (5 b^3 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{96 c^3}+\frac{\left (5 b^3 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{288 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \cos ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac{5 b^{5/2} \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{6 c^3}+\frac{5 b^{5/2} \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{6 c^3}+\frac{\left (5 b^2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{48 c^3}+\frac{\left (5 b^2 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{144 c^3}+\frac{\left (5 b^2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{48 c^3}+\frac{\left (5 b^2 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{144 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \cos ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{5/2}+\frac{15 b^{5/2} \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{5 b^{5/2} \sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{144 c^3}+\frac{15 b^{5/2} \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{16 c^3}+\frac{5 b^{5/2} \sqrt{\frac{\pi }{6}} S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{144 c^3}\\ \end{align*}

Mathematica [C]  time = 16.464, size = 1002, normalized size = 2.8 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcCos[c*x])^(5/2),x]

[Out]

(a^2*Sqrt[a + b*ArcCos[c*x]]*(9*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcCos
[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b] + Sqrt
[3]*(Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcCos[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[((-I)*(
a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcCos[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[(a + b*ArcCos
[c*x])^2/b^2]) + (a*b*(-18*Sqrt[a + b*ArcCos[c*x]]*(3*Sqrt[1 - c^2*x^2] - 2*c*x*ArcCos[c*x]) + 9*Sqrt[b^(-1)]*
Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) - 9*Sqrt[b^
(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-2*a*Cos[a/b] + 3*b*Sin[a/b]) + Sq
rt[b^(-1)]*Sqrt[6*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(b*Cos[(3*a)/b] + 2*a*Sin[(3*a
)/b]) - Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-2*a*Cos[(3*a)/b] +
b*Sin[(3*a)/b]) - 6*Sqrt[a + b*ArcCos[c*x]]*(-2*ArcCos[c*x]*Cos[3*ArcCos[c*x]] + Sin[3*ArcCos[c*x]])))/(72*c^
3) - ((-54*Sqrt[a + b*ArcCos[c*x]]*(2*Sqrt[1 - c^2*x^2]*(a - 5*b*ArcCos[c*x]) + b*c*x*(-15 + 4*ArcCos[c*x]^2))
)/Sqrt[b^(-1)] + 27*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*((4*a^2 - 15*b^2)*Cos
[a/b] - 12*a*b*Sin[a/b]) - 27*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-12*a*b*Co
s[a/b] + (-4*a^2 + 15*b^2)*Sin[a/b]) + Sqrt[6*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]]]*((
12*a^2 - 5*b^2)*Cos[(3*a)/b] - 12*a*b*Sin[(3*a)/b]) - Sqrt[6*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*A
rcCos[c*x]]]*(-12*a*b*Cos[(3*a)/b] + (-12*a^2 + 5*b^2)*Sin[(3*a)/b]) - (6*Sqrt[a + b*ArcCos[c*x]]*(b*(-5 + 12*
ArcCos[c*x]^2)*Cos[3*ArcCos[c*x]] + 2*(a - 5*b*ArcCos[c*x])*Sin[3*ArcCos[c*x]]))/Sqrt[b^(-1)])/(864*Sqrt[b^(-1
)]*c^3)

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Maple [B]  time = 0.171, size = 792, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arccos(c*x))^(5/2),x)

[Out]

1/864/c^3*(5*3^(1/2)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(3*a/b)*FresnelC(2^(1/2)/Pi^(1/2)
*3^(1/2)/(1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*b^3+5*3^(1/2)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(
1/2)*sin(3*a/b)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*b^3+405*(1/b)^(1/2)*P
i^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)
/b)*b^3+405*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2
)*(a+b*arccos(c*x))^(1/2)/b)*b^3+216*arccos(c*x)^3*cos((a+b*arccos(c*x))/b-a/b)*b^3+72*arccos(c*x)^3*cos(3*(a+
b*arccos(c*x))/b-3*a/b)*b^3+648*arccos(c*x)^2*cos((a+b*arccos(c*x))/b-a/b)*a*b^2-540*arccos(c*x)^2*sin((a+b*ar
ccos(c*x))/b-a/b)*b^3+216*arccos(c*x)^2*cos(3*(a+b*arccos(c*x))/b-3*a/b)*a*b^2-60*arccos(c*x)^2*sin(3*(a+b*arc
cos(c*x))/b-3*a/b)*b^3+648*arccos(c*x)*cos((a+b*arccos(c*x))/b-a/b)*a^2*b-810*arccos(c*x)*cos((a+b*arccos(c*x)
)/b-a/b)*b^3-1080*arccos(c*x)*sin((a+b*arccos(c*x))/b-a/b)*a*b^2+216*arccos(c*x)*cos(3*(a+b*arccos(c*x))/b-3*a
/b)*a^2*b-30*arccos(c*x)*cos(3*(a+b*arccos(c*x))/b-3*a/b)*b^3-120*arccos(c*x)*sin(3*(a+b*arccos(c*x))/b-3*a/b)
*a*b^2+216*cos((a+b*arccos(c*x))/b-a/b)*a^3-810*cos((a+b*arccos(c*x))/b-a/b)*a*b^2-540*sin((a+b*arccos(c*x))/b
-a/b)*a^2*b+72*cos(3*(a+b*arccos(c*x))/b-3*a/b)*a^3-30*cos(3*(a+b*arccos(c*x))/b-3*a/b)*a*b^2-60*sin(3*(a+b*ar
ccos(c*x))/b-3*a/b)*a^2*b)/(a+b*arccos(c*x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arccos \left (c x\right ) + a\right )}^{\frac{5}{2}} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccos(c*x))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arccos(c*x) + a)^(5/2)*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccos(c*x))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*acos(c*x))**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 4.66928, size = 3471, normalized size = 9.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccos(c*x))^(5/2),x, algorithm="giac")

[Out]

-3/16*sqrt(2)*sqrt(pi)*a*b^4*i*erf(-1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*ar
ccos(c*x) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) - 3/16*sqrt(2)*sqrt(pi)
*a*b^4*i*erf(1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs
(b))/b)*e^(-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*c^3) - 1/24*sqrt(pi)*a*b^(7/2)*i*erf(-1/2*sqrt(6)*
sqrt(b*arccos(c*x) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e^(3*a*i/b)/((sqrt(6)*
b^3*i/abs(b) + sqrt(6)*b^2)*c^3) - 1/8*sqrt(2)*sqrt(pi)*a^2*b^3*erf(-1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqr
t(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(
b)))*c^3) + 3/16*sqrt(2)*sqrt(pi)*a*b^3*i*erf(-1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2
)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) + 1/8*sqrt(2)*
sqrt(pi)*a^2*b^3*erf(1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*
sqrt(abs(b))/b)*e^(-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*c^3) + 3/16*sqrt(2)*sqrt(pi)*a*b^3*i*erf(1
/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-a*
i/b)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*c^3) - 1/24*sqrt(pi)*a*b^(7/2)*i*erf(1/2*sqrt(6)*sqrt(b*arccos(c*x
) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e^(-3*a*i/b)/((sqrt(6)*b^3*i/abs(b) - s
qrt(6)*b^2)*c^3) - 1/12*sqrt(pi)*a^2*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)*i/abs(b) - 1/2*s
qrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e^(3*a*i/b)/((sqrt(6)*b^3*i/abs(b) + sqrt(6)*b^2)*c^3) + 1/24*sqrt(pi)
*a*b^(5/2)*i*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/s
qrt(b))*e^(3*a*i/b)/((sqrt(6)*b^2*i/abs(b) + sqrt(6)*b)*c^3) + 1/8*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*sqrt(2)*s
qrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*i/
sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) - 15/64*sqrt(2)*sqrt(pi)*b^4*erf(-1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sq
rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b
)))*c^3) - 1/8*sqrt(2)*sqrt(pi)*a^2*b^2*erf(1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*s
qrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*c^3) + 15/64*sqrt(2)*
sqrt(pi)*b^4*erf(1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt
(abs(b))/b)*e^(-a*i/b)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*c^3) + 1/12*sqrt(pi)*a^2*b^(5/2)*erf(1/2*sqrt(6)
*sqrt(b*arccos(c*x) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e^(-3*a*i/b)/((sqrt(6
)*b^3*i/abs(b) - sqrt(6)*b^2)*c^3) + 1/24*sqrt(pi)*a*b^(5/2)*i*erf(1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)
*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e^(-3*a*i/b)/((sqrt(6)*b^2*i/abs(b) - sqrt(6)*b)*c^3)
+ 5/144*sqrt(b*arccos(c*x) + a)*b^2*i*arccos(c*x)*e^(3*i*arccos(c*x))/c^3 + 1/24*sqrt(b*arccos(c*x) + a)*b^2*
arccos(c*x)^2*e^(3*i*arccos(c*x))/c^3 + 5/16*sqrt(b*arccos(c*x) + a)*b^2*i*arccos(c*x)*e^(i*arccos(c*x))/c^3 +
1/8*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)^2*e^(i*arccos(c*x))/c^3 - 5/16*sqrt(b*arccos(c*x) + a)*b^2*i*arcc
os(c*x)*e^(-i*arccos(c*x))/c^3 + 1/8*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)^2*e^(-i*arccos(c*x))/c^3 - 5/144*
sqrt(b*arccos(c*x) + a)*b^2*i*arccos(c*x)*e^(-3*i*arccos(c*x))/c^3 + 1/24*sqrt(b*arccos(c*x) + a)*b^2*arccos(c
*x)^2*e^(-3*i*arccos(c*x))/c^3 + 1/24*sqrt(pi)*a^2*b^2*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)*i/abs(
b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e^(3*a*i/b)/((sqrt(6)*b^(5/2)*i/abs(b) + sqrt(6)*b^(3/2))*c^
3) - 1/24*sqrt(pi)*a^2*b^2*erf(1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arcco
s(c*x) + a)/sqrt(b))*e^(-3*a*i/b)/((sqrt(6)*b^(5/2)*i/abs(b) - sqrt(6)*b^(3/2))*c^3) + 1/24*sqrt(pi)*a^2*b^(3/
2)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e^
(3*a*i/b)/((sqrt(6)*b^2*i/abs(b) + sqrt(6)*b)*c^3) - 5/288*sqrt(pi)*b^(7/2)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x
) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e^(3*a*i/b)/((sqrt(6)*b^2*i/abs(b) + sq
rt(6)*b)*c^3) - 1/24*sqrt(pi)*a^2*b^(3/2)*erf(1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(
6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e^(-3*a*i/b)/((sqrt(6)*b^2*i/abs(b) - sqrt(6)*b)*c^3) + 5/288*sqrt(pi)*b^(
7/2)*erf(1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e
^(-3*a*i/b)/((sqrt(6)*b^2*i/abs(b) - sqrt(6)*b)*c^3) + 5/144*sqrt(b*arccos(c*x) + a)*a*b*i*e^(3*i*arccos(c*x))
/c^3 + 1/12*sqrt(b*arccos(c*x) + a)*a*b*arccos(c*x)*e^(3*i*arccos(c*x))/c^3 + 5/16*sqrt(b*arccos(c*x) + a)*a*b
*i*e^(i*arccos(c*x))/c^3 + 1/4*sqrt(b*arccos(c*x) + a)*a*b*arccos(c*x)*e^(i*arccos(c*x))/c^3 - 5/16*sqrt(b*arc
cos(c*x) + a)*a*b*i*e^(-i*arccos(c*x))/c^3 + 1/4*sqrt(b*arccos(c*x) + a)*a*b*arccos(c*x)*e^(-i*arccos(c*x))/c^
3 - 5/144*sqrt(b*arccos(c*x) + a)*a*b*i*e^(-3*i*arccos(c*x))/c^3 + 1/12*sqrt(b*arccos(c*x) + a)*a*b*arccos(c*x
)*e^(-3*i*arccos(c*x))/c^3 + 1/24*sqrt(b*arccos(c*x) + a)*a^2*e^(3*i*arccos(c*x))/c^3 - 5/288*sqrt(b*arccos(c*
x) + a)*b^2*e^(3*i*arccos(c*x))/c^3 + 1/8*sqrt(b*arccos(c*x) + a)*a^2*e^(i*arccos(c*x))/c^3 - 15/32*sqrt(b*arc
cos(c*x) + a)*b^2*e^(i*arccos(c*x))/c^3 + 1/8*sqrt(b*arccos(c*x) + a)*a^2*e^(-i*arccos(c*x))/c^3 - 15/32*sqrt(
b*arccos(c*x) + a)*b^2*e^(-i*arccos(c*x))/c^3 + 1/24*sqrt(b*arccos(c*x) + a)*a^2*e^(-3*i*arccos(c*x))/c^3 - 5/
288*sqrt(b*arccos(c*x) + a)*b^2*e^(-3*i*arccos(c*x))/c^3