3.178 \(\int x^2 (a+b \cos ^{-1}(c x))^{3/2} \, dx\)

Optimal. Leaf size=313 \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}-\frac{\sqrt{\frac{\pi }{6}} b^{3/2} \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{6 c}-\frac{b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{3 c^3}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2} \]

[Out]

-(b*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(3*c^3) - (b*x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(6*
c) + (x^3*(a + b*ArcCos[c*x])^(3/2))/3 + (3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos
[c*x]])/Sqrt[b]])/(8*c^3) + (b^(3/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqr
t[b]])/(24*c^3) - (3*b^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^
3) - (b^(3/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.948978, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {4630, 4708, 4678, 4624, 3306, 3305, 3351, 3304, 3352, 4636, 4406} \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}-\frac{\sqrt{\frac{\pi }{6}} b^{3/2} \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{6 c}-\frac{b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{3 c^3}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCos[c*x])^(3/2),x]

[Out]

-(b*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(3*c^3) - (b*x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(6*
c) + (x^3*(a + b*ArcCos[c*x])^(3/2))/3 + (3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos
[c*x]])/Sqrt[b]])/(8*c^3) + (b^(3/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqr
t[b]])/(24*c^3) - (3*b^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^
3) - (b^(3/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*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 4624

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

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 4636

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Cos[x]^m*Sin[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2} \, dx &=\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac{1}{2} (b c) \int \frac{x^3 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}-\frac{1}{12} b^2 \int \frac{x^2}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx+\frac{b \int \frac{x \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{1-c^2 x^2}} \, dx}{3 c}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{12 c^3}-\frac{b^2 \int \frac{1}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx}{6 c^2}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}-\frac{b \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cos ^{-1}(c x)\right )}{6 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 \sqrt{a+b x}}+\frac{\sin (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{48 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{48 c^3}+\frac{\left (b \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cos ^{-1}(c x)\right )}{6 c^3}-\frac{\left (b \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \cos ^{-1}(c x)\right )}{6 c^3}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac{\left (b \cos \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 )}{3 c^3}+\frac{\left (b^2 \cos \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 )}{48 c^3}+\frac{\left (b^2 \cos \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 )}{48 c^3}-\frac{\left (b \sin \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 )}{3 c^3}-\frac{\left (b^2 \sin \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 )}{48 c^3}-\frac{\left (b^2 \sin \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 )}{48 c^3}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac{b^{3/2} \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{3 c^3}-\frac{b^{3/2} \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{3 c^3}+\frac{\left (b \cos \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 )}{24 c^3}+\frac{\left (b \cos \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 )}{24 c^3}-\frac{\left (b \sin \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 )}{24 c^3}-\frac{\left (b \sin \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 )}{24 c^3}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{3 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{6 c}+\frac{1}{3} x^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac{3 b^{3/2} \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c^3}+\frac{b^{3/2} \sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}-\frac{3 b^{3/2} \sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{8 c^3}-\frac{b^{3/2} \sqrt{\frac{\pi }{6}} C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{24 c^3}\\ \end{align*}

Mathematica [C]  time = 9.59482, size = 589, normalized size = 1.88 \[ \frac{b \left (-18 \left (3 \sqrt{1-c^2 x^2}-2 c x \cos ^{-1}(c x)\right ) \sqrt{a+b \cos ^{-1}(c x)}-9 \sqrt{2 \pi } \sqrt{\frac{1}{b}} \left (3 b \sin \left (\frac{a}{b}\right )-2 a \cos \left (\frac{a}{b}\right )\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}\right )-\sqrt{6 \pi } \sqrt{\frac{1}{b}} \left (b \sin \left (\frac{3 a}{b}\right )-2 a \cos \left (\frac{3 a}{b}\right )\right ) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}\right )+9 \sqrt{2 \pi } \sqrt{\frac{1}{b}} \left (2 a \sin \left (\frac{a}{b}\right )+3 b \cos \left (\frac{a}{b}\right )\right ) S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right )+\sqrt{6 \pi } \sqrt{\frac{1}{b}} \left (2 a \sin \left (\frac{3 a}{b}\right )+b \cos \left (\frac{3 a}{b}\right )\right ) S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right )-6 \left (\sin \left (3 \cos ^{-1}(c x)\right )-2 \cos ^{-1}(c x) \cos \left (3 \cos ^{-1}(c x)\right )\right ) \sqrt{a+b \cos ^{-1}(c x)}\right )}{144 c^3}+\frac{a e^{-\frac{3 i a}{b}} \sqrt{a+b \cos ^{-1}(c x)} \left (9 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+9 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+\sqrt{3} \left (\sqrt{\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{3 i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )\right )\right )}{72 c^3 \sqrt{\frac{\left (a+b \cos ^{-1}(c x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcCos[c*x])^(3/2),x]

[Out]

(a*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]) + (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]) + Sqrt[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*S
in[(3*a)/b]) - 6*Sqrt[a + b*ArcCos[c*x]]*(-2*ArcCos[c*x]*Cos[3*ArcCos[c*x]] + Sin[3*ArcCos[c*x]])))/(144*c^3)

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Maple [B]  time = 0.148, size = 541, normalized size = 1.7 \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))^(3/2),x)

[Out]

1/144/c^3*(3^(1/2)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(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^2-3^(1/2)*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)
*sin(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^2+27*(1/b)^(1/2)*Pi^(1/
2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*b
^2-27*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b
*arccos(c*x))^(1/2)/b)*b^2+36*arccos(c*x)^2*cos((a+b*arccos(c*x))/b-a/b)*b^2+12*arccos(c*x)^2*cos(3*(a+b*arcco
s(c*x))/b-3*a/b)*b^2+72*arccos(c*x)*cos((a+b*arccos(c*x))/b-a/b)*a*b-54*arccos(c*x)*sin((a+b*arccos(c*x))/b-a/
b)*b^2+24*arccos(c*x)*cos(3*(a+b*arccos(c*x))/b-3*a/b)*a*b-6*arccos(c*x)*sin(3*(a+b*arccos(c*x))/b-3*a/b)*b^2+
36*cos((a+b*arccos(c*x))/b-a/b)*a^2-54*sin((a+b*arccos(c*x))/b-a/b)*a*b+12*cos(3*(a+b*arccos(c*x))/b-3*a/b)*a^
2-6*sin(3*(a+b*arccos(c*x))/b-3*a/b)*a*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{3}{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))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*arccos(c*x) + a)^(3/2)*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))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{acos}{\left (c x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*acos(c*x))**(3/2),x)

[Out]

Integral(x**2*(a + b*acos(c*x))**(3/2), x)

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Giac [B]  time = 3.14784, size = 1800, normalized size = 5.75 \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))^(3/2),x, algorithm="giac")

[Out]

-3/32*sqrt(2)*sqrt(pi)*b^4*i*erf(-1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcc
os(c*x) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) - 3/32*sqrt(2)*sqrt(pi)*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) - 1/48*sqrt(pi)*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) - 1/16*sqrt(2)*sqrt(pi)*a*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) + 1/16*sqrt(2)*sqrt(pi)*a*b^3*erf(1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*a
rccos(c*x) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*c^3) - 1/48*sqrt(pi)*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) - 1/24*sqrt(pi)*a*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arcco
s(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) + 1/16*sqrt(2)*sqrt(pi)*a*b^2*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) - 1/1
6*sqrt(2)*sqrt(pi)*a*b^2*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^(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^(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) - 1/24*sqrt(pi)*a*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) + 1/48*sqrt(b*arccos(c*x) +
a)*b*i*e^(3*i*arccos(c*x))/c^3 + 1/24*sqrt(b*arccos(c*x) + a)*b*arccos(c*x)*e^(3*i*arccos(c*x))/c^3 + 3/16*sq
rt(b*arccos(c*x) + a)*b*i*e^(i*arccos(c*x))/c^3 + 1/8*sqrt(b*arccos(c*x) + a)*b*arccos(c*x)*e^(i*arccos(c*x))/
c^3 - 3/16*sqrt(b*arccos(c*x) + a)*b*i*e^(-i*arccos(c*x))/c^3 + 1/8*sqrt(b*arccos(c*x) + a)*b*arccos(c*x)*e^(-
i*arccos(c*x))/c^3 - 1/48*sqrt(b*arccos(c*x) + a)*b*i*e^(-3*i*arccos(c*x))/c^3 + 1/24*sqrt(b*arccos(c*x) + a)*
b*arccos(c*x)*e^(-3*i*arccos(c*x))/c^3 + 1/24*sqrt(b*arccos(c*x) + a)*a*e^(3*i*arccos(c*x))/c^3 + 1/8*sqrt(b*a
rccos(c*x) + a)*a*e^(i*arccos(c*x))/c^3 + 1/8*sqrt(b*arccos(c*x) + a)*a*e^(-i*arccos(c*x))/c^3 + 1/24*sqrt(b*a
rccos(c*x) + a)*a*e^(-3*i*arccos(c*x))/c^3