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

Optimal. Leaf size=292 \[ -\frac{\sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}-\frac{\sqrt{6 \pi } \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{5/2} c^3}+\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}+\frac{\sqrt{6 \pi } \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{5/2} c^3}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{4 x^3}{b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \]

[Out]

(2*x^2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcCos[c*x])^(3/2)) - (8*x)/(3*b^2*c^2*Sqrt[a + b*ArcCos[c*x]]) + (4*x
^3)/(b^2*Sqrt[a + b*ArcCos[c*x]]) + (Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]
])/(3*b^(5/2)*c^3) + (Sqrt[6*Pi]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(b^(5/2)
*c^3) - (Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*c^3) - (Sqrt[6
*Pi]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(5/2)*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.945343, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4634, 4720, 4636, 4406, 3306, 3305, 3351, 3304, 3352, 4624} \[ -\frac{\sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}-\frac{\sqrt{6 \pi } \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{5/2} c^3}+\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}+\frac{\sqrt{6 \pi } \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{5/2} c^3}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{4 x^3}{b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*ArcCos[c*x])^(5/2),x]

[Out]

(2*x^2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcCos[c*x])^(3/2)) - (8*x)/(3*b^2*c^2*Sqrt[a + b*ArcCos[c*x]]) + (4*x
^3)/(b^2*Sqrt[a + b*ArcCos[c*x]]) + (Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]
])/(3*b^(5/2)*c^3) + (Sqrt[6*Pi]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(b^(5/2)
*c^3) - (Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*c^3) - (Sqrt[6
*Pi]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(5/2)*c^3)

Rule 4634

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

Rule 4720

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

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]

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 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]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (a+b \cos ^{-1}(c x)\right )^{5/2}} \, dx &=\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{4 \int \frac{x}{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \, dx}{3 b c}+\frac{(2 c) \int \frac{x^3}{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \, dx}{b}\\ &=\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{4 x^3}{b^2 \sqrt{a+b \cos ^{-1}(c x)}}-\frac{12 \int \frac{x^2}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx}{b^2}+\frac{8 \int \frac{1}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx}{3 b^2 c^2}\\ &=\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{4 x^3}{b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 \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 )}{3 b^3 c^3}+\frac{12 \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^3}\\ &=\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{4 x^3}{b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{12 \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 )}{b^2 c^3}-\frac{\left (8 \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 )}{3 b^3 c^3}+\frac{\left (8 \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 )}{3 b^3 c^3}\\ &=\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{4 x^3}{b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^3}-\frac{\left (16 \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 b^3 c^3}+\frac{\left (16 \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 b^3 c^3}\\ &=\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{4 x^3}{b^2 \sqrt{a+b \cos ^{-1}(c x)}}-\frac{8 \sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}+\frac{8 \sqrt{2 \pi } C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{3 b^{5/2} c^3}+\frac{\left (3 \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 )}{b^2 c^3}+\frac{\left (3 \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 )}{b^2 c^3}-\frac{\left (3 \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 )}{b^2 c^3}-\frac{\left (3 \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 )}{b^2 c^3}\\ &=\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{4 x^3}{b^2 \sqrt{a+b \cos ^{-1}(c x)}}-\frac{8 \sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}+\frac{8 \sqrt{2 \pi } C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{3 b^{5/2} c^3}+\frac{\left (6 \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 )}{b^3 c^3}+\frac{\left (6 \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 )}{b^3 c^3}-\frac{\left (6 \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 )}{b^3 c^3}-\frac{\left (6 \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 )}{b^3 c^3}\\ &=\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{4 x^3}{b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}+\frac{\sqrt{6 \pi } \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{5/2} c^3}-\frac{\sqrt{2 \pi } C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{3 b^{5/2} c^3}-\frac{\sqrt{6 \pi } C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{b^{5/2} c^3}\\ \end{align*}

Mathematica [C]  time = 2.57622, size = 322, normalized size = 1.1 \[ -\frac{-\left (a+b \cos ^{-1}(c x)\right ) \left (-e^{-\frac{i a}{b}} \sqrt{-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )-e^{\frac{i a}{b}} \sqrt{\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+e^{-i \cos ^{-1}(c x)}+e^{i \cos ^{-1}(c x)}\right )-3 \left (a+b \cos ^{-1}(c x)\right ) \left (-\sqrt{3} e^{-\frac{3 i a}{b}} \sqrt{-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )-\sqrt{3} e^{\frac{3 i a}{b}} \sqrt{\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{3 i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+e^{-3 i \cos ^{-1}(c x)}+e^{3 i \cos ^{-1}(c x)}\right )-b \sqrt{1-c^2 x^2}+b \left (-\sin \left (3 \cos ^{-1}(c x)\right )\right )}{6 b^2 c^3 \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-(b*Sqrt[1 - c^2*x^2]) - (a + b*ArcCos[c*x])*(E^((-I)*ArcCos[c*x]) + E^(I*ArcCos[c*x]) - (Sqrt[((-I)*(a + b*
ArcCos[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b])/E^((I*a)/b) - E^((I*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x
]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c*x]))/b]) - 3*(a + b*ArcCos[c*x])*(E^((-3*I)*ArcCos[c*x]) + E^((3*I)*ArcCo
s[c*x]) - (Sqrt[3]*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcCos[c*x]))/b])/E^(((3*I)*a)
/b) - Sqrt[3]*E^(((3*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcCos[c*x]))/b]) - b*S
in[3*ArcCos[c*x]])/(6*b^2*c^3*(a + b*ArcCos[c*x])^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.148, size = 659, normalized size = 2.3 \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]

1/6/c^3/b^2*(6*arccos(c*x)*3^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*(1/b)^(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-6*arccos(c*x)*3^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*ar
ccos(c*x))^(1/2)*(1/b)^(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*arccos(c*x)*2^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*(1/b)^(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*arccos(c*x)*2^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*(1/b)^(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+6*3^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arccos(
c*x))^(1/2)*(1/b)^(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)*a-
6*3^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*(1/b)^(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)*a+2*2^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*(1/b)^(1/2)*cos(a/b)*Fresnel
S(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*a-2*2^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*(1/b)^(
1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*a+2*arccos(c*x)*cos((a+b*arccos
(c*x))/b-a/b)*b+6*arccos(c*x)*cos(3*(a+b*arccos(c*x))/b-3*a/b)*b+sin((a+b*arccos(c*x))/b-a/b)*b+2*cos((a+b*arc
cos(c*x))/b-a/b)*a+sin(3*(a+b*arccos(c*x))/b-3*a/b)*b+6*cos(3*(a+b*arccos(c*x))/b-3*a/b)*a)/(a+b*arccos(c*x))^
(3/2)

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (a + b \operatorname{acos}{\left (c x \right )}\right )^{\frac{5}{2}}}\, dx \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]

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

________________________________________________________________________________________

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

[Out]

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