3.2.83 \(\int x^2 (a+b \text {ArcCos}(c x))^{5/2} \, dx\) [183]

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

[Out]

1/3*x^3*(a+b*arccos(c*x))^(5/2)+5/864*b^(5/2)*cos(3*a/b)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(
1/2))*6^(1/2)*Pi^(1/2)/c^3+5/864*b^(5/2)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(3*a/b)
*6^(1/2)*Pi^(1/2)/c^3+15/32*b^(5/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*2^(1/2
)*Pi^(1/2)/c^3+15/32*b^(5/2)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1
/2)/c^3-5/9*b*(a+b*arccos(c*x))^(3/2)*(-c^2*x^2+1)^(1/2)/c^3-5/18*b*x^2*(a+b*arccos(c*x))^(3/2)*(-c^2*x^2+1)^(
1/2)/c-5/6*b^2*x*(a+b*arccos(c*x))^(1/2)/c^2-5/36*b^2*x^3*(a+b*arccos(c*x))^(1/2)

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Rubi [A]
time = 0.83, antiderivative size = 358, normalized size of antiderivative = 1.00, 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 = {4726, 4796, 4768, 4716, 4810, 3387, 3386, 3432, 3385, 3433, 3393} \begin {gather*} \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 \text {ArcCos}(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 \text {ArcCos}(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 \text {ArcCos}(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 \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{144 c^3}-\frac {5 b^2 x \sqrt {a+b \text {ArcCos}(c x)}}{6 c^2}-\frac {5}{36} b^2 x^3 \sqrt {a+b \text {ArcCos}(c x)}-\frac {5 b x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^{3/2}}{18 c}-\frac {5 b \sqrt {1-c^2 x^2} (a+b \text {ArcCos}(c x))^{3/2}}{9 c^3}+\frac {1}{3} x^3 (a+b \text {ArcCos}(c x))^{5/2} \end {gather*}

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 3385

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 3386

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 3387

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 3393

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])
)

Rule 3432

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

Rule 3433

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

Rule 4716

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 4726

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 4768

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 4796

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

Rule 4810

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

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 12.61, size = 1002, normalized size = 2.80 \begin {gather*} \frac {a^2 e^{-\frac {3 i a}{b}} \sqrt {a+b \text {ArcCos}(c x)} \left (9 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \text {ArcCos}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {i (a+b \text {ArcCos}(c x))}{b}\right )+9 e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcCos}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {i (a+b \text {ArcCos}(c x))}{b}\right )+\sqrt {3} \left (\sqrt {\frac {i (a+b \text {ArcCos}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {3 i (a+b \text {ArcCos}(c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcCos}(c x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {3 i (a+b \text {ArcCos}(c x))}{b}\right )\right )\right )}{72 c^3 \sqrt {\frac {(a+b \text {ArcCos}(c x))^2}{b^2}}}+\frac {a b \left (-18 \sqrt {a+b \text {ArcCos}(c x)} \left (3 \sqrt {1-c^2 x^2}-2 c x \text {ArcCos}(c x)\right )+9 \sqrt {\frac {1}{b}} \sqrt {2 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )-9 \sqrt {\frac {1}{b}} \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}\right ) \left (-2 a \cos \left (\frac {a}{b}\right )+3 b \sin \left (\frac {a}{b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {6 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}\right ) \left (b \cos \left (\frac {3 a}{b}\right )+2 a \sin \left (\frac {3 a}{b}\right )\right )-\sqrt {\frac {1}{b}} \sqrt {6 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}\right ) \left (-2 a \cos \left (\frac {3 a}{b}\right )+b \sin \left (\frac {3 a}{b}\right )\right )-6 \sqrt {a+b \text {ArcCos}(c x)} (-2 \text {ArcCos}(c x) \cos (3 \text {ArcCos}(c x))+\sin (3 \text {ArcCos}(c x)))\right )}{72 c^3}-\frac {-\frac {54 \sqrt {a+b \text {ArcCos}(c x)} \left (2 \sqrt {1-c^2 x^2} (a-5 b \text {ArcCos}(c x))+b c x \left (-15+4 \text {ArcCos}(c x)^2\right )\right )}{\sqrt {\frac {1}{b}}}+27 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}\right ) \left (\left (4 a^2-15 b^2\right ) \cos \left (\frac {a}{b}\right )-12 a b \sin \left (\frac {a}{b}\right )\right )-27 \sqrt {2 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}\right ) \left (-12 a b \cos \left (\frac {a}{b}\right )+\left (-4 a^2+15 b^2\right ) \sin \left (\frac {a}{b}\right )\right )+\sqrt {6 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}\right ) \left (\left (12 a^2-5 b^2\right ) \cos \left (\frac {3 a}{b}\right )-12 a b \sin \left (\frac {3 a}{b}\right )\right )-\sqrt {6 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}\right ) \left (-12 a b \cos \left (\frac {3 a}{b}\right )+\left (-12 a^2+5 b^2\right ) \sin \left (\frac {3 a}{b}\right )\right )-\frac {6 \sqrt {a+b \text {ArcCos}(c x)} \left (b \left (-5+12 \text {ArcCos}(c x)^2\right ) \cos (3 \text {ArcCos}(c x))+2 (a-5 b \text {ArcCos}(c x)) \sin (3 \text {ArcCos}(c x))\right )}{\sqrt {\frac {1}{b}}}}{864 \sqrt {\frac {1}{b}} c^3} \end {gather*}

Antiderivative was successfully verified.

[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] Leaf count of result is larger than twice the leaf count of optimal. \(797\) vs. \(2(278)=556\).
time = 0.50, size = 798, normalized size = 2.23

method	result	size
default	\(\frac {5 \sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b^{3}-5 \sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b^{3}+405 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b^{3}-405 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b^{3}+216 \arccos \left (c x \right )^{3} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+72 \arccos \left (c x \right )^{3} \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{3}+648 \arccos \left (c x \right )^{2} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+540 \arccos \left (c x \right )^{2} \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+216 \arccos \left (c x \right )^{2} \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a \,b^{2}+60 \arccos \left (c x \right )^{2} \sin \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{3}+648 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} b -810 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+1080 \arccos \left (c x \right ) \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+216 \arccos \left (c x \right ) \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a^{2} b -30 \arccos \left (c x \right ) \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{3}+120 \arccos \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a \,b^{2}+216 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{3}-810 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+540 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} b +72 \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a^{3}-30 \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a \,b^{2}+60 \sin \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a^{2} b}{864 c^{3} \sqrt {a +b \arccos \left (c x \right )}}\)	\(798\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/864/c^3/(a+b*arccos(c*x))^(1/2)*(5*(-3/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(3*a/b)*FresnelC
(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*b^3-5*(-3/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*
x))^(1/2)*sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/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)*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*arccos(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*arccos(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*arcc
os(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*arcco
s(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*arccos(c*x))/b+3*a/b)*a^2*b)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}

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)

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Giac [C] Result contains complex when optimal does not.
time = 2.05, size = 2778, normalized size = 7.76 \begin {gather*} \text {Too large to display} \end {gather*}

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]

-1/576*(72*I*sqrt(2)*sqrt(pi)*a^3*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sq
rt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b))) - 72*I*sqrt(2)*sqrt(pi
)*a^3*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(ab
s(b))/b)*e^(-I*a/b)/(-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b))) - 216*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*I*sqrt(2)
*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b^2/s
qrt(abs(b)) + b*sqrt(abs(b))) - 216*sqrt(2)*sqrt(pi)*a^2*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(ab
s(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))
- 24*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)^2*e^(3*I*arccos(c*x)) - 72*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x
)^2*e^(I*arccos(c*x)) - 72*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)^2*e^(-I*arccos(c*x)) - 24*sqrt(b*arccos(c*x
) + a)*b^2*arccos(c*x)^2*e^(-3*I*arccos(c*x)) - 144*I*sqrt(pi)*a^3*b*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/
sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*b^(3/2) + I*sqrt(6)*b^(5/
2)/abs(b)) + 144*I*sqrt(pi)*a^3*b*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcc
os(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*b^(3/2) - I*sqrt(6)*b^(5/2)/abs(b)) + 144*I*sqrt(pi)*a^3*sq
rt(b)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))
*e^(3*I*a/b)/(sqrt(6)*b + I*sqrt(6)*b^2/abs(b)) - 144*sqrt(pi)*a^2*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x)
+ a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*b + I*sqrt(6)*b^2/a
bs(b)) + 216*sqrt(2)*sqrt(pi)*a^2*b*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt
(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b))) + 135*sqrt(2)*sqrt(pi)*b^3*erf
(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(
I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b))) + 216*sqrt(2)*sqrt(pi)*a^2*b*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a
)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b/sqrt(abs(b)) + sqrt(abs(
b))) + 135*sqrt(2)*sqrt(pi)*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*ar
ccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b/sqrt(abs(b)) + sqrt(abs(b))) - 144*I*sqrt(pi)*a^3*sqrt(b)*erf(
-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a
/b)/(sqrt(6)*b - I*sqrt(6)*b^2/abs(b)) - 144*sqrt(pi)*a^2*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqr
t(b) + 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*b - I*sqrt(6)*b^2/abs(b)) -
48*sqrt(b*arccos(c*x) + a)*a*b*arccos(c*x)*e^(3*I*arccos(c*x)) - 20*I*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)
*e^(3*I*arccos(c*x)) - 144*sqrt(b*arccos(c*x) + a)*a*b*arccos(c*x)*e^(I*arccos(c*x)) - 180*I*sqrt(b*arccos(c*x
) + a)*b^2*arccos(c*x)*e^(I*arccos(c*x)) - 144*sqrt(b*arccos(c*x) + a)*a*b*arccos(c*x)*e^(-I*arccos(c*x)) + 18
0*I*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)*e^(-I*arccos(c*x)) - 48*sqrt(b*arccos(c*x) + a)*a*b*arccos(c*x)*e^
(-3*I*arccos(c*x)) + 20*I*sqrt(b*arccos(c*x) + a)*b^2*arccos(c*x)*e^(-3*I*arccos(c*x)) - 144*I*sqrt(pi)*a^3*er
f(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*
a/b)/(sqrt(6)*sqrt(b) + I*sqrt(6)*b^(3/2)/abs(b)) + 144*sqrt(pi)*a^2*b*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a
)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*sqrt(b) + I*sqrt(6)*b^(
3/2)/abs(b)) - 36*I*sqrt(pi)*a*b^2*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arc
cos(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*sqrt(b) + I*sqrt(6)*b^(3/2)/abs(b)) - 144*I*sqrt(pi)*a^3*er
f(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^
(I*a/b)/(I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b))) + 144*I*sqrt(pi)*a^3*erf(1/2*I*sqrt(2)*sqrt(b*arccos
(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*sqrt(2)*b/sqrt(ab
s(b)) + sqrt(2)*sqrt(abs(b))) + 144*I*sqrt(pi)*a^3*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) + 1/2*I*sq
rt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*sqrt(b) - I*sqrt(6)*b^(3/2)/abs(b)) + 144*
sqrt(pi)*a^2*b*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b
)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*sqrt(b) - I*sqrt(6)*b^(3/2)/abs(b)) + 36*I*sqrt(pi)*a*b^2*erf(-1/2*sqrt(6)*sqr
t(b*arccos(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*sqr
t(b) - I*sqrt(6)*b^(3/2)/abs(b)) + 144*I*sqrt(pi)*a^3*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) - 1/2*I
*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b)...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*acos(c*x))^(5/2),x)

[Out]

int(x^2*(a + b*acos(c*x))^(5/2), x)

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