3.2.98 \(\int \frac {x^2}{(a+b \text {ArcCos}(c x))^{5/2}} \, dx\) [198]

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

[Out]

1/3*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)/c^3-1/3*Fresn
elC(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)/b^(5/2)/c^3+cos(3*a/b)*Fresnel
S(6^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/b^(5/2)/c^3-FresnelC(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)/b^(5/2)/c^3+2/3*x^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcc
os(c*x))^(3/2)-8/3*x/b^2/c^2/(a+b*arccos(c*x))^(1/2)+4*x^3/b^2/(a+b*arccos(c*x))^(1/2)

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Rubi [A]
time = 0.64, antiderivative size = 292, normalized size of antiderivative = 1.00, 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 = {4730, 4808, 4732, 4491, 3387, 3386, 3432, 3385, 3433, 4720} \begin {gather*} -\frac {\sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(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 \text {ArcCos}(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 \text {ArcCos}(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 \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{b^{5/2} c^3}-\frac {8 x}{3 b^2 c^2 \sqrt {a+b \text {ArcCos}(c x)}}+\frac {4 x^3}{b^2 \sqrt {a+b \text {ArcCos}(c x)}}+\frac {2 x^2 \sqrt {1-c^2 x^2}}{3 b c (a+b \text {ArcCos}(c x))^{3/2}} \end {gather*}

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 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 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 4491

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 4720

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

Rule 4730

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(-x^m)*Sqrt[1 - c^2*x^2]*((a + b*Arc
Cos[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 4732

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

Rule 4808

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

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 \text {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 \text {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 \text {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 ) \text {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 ) \text {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 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^3}+\frac {3 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \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 )}{b^2 c^3}-\frac {\left (3 \sin \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 )}{b^2 c^3}-\frac {\left (3 \sin \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 )}{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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.77, size = 322, normalized size = 1.10 \begin {gather*} -\frac {-b \sqrt {1-c^2 x^2}-(a+b \text {ArcCos}(c x)) \left (e^{-i \text {ArcCos}(c x)}+e^{i \text {ArcCos}(c x)}-e^{-\frac {i a}{b}} \sqrt {-\frac {i (a+b \text {ArcCos}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {i (a+b \text {ArcCos}(c x))}{b}\right )-e^{\frac {i a}{b}} \sqrt {\frac {i (a+b \text {ArcCos}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {i (a+b \text {ArcCos}(c x))}{b}\right )\right )-3 (a+b \text {ArcCos}(c x)) \left (e^{-3 i \text {ArcCos}(c x)}+e^{3 i \text {ArcCos}(c x)}-\sqrt {3} e^{-\frac {3 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcCos}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {3 i (a+b \text {ArcCos}(c x))}{b}\right )-\sqrt {3} e^{\frac {3 i a}{b}} \sqrt {\frac {i (a+b \text {ArcCos}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {3 i (a+b \text {ArcCos}(c x))}{b}\right )\right )-b \sin (3 \text {ArcCos}(c x))}{6 b^2 c^3 (a+b \text {ArcCos}(c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(-(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)*A
rcCos[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*Sin[3*ArcCos[c*x]])/(b^2*c^3*(a + b*ArcCos[c*x])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(672\) vs. \(2(236)=472\).
time = 0.52, size = 673, normalized size = 2.30

method	result	size
default	\(\frac {-2 \arccos \left (c x \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \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 -2 \arccos \left (c x \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \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 -6 \arccos \left (c x \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \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 ) \sqrt {-\frac {3}{b}}\, b -6 \arccos \left (c x \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \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 ) \sqrt {-\frac {3}{b}}\, b -2 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \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 ) a -2 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \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 ) a -6 \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \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 ) \sqrt {-\frac {3}{b}}\, a -6 \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \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 ) \sqrt {-\frac {3}{b}}\, a +2 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b +6 \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 -\sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b +2 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a -\sin \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b +6 \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a}{6 c^{3} b^{2} \left (a +b \arccos \left (c x \right )\right )^{\frac {3}{2}}}\)	\(673\)

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]

1/6/c^3/b^2*(-2*arccos(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(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*arccos(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(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-6*arccos(c*x)*2^(1/2)*Pi^(
1/2)*(a+b*arccos(c*x))^(1/2)*cos(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(-
3/b)^(1/2)*b-6*arccos(c*x)*2^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3
/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(-3/b)^(1/2)*b-2*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)*co
s(a/b)*FresnelS(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)*(-1/b)^(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)*a-6*2^(1/2)*Pi
^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*
(-3/b)^(1/2)*a-6*2^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*
(a+b*arccos(c*x))^(1/2)/b)*(-3/b)^(1/2)*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*arccos(c*x))/b+a/b)*a-sin(-3*(a+b*arcc
os(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)

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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(x^2/(b*arccos(c*x) + a)^(5/2), 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 \frac {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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not 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="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {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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