\(\int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx\) [199]

Optimal result

Integrand size = 14, antiderivative size = 180 \[ \int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx=\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \arccos (c x)}}+\frac {8 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} c^2}-\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} c^2} \] Output:

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

Mathematica [F]

\[ \int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx=\int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.46 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {5145, 5153, 5223, 5147, 25, 4906, 27, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx\)

\(\Big \downarrow \) 5145

\(\displaystyle -\frac {2 \int \frac {1}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}dx}{3 b c}+\frac {4 c \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}dx}{3 b}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 5153

\(\displaystyle \frac {4 c \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))^{3/2}}dx}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 5223

\(\displaystyle \frac {4 c \left (\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}-\frac {4 \int \frac {x}{\sqrt {a+b \arccos (c x)}}dx}{b c}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 5147

\(\displaystyle \frac {4 c \left (\frac {4 \int -\frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c^3}+\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 c \left (\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}-\frac {4 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c^3}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {4 c \left (\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}-\frac {4 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{2 \sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c^3}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 c \left (\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c^3}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4 c \left (\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}-\frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c^3}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 3787

\(\displaystyle \frac {4 c \left (\frac {2 \left (-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\cos \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))\right )}{b^2 c^3}+\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 c \left (\frac {2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))\right )}{b^2 c^3}+\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4 c \left (\frac {2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))\right )}{b^2 c^3}+\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 3785

\(\displaystyle \frac {4 c \left (\frac {2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )d\sqrt {a+b \arccos (c x)}\right )}{b^2 c^3}+\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {4 c \left (\frac {2 \left (2 \cos \left (\frac {2 a}{b}\right ) \int \sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )d\sqrt {a+b \arccos (c x)}-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )d\sqrt {a+b \arccos (c x)}\right )}{b^2 c^3}+\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {4 c \left (\frac {2 \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )d\sqrt {a+b \arccos (c x)}\right )}{b^2 c^3}+\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {4 c \left (\frac {2 \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2 c^3}+\frac {2 x^2}{b c \sqrt {a+b \arccos (c x)}}\right )}{3 b}-\frac {4}{3 b^2 c^2 \sqrt {a+b \arccos (c x)}}+\frac {2 x \sqrt {1-c^2 x^2}}{3 b c (a+b \arccos (c x))^{3/2}}\)

Input:

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

Output:

(2*x*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcCos[c*x])^(3/2)) - 4/(3*b^2*c^2*S 
qrt[a + b*ArcCos[c*x]]) + (4*c*((2*x^2)/(b*c*Sqrt[a + b*ArcCos[c*x]]) + (2 
*(Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt 
[b]*Sqrt[Pi])] - Sqrt[b]*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sq 
rt[b]*Sqrt[Pi])]*Sin[(2*a)/b]))/(b^2*c^3)))/(3*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[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 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[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 3832

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 3833

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 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG 
tQ[p, 0]
 

rule 5145

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

rule 5147

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- 
(b*c^(m + 1))^(-1)   Subst[Int[x^n*Cos[-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 5153

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

rule 5223

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] + Simp[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(339\) vs. \(2(142)=284\).

Time = 0.15 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.89

method	result	size
default	\(\frac {-8 \arccos \left (c x \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b -8 \arccos \left (c x \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b -8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) a -8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) a +4 \arccos \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b -\sin \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b +4 \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a}{3 c^{2} b^{2} \left (a +b \arccos \left (c x \right )\right )^{\frac {3}{2}}}\)	\(340\)

Input:

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

Output:

1/3/c^2/b^2*(-8*arccos(c*x)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)* 
cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x))^(1/2 
)/b)*b-8*arccos(c*x)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(2*a 
/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*b- 
8*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(2*a/b)*FresnelS(2*2^(1 
/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*a-8*(-1/b)^(1/2)*Pi^( 
1/2)*(a+b*arccos(c*x))^(1/2)*sin(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b) 
^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*a+4*arccos(c*x)*cos(-2*(a+b*arccos(c*x)) 
/b+2*a/b)*b-sin(-2*(a+b*arccos(c*x))/b+2*a/b)*b+4*cos(-2*(a+b*arccos(c*x)) 
/b+2*a/b)*a)/(a+b*arccos(c*x))^(3/2)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx=\int \frac {x}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx=\int { \frac {x}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx=\int { \frac {x}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x}{(a+b \arccos (c x))^{5/2}} \, dx=\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x}{\mathit {acos} \left (c x \right )^{3} b^{3}+3 \mathit {acos} \left (c x \right )^{2} a \,b^{2}+3 \mathit {acos} \left (c x \right ) a^{2} b +a^{3}}d x \] Input:

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

Output:

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