3.9.99 \(\int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx\) [899]

3.9.99.1 Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{c x}+\frac {5}{2} a \text {Int}\left (\frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}},x\right ) \]

output

-arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/c/x+5/2*a*Unintegrable(arctan(a*x)^ 
(3/2)/x/(a^2*c*x^2+c)^(1/2),x)
 

3.9.99.2 Mathematica [N/A]

Not integrable

Time = 2.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx \]

input

Integrate[ArcTan[a*x]^(5/2)/(x^2*Sqrt[c + a^2*c*x^2]),x]
 

output

Integrate[ArcTan[a*x]^(5/2)/(x^2*Sqrt[c + a^2*c*x^2]), x]
 

3.9.99.3 Rubi [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5479, 5560}

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.

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

\(\Big \downarrow \) 5479

\(\displaystyle \frac {5}{2} a \int \frac {\arctan (a x)^{3/2}}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^{5/2} \sqrt {a^2 c x^2+c}}{c x}\)

\(\Big \downarrow \) 5560

\(\displaystyle \frac {5}{2} a \int \frac {\arctan (a x)^{3/2}}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^{5/2} \sqrt {a^2 c x^2+c}}{c x}\)

input

Int[ArcTan[a*x]^(5/2)/(x^2*Sqrt[c + a^2*c*x^2]),x]
 

output

$Aborted
 

3.9.99.3.1 Defintions of rubi rules used

rule 5479

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + 
 b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1)))   Int[(f*x) 
^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, 
c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] 
&& NeQ[m, -1]
 

rule 5560

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Unintegrab 
le[u*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || M 
atchQ[u, ((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_. 
)*x)^(m_.)*((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[ 
u, ((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x) 
^(m_.)*((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, f, m, q}, x]])
 

3.9.99.4 Maple [N/A] (verified)

Not integrable

Time = 5.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

\[\int \frac {\arctan \left (a x \right )^{\frac {5}{2}}}{x^{2} \sqrt {a^{2} c \,x^{2}+c}}d x\]

input

int(arctan(a*x)^(5/2)/x^2/(a^2*c*x^2+c)^(1/2),x)
 

output

int(arctan(a*x)^(5/2)/x^2/(a^2*c*x^2+c)^(1/2),x)
 

3.9.99.5 Fricas [F(-2)]

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

input

integrate(arctan(a*x)^(5/2)/x^2/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")
 

output

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

3.9.99.6 Sympy [N/A]

Not integrable

Time = 171.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

input

integrate(atan(a*x)**(5/2)/x**2/(a**2*c*x**2+c)**(1/2),x)
 

output

Integral(atan(a*x)**(5/2)/(x**2*sqrt(c*(a**2*x**2 + 1))), x)
 

3.9.99.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate(arctan(a*x)^(5/2)/x^2/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")
 

output

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

3.9.99.8 Giac [N/A]

Not integrable

Time = 223.40 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{\sqrt {a^{2} c x^{2} + c} x^{2}} \,d x } \]

input

integrate(arctan(a*x)^(5/2)/x^2/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")
 

output

sage0*x
 

3.9.99.9 Mupad [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{x^2\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

input

int(atan(a*x)^(5/2)/(x^2*(c + a^2*c*x^2)^(1/2)),x)
 

output

int(atan(a*x)^(5/2)/(x^2*(c + a^2*c*x^2)^(1/2)), x)