\(\int \frac {1}{x^3 (c+a^2 c x^2)^{3/2} \arctan (a x)^{5/2}} \, dx\) [1100]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}+\frac {4}{a^2 c x^4 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+\frac {16}{3 c x^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+\frac {16 \text {Int}\left (\frac {1}{x^5 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}},x\right )}{a^2}+\frac {92}{3} \text {Int}\left (\frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}},x\right )+16 a^2 \text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}},x\right ) \]

[Out]

-2/3/a/c/x^3/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2)+4/a^2/c/x^4/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2)+16/3/c/x^
2/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2)+16*Unintegrable(1/x^5/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)/a^2+92/
3*Unintegrable(1/x^3/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)+16*a^2*Unintegrable(1/x/(a^2*c*x^2+c)^(3/2)/arct
an(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx \]

[In]

Int[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]

[Out]

-2/(3*a*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 4/(a^2*c*x^4*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 1
6/(3*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (16*Defer[Int][1/(x^5*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*
x]]), x])/a^2 + (92*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3 + 16*a^2*Defer[Int][1/(x
*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {2 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx}{a}-\frac {1}{3} (8 a) \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2}{3 a c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}+\frac {4}{a^2 c x^4 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+\frac {16}{3 c x^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+\frac {32}{3} \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+20 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {16 \int \frac {1}{x^5 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{a^2}+\left (16 a^2\right ) \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 14.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx \]

[In]

Integrate[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]

Maple [N/A] (verified)

Not integrable

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

\[\int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x^3/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/x**3/(a**2*c*x**2+c)**(3/2)/atan(a*x)**(5/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/x^3/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x^3/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[In]

int(1/(x^3*atan(a*x)^(5/2)*(c + a^2*c*x^2)^(3/2)),x)

[Out]

int(1/(x^3*atan(a*x)^(5/2)*(c + a^2*c*x^2)^(3/2)), x)