\(\int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx\) [901]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=-\frac {5 a^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{8 c x}-\frac {5 a \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{12 c x^2}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 c x}+\frac {5}{16} a^3 \text {Int}\left (\frac {1}{x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )-\frac {25}{12} a^3 \text {Int}\left (\frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}},x\right ) \]

[Out]

-5/12*a*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/c/x^2-1/3*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/c/x^3+2/3*a^2*ar
ctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/c/x-5/8*a^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/c/x-25/12*a^3*Unintegrab
le(arctan(a*x)^(3/2)/x/(a^2*c*x^2+c)^(1/2),x)+5/16*a^3*Unintegrable(1/x/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),
x)

Rubi [N/A]

Not integrable

Time = 0.47 (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 = {} \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \]

[In]

Int[ArcTan[a*x]^(5/2)/(x^4*Sqrt[c + a^2*c*x^2]),x]

[Out]

(-5*a^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*c*x) - (5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*c*x^2
) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x)
+ (5*a^3*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/16 - (25*a^3*Defer[Int][ArcTan[a*x]^(3/2)
/(x*Sqrt[c + a^2*c*x^2]), x])/12

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 c x^3}+\frac {1}{6} (5 a) \int \frac {\arctan (a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (2 a^2\right ) \int \frac {\arctan (a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {5 a \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{12 c x^2}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 c x}+\frac {1}{8} \left (5 a^2\right ) \int \frac {\sqrt {\arctan (a x)}}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{12} \left (5 a^3\right ) \int \frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (5 a^3\right ) \int \frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {5 a^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{8 c x}-\frac {5 a \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{12 c x^2}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 c x}+\frac {1}{16} \left (5 a^3\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx-\frac {1}{12} \left (5 a^3\right ) \int \frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (5 a^3\right ) \int \frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 17.76 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \]

[In]

Integrate[ArcTan[a*x]^(5/2)/(x^4*Sqrt[c + a^2*c*x^2]),x]

[Out]

Integrate[ArcTan[a*x]^(5/2)/(x^4*Sqrt[c + a^2*c*x^2]), x]

Maple [N/A] (verified)

Not integrable

Time = 5.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

\[\int \frac {\arctan \left (a x \right )^{\frac {5}{2}}}{x^{4} \sqrt {a^{2} c \,x^{2}+c}}d x\]

[In]

int(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 177.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{x^{4} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate(atan(a*x)**(5/2)/x**4/(a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(atan(a*x)**(5/2)/(x**4*sqrt(c*(a**2*x**2 + 1))), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 227.33 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{\sqrt {a^{2} c x^{2} + c} x^{4}} \,d x } \]

[In]

integrate(arctan(a*x)^(5/2)/x^4/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{x^4\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

int(atan(a*x)^(5/2)/(x^4*(c + a^2*c*x^2)^(1/2)),x)

[Out]

int(atan(a*x)^(5/2)/(x^4*(c + a^2*c*x^2)^(1/2)), x)