\(\int \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx\) [881]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=-\frac {5 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{4 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}+\frac {15}{8} c \text {Int}\left (\frac {\sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}},x\right )+\frac {1}{2} c \text {Int}\left (\frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}},x\right ) \]

[Out]

-5/4*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/a+1/2*x*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)+1/2*c*Unintegrable(ar
ctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)+15/8*c*Unintegrable(arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 23, 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 = {} \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\int \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

(-5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(4*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/2 + (15*c*Defer[I
nt][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/8 + (c*Defer[Int][ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/2

Rubi steps \begin{align*} \text {integral}& = -\frac {5 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{4 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}+\frac {1}{2} c \int \frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} (15 c) \int \frac {\sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\int \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.58 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(arctan(a*x)^(5/2)*(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 [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(arctan(a*x)^(5/2)*(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 [F(-2)]

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

[In]

integrate(arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),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 = 0.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,\sqrt {c\,a^2\,x^2+c} \,d x \]

[In]

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

[Out]

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