\(\int x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx\) [880]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\frac {5 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{8 a^2}-\frac {5 x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{12 a}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}}{3 a^2 c}-\frac {5 c \text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )}{16 a}-\frac {5 c \text {Int}\left (\frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}},x\right )}{12 a} \]

[Out]

1/3*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(5/2)/a^2/c-5/12*x*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/a+5/8*(a^2*c*x^2+
c)^(1/2)*arctan(a*x)^(1/2)/a^2-5/12*c*Unintegrable(arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x)/a-5/16*c*Unintegra
ble(1/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)/a

Rubi [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 24, 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 = {} \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\int x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\int x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.50 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[In]

integrate(x*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 = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,\sqrt {c\,a^2\,x^2+c} \,d x \]

[In]

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

[Out]

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