\(\int x (c+a^2 c x^2)^{3/2} \arctan (a x)^{5/2} \, dx\) [885]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx=\frac {9 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{32 a^2}+\frac {\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{16 a^2}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{16 a}-\frac {x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{8 a}+\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}}{5 a^2 c}-\frac {9 c^2 \text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )}{64 a}-\frac {c \text {Int}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}},x\right )}{32 a}-\frac {3 c^2 \text {Int}\left (\frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}},x\right )}{16 a} \]

[Out]

-1/8*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(3/2)/a+1/5*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(5/2)/a^2/c-3/16*c*x*arctan
(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/a+1/16*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2)/a^2+9/32*c*(a^2*c*x^2+c)^(1/2)*ar
ctan(a*x)^(1/2)/a^2-3/16*c^2*Unintegrable(arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x)/a-9/64*c^2*Unintegrable(1/(
a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)/a-1/32*c*Unintegrable((a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)/a

Rubi [N/A]

Not integrable

Time = 0.20 (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 = {} \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx=\int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

(9*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(32*a^2) + ((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(16*a^2) - (3
*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(16*a) - (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(8*a) + ((c +
a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/(5*a^2*c) - (9*c^2*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]),
x])/(64*a) - (c*Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/(32*a) - (3*c^2*Defer[Int][ArcTan[a*x]^(
3/2)/Sqrt[c + a^2*c*x^2], x])/(16*a)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}}{5 a^2 c}-\frac {\int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx}{2 a} \\ & = \frac {\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{16 a^2}-\frac {x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{8 a}+\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}}{5 a^2 c}-\frac {c \int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}} \, dx}{32 a}-\frac {(3 c) \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx}{8 a} \\ & = \frac {9 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{32 a^2}+\frac {\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{16 a^2}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{16 a}-\frac {x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{8 a}+\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}}{5 a^2 c}-\frac {c \int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}} \, dx}{32 a}-\frac {\left (9 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx}{64 a}-\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{16 a} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx=\int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.93 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[In]

integrate(x*(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 = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]

[In]

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

[Out]

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