\(\int x (c+a^2 c x^2)^2 \arctan (a x)^{5/2} \, dx\) [845]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\frac {c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}{12 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}{32 a^2}-\frac {c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}{9 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{12 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^{5/2}}{6 a^2}-\frac {c^2 \text {Int}\left (\frac {1}{\sqrt {\arctan (a x)}},x\right )}{24 a}-\frac {c \text {Int}\left (\frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}},x\right )}{64 a}-\frac {2 c^2 \text {Int}\left (\arctan (a x)^{3/2},x\right )}{9 a} \]

[Out]

-1/9*c^2*x*(a^2*x^2+1)*arctan(a*x)^(3/2)/a-1/12*c^2*x*(a^2*x^2+1)^2*arctan(a*x)^(3/2)/a+1/6*c^2*(a^2*x^2+1)^3*
arctan(a*x)^(5/2)/a^2+1/12*c^2*(a^2*x^2+1)*arctan(a*x)^(1/2)/a^2+1/32*c^2*(a^2*x^2+1)^2*arctan(a*x)^(1/2)/a^2-
2/9*c^2*Unintegrable(arctan(a*x)^(3/2),x)/a-1/24*c^2*Unintegrable(1/arctan(a*x)^(1/2),x)/a-1/64*c*Unintegrable
((a^2*c*x^2+c)/arctan(a*x)^(1/2),x)/a

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 22, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(12*a^2) + (c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(32*a^2) - (c^2*x*(1
+ a^2*x^2)*ArcTan[a*x]^(3/2))/(9*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(12*a) + (c^2*(1 + a^2*x^2)^3*
ArcTan[a*x]^(5/2))/(6*a^2) - (c^2*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/(24*a) - (c*Defer[Int][(c + a^2*c*x^2)/S
qrt[ArcTan[a*x]], x])/(64*a) - (2*c^2*Defer[Int][ArcTan[a*x]^(3/2), x])/(9*a)

Rubi steps \begin{align*} \text {integral}& = \frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^{5/2}}{6 a^2}-\frac {5 \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2} \, dx}{12 a} \\ & = \frac {c^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}{32 a^2}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{12 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^{5/2}}{6 a^2}-\frac {c \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx}{64 a}-\frac {c \int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx}{3 a} \\ & = \frac {c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}{12 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}{32 a^2}-\frac {c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}{9 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{12 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^{5/2}}{6 a^2}-\frac {c \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx}{64 a}-\frac {c^2 \int \frac {1}{\sqrt {\arctan (a x)}} \, dx}{24 a}-\frac {\left (2 c^2\right ) \int \arctan (a x)^{3/2} \, dx}{9 a} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.98 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x*(a^2*c*x^2+c)^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 [N/A]

Not integrable

Time = 51.42 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=c^{2} \left (\int x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int 2 a^{2} x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int a^{4} x^{5} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**2*(Integral(x*atan(a*x)**(5/2), x) + Integral(2*a**2*x**3*atan(a*x)**(5/2), x) + Integral(a**4*x**5*atan(a*
x)**(5/2), x))

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 109.50 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

[In]

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

[Out]

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