\(\int (c+a^2 c x^2)^2 \arctan (a x)^{5/2} \, dx\) [846]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=-\frac {c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}{3 a}-\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{8 a}+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^{5/2}+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{5/2}+\frac {1}{2} c^2 \text {Int}\left (\sqrt {\arctan (a x)},x\right )+\frac {3}{16} c \text {Int}\left (\left (c+a^2 c x^2\right ) \sqrt {\arctan (a x)},x\right )+\frac {8}{15} c^2 \text {Int}\left (\arctan (a x)^{5/2},x\right ) \]

[Out]

-1/3*c^2*(a^2*x^2+1)*arctan(a*x)^(3/2)/a-1/8*c^2*(a^2*x^2+1)^2*arctan(a*x)^(3/2)/a+4/15*c^2*x*(a^2*x^2+1)*arct
an(a*x)^(5/2)+1/5*c^2*x*(a^2*x^2+1)^2*arctan(a*x)^(5/2)+8/15*c^2*Unintegrable(arctan(a*x)^(5/2),x)+1/2*c^2*Uni
ntegrable(arctan(a*x)^(1/2),x)+3/16*c*Unintegrable((a^2*c*x^2+c)*arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 21, 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 = {} \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

-1/3*(c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/a - (c^2*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(8*a) + (4*c^2*x*(1 + a
^2*x^2)*ArcTan[a*x]^(5/2))/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(5/2))/5 + (c^2*Defer[Int][Sqrt[ArcTan[a*x]
], x])/2 + (3*c*Defer[Int][(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/16 + (8*c^2*Defer[Int][ArcTan[a*x]^(5/2), x]
)/15

Rubi steps \begin{align*} \text {integral}& = -\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{8 a}+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{5/2}+\frac {1}{16} (3 c) \int \left (c+a^2 c x^2\right ) \sqrt {\arctan (a x)} \, dx+\frac {1}{5} (4 c) \int \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx \\ & = -\frac {c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}{3 a}-\frac {c^2 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{8 a}+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^{5/2}+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{5/2}+\frac {1}{16} (3 c) \int \left (c+a^2 c x^2\right ) \sqrt {\arctan (a x)} \, dx+\frac {1}{2} c^2 \int \sqrt {\arctan (a x)} \, dx+\frac {1}{15} \left (8 c^2\right ) \int \arctan (a x)^{5/2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.49 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((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 = 40.92 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=c^{2} \left (\int 2 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((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 = 111.91 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate((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 = 21, normalized size of antiderivative = 1.00 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

[In]

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

[Out]

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