\(\int (c+a^2 c x^2)^3 \arctan (a x)^{5/2} \, dx\) [852]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=-\frac {2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}{7 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{28 a}-\frac {5 c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}}{84 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^{5/2}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{5/2}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{5/2}+\frac {3}{7} c^3 \text {Int}\left (\sqrt {\arctan (a x)},x\right )+\frac {9}{56} c^2 \text {Int}\left (\left (c+a^2 c x^2\right ) \sqrt {\arctan (a x)},x\right )+\frac {5}{56} c \text {Int}\left (\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)},x\right )+\frac {16}{35} c^3 \text {Int}\left (\arctan (a x)^{5/2},x\right ) \]

[Out]

-2/7*c^3*(a^2*x^2+1)*arctan(a*x)^(3/2)/a-3/28*c^3*(a^2*x^2+1)^2*arctan(a*x)^(3/2)/a-5/84*c^3*(a^2*x^2+1)^3*arc
tan(a*x)^(3/2)/a+8/35*c^3*x*(a^2*x^2+1)*arctan(a*x)^(5/2)+6/35*c^3*x*(a^2*x^2+1)^2*arctan(a*x)^(5/2)+1/7*c^3*x
*(a^2*x^2+1)^3*arctan(a*x)^(5/2)+16/35*c^3*Unintegrable(arctan(a*x)^(5/2),x)+3/7*c^3*Unintegrable(arctan(a*x)^
(1/2),x)+9/56*c^2*Unintegrable((a^2*c*x^2+c)*arctan(a*x)^(1/2),x)+5/56*c*Unintegrable((a^2*c*x^2+c)^2*arctan(a
*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.10 (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 = {} \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

(-2*c^3*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(7*a) - (3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(28*a) - (5*c^3*(1
+ a^2*x^2)^3*ArcTan[a*x]^(3/2))/(84*a) + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(5/2))/35 + (6*c^3*x*(1 + a^2*x^2)
^2*ArcTan[a*x]^(5/2))/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(5/2))/7 + (3*c^3*Defer[Int][Sqrt[ArcTan[a*x]],
x])/7 + (9*c^2*Defer[Int][(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/56 + (5*c*Defer[Int][(c + a^2*c*x^2)^2*Sqrt[A
rcTan[a*x]], x])/56 + (16*c^3*Defer[Int][ArcTan[a*x]^(5/2), x])/35

Rubi steps \begin{align*} \text {integral}& = -\frac {5 c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}}{84 a}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{5/2}+\frac {1}{56} (5 c) \int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx+\frac {1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx \\ & = -\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{28 a}-\frac {5 c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}}{84 a}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{5/2}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{5/2}+\frac {1}{56} (5 c) \int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx+\frac {1}{56} \left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \sqrt {\arctan (a x)} \, dx+\frac {1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx \\ & = -\frac {2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}{7 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{28 a}-\frac {5 c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}}{84 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^{5/2}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{5/2}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{5/2}+\frac {1}{56} (5 c) \int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx+\frac {1}{56} \left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \sqrt {\arctan (a x)} \, dx+\frac {1}{7} \left (3 c^3\right ) \int \sqrt {\arctan (a x)} \, dx+\frac {1}{35} \left (16 c^3\right ) \int \arctan (a x)^{5/2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.52 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((a^2*c*x^2+c)^3*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 = 85.96 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.14 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=c^{3} \left (\int 3 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int a^{6} x^{6} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(3*a**2*x**2*atan(a*x)**(5/2), x) + Integral(3*a**4*x**4*atan(a*x)**(5/2), x) + Integral(a**6*x*
*6*atan(a*x)**(5/2), x) + Integral(atan(a*x)**(5/2), x))

Maxima [F(-2)]

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

[In]

integrate((a^2*c*x^2+c)^3*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 = 108.74 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^3*arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]

[In]

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

[Out]

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