\(\int (c+a^2 c x^2) \arctan (a x)^{3/2} \, dx\) [759]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx=-\frac {c \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}{4 a}+\frac {1}{3} c x \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}+\frac {1}{8} c \text {Int}\left (\frac {1}{\sqrt {\arctan (a x)}},x\right )+\frac {2}{3} c \text {Int}\left (\arctan (a x)^{3/2},x\right ) \]

[Out]

1/3*c*x*(a^2*x^2+1)*arctan(a*x)^(3/2)-1/4*c*(a^2*x^2+1)*arctan(a*x)^(1/2)/a+2/3*c*Unintegrable(arctan(a*x)^(3/
2),x)+1/8*c*Unintegrable(1/arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 19, 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 = {} \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx=\int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.42 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx=\int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.59 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 3.85 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx=c \left (\int a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 77.83 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.16 \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx=\int {\mathrm {atan}\left (a\,x\right )}^{3/2}\,\left (c\,a^2\,x^2+c\right ) \,d x \]

[In]

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

[Out]

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