\(\int \frac {x^3 \arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx\) [856]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=-\frac {2 x \arctan (a x)^{7/2}}{7 a^3 c}+\frac {\text {Int}\left (x \arctan (a x)^{5/2},x\right )}{a^2 c}+\frac {2 \text {Int}\left (\arctan (a x)^{7/2},x\right )}{7 a^3 c} \]

[Out]

-2/7*x*arctan(a*x)^(7/2)/a^3/c+Unintegrable(x*arctan(a*x)^(5/2),x)/a^2/c+2/7*Unintegrable(arctan(a*x)^(7/2),x)
/a^3/c

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int \frac {x^3 \arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\int \frac {x^3 \arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx \]

[In]

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

[Out]

(-2*x*ArcTan[a*x]^(7/2))/(7*a^3*c) + Defer[Int][x*ArcTan[a*x]^(5/2), x]/(a^2*c) + (2*Defer[Int][ArcTan[a*x]^(7
/2), x])/(7*a^3*c)

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {x \arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int x \arctan (a x)^{5/2} \, dx}{a^2 c} \\ & = -\frac {2 x \arctan (a x)^{7/2}}{7 a^3 c}+\frac {2 \int \arctan (a x)^{7/2} \, dx}{7 a^3 c}+\frac {\int x \arctan (a x)^{5/2} \, dx}{a^2 c} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.69 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\int \frac {x^3 \arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.68 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

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

[Out]

Integral(x**3*atan(a*x)**(5/2)/(a**2*x**2 + 1), x)/c

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 108.10 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{\frac {5}{2}}}{a^{2} c x^{2} + c} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{c\,a^2\,x^2+c} \,d x \]

[In]

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

[Out]

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