\(\int \frac {x^2 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx\) [895]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {x^2 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {5 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{4 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{2 a^2 c}+\frac {15 \text {Int}\left (\frac {\sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}},x\right )}{8 a^2}-\frac {\text {Int}\left (\frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}},x\right )}{2 a^2} \]

[Out]

-5/4*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/a^3/c+1/2*x*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/a^2/c-1/2*Uninteg
rable(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)/a^2+15/8*Unintegrable(arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(1/2),x)/
a^2

Rubi [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {x^2 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^2 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^2 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 4.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

int(x^2*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(x^2*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)

Fricas [F(-2)]

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

[In]

integrate(x^2*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(x^2*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 227.56 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^2 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{\frac {5}{2}}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]

[In]

integrate(x^2*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

int((x^2*atan(a*x)^(5/2))/(c + a^2*c*x^2)^(1/2),x)

[Out]

int((x^2*atan(a*x)^(5/2))/(c + a^2*c*x^2)^(1/2), x)