\(\int \frac {x^3 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx\) [894]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {5 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{8 a^4 c}-\frac {5 x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{12 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 a^2 c}-\frac {5 \text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )}{16 a^3}+\frac {25 \text {Int}\left (\frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}},x\right )}{12 a^3} \]

[Out]

-5/12*x*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/a^3/c-2/3*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/a^4/c+1/3*x^2*ar
ctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/a^2/c+5/8*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/a^4/c+25/12*Unintegrable(a
rctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x)/a^3-5/16*Unintegrable(1/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)/a^3

Rubi [N/A]

Not integrable

Time = 0.32 (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 = {} \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

(5*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*a^4*c) - (5*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*a^3*c) -
(2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*a^2*c) -
(5*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(16*a^3) + (25*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt
[c + a^2*c*x^2], x])/(12*a^3)

Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 a^2 c}-\frac {2 \int \frac {x \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {5 \int \frac {x^2 \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{6 a} \\ & = -\frac {5 x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{12 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 a^2 c}+\frac {5 \int \frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^3}+\frac {5 \int \frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {5 \int \frac {x \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2} \\ & = \frac {5 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{8 a^4 c}-\frac {5 x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{12 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}}{3 a^2 c}-\frac {5 \int \frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx}{16 a^3}+\frac {5 \int \frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^3}+\frac {5 \int \frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 6.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

int(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x)

Fricas [F(-2)]

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

[In]

integrate(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(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. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

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

[In]

integrate(x^3*arctan(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

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

[Out]

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