\(\int (c+a^2 c x^2)^{5/2} \arctan (a x)^{3/2} \, dx\) [810]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=-\frac {15 c^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{32 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{48 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}}{20 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}+\frac {15}{64} c^3 \text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )+\frac {5}{96} c^2 \text {Int}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}},x\right )+\frac {1}{40} c \text {Int}\left (\frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}},x\right )+\frac {5}{16} c^3 \text {Int}\left (\frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}},x\right ) \]

[Out]

5/24*c*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(3/2)+1/6*x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(3/2)+5/16*c^2*x*arctan(a
*x)^(3/2)*(a^2*c*x^2+c)^(1/2)-5/48*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2)/a-1/20*(a^2*c*x^2+c)^(5/2)*arctan(a
*x)^(1/2)/a-15/32*c^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/a+5/16*c^3*Unintegrable(arctan(a*x)^(3/2)/(a^2*c*x
^2+c)^(1/2),x)+1/40*c*Unintegrable((a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2),x)+15/64*c^3*Unintegrable(1/(a^2*c*x^
2+c)^(1/2)/arctan(a*x)^(1/2),x)+5/96*c^2*Unintegrable((a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 23, 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 = {} \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx \]

[In]

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

[Out]

(-15*c^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(32*a) - (5*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(48*a)
- ((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]])/(20*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/16 + (5*c
*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/6 + (15*c^3*Defer
[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/64 + (5*c^2*Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*
x]], x])/96 + (c*Defer[Int][(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x])/40 + (5*c^3*Defer[Int][ArcTan[a*x]^(3
/2)/Sqrt[c + a^2*c*x^2], x])/16

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}}{20 a}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}+\frac {1}{40} c \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx \\ & = -\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{48 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}}{20 a}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}+\frac {1}{40} c \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{96} \left (5 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{8} \left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx \\ & = -\frac {15 c^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{32 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{48 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}}{20 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}+\frac {1}{40} c \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{96} \left (5 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}} \, dx+\frac {1}{64} \left (15 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx+\frac {1}{16} \left (5 c^3\right ) \int \frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 13.62 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^(3/2),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 = 21, normalized size of antiderivative = 0.91 \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\int {\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

[In]

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

[Out]

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