\(\int (c+a^2 c x^2)^{3/2} \arctan (a x)^{5/2} \, dx\) [886]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx=-\frac {15 c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{16 a}-\frac {5 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{24 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}+\frac {45}{32} c^2 \text {Int}\left (\frac {\sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}},x\right )+\frac {5}{16} c \text {Int}\left (\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)},x\right )+\frac {3}{8} c^2 \text {Int}\left (\frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}},x\right ) \]

[Out]

-5/24*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(3/2)/a+1/4*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(5/2)-15/16*c*arctan(a*x)^
(3/2)*(a^2*c*x^2+c)^(1/2)/a+3/8*c*x*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)+3/8*c^2*Unintegrable(arctan(a*x)^(5/
2)/(a^2*c*x^2+c)^(1/2),x)+45/32*c^2*Unintegrable(arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(1/2),x)+5/16*c*Unintegrable(
(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.13 (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 = {} \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx=\int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

(-15*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(16*a) - (5*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(24*a) + (3
*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/4 + (45*c^2*Defer[
Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/32 + (5*c*Defer[Int][Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]
)/16 + (3*c^2*Defer[Int][ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/8

Rubi steps \begin{align*} \text {integral}& = -\frac {5 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{24 a}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}+\frac {1}{16} (5 c) \int \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx \\ & = -\frac {15 c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{16 a}-\frac {5 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{24 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}+\frac {1}{16} (5 c) \int \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx+\frac {1}{8} \left (3 c^2\right ) \int \frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{32} \left (45 c^2\right ) \int \frac {\sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.74 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx=\int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.87 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(5/2),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 = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2} \, dx=\int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]

[In]

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

[Out]

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