\(\int (c+a^2 c x^2)^{5/2} \arctan (a x)^{5/2} \, dx\) [891]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=-\frac {25 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{32 a}-\frac {25 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{144 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}+\frac {75}{64} c^3 \text {Int}\left (\frac {\sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}},x\right )+\frac {25}{96} c^2 \text {Int}\left (\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)},x\right )+\frac {1}{8} c \text {Int}\left (\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)},x\right )+\frac {5}{16} c^3 \text {Int}\left (\frac {\arctan (a x)^{5/2}}{\sqrt {c+a^2 c x^2}},x\right ) \]

[Out]

-25/144*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(3/2)/a-1/12*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(3/2)/a+5/24*c*x*(a^2*c
*x^2+c)^(3/2)*arctan(a*x)^(5/2)+1/6*x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(5/2)-25/32*c^2*arctan(a*x)^(3/2)*(a^2*c
*x^2+c)^(1/2)/a+5/16*c^2*x*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)+5/16*c^3*Unintegrable(arctan(a*x)^(5/2)/(a^2*
c*x^2+c)^(1/2),x)+1/8*c*Unintegrable((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2),x)+75/64*c^3*Unintegrable(arctan(a*
x)^(1/2)/(a^2*c*x^2+c)^(1/2),x)+25/96*c^2*Unintegrable((a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.18 (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 = {} \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

(-25*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(32*a) - (25*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(144*a
) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/(12*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/16 + (5
*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/6 + (75*c^3*Def
er[Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/64 + (25*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])/8 + (5*c^3*Defer[Int][ArcTan[a*x]^(5/2
)/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} \arctan (a x)^{3/2}}{12 a}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}+\frac {1}{8} c \int \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)^{5/2} \, dx \\ & = -\frac {25 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{144 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}}{12 a}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}+\frac {1}{8} c \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx+\frac {1}{96} \left (25 c^2\right ) \int \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)^{5/2} \, dx \\ & = -\frac {25 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{32 a}-\frac {25 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{144 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}+\frac {1}{8} c \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx+\frac {1}{96} \left (25 c^2\right ) \int \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)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{64} \left (75 c^3\right ) \int \frac {\sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.66 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[In]

integrate((a^2*c*x^2+c)^(5/2)*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 = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

[In]

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

[Out]

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